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Department of Mathematics
 Numerical analysis and scientific computing
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Areas of expertise
Research seminars, phd research opportunities, research publications.
Our researchers develop and analyse algorithms that compute numerical approximations and apply them to realworld problems.
Numerical analysis is the branch of rigorous mathematics that concerns the development and analysis of methods to compute numerical approximations to the solutions of mathematical problems. It is a broadly based discipline that sits at the interface between mathematical analysis and scientific computing. Scientific computing describes the use of numerical simulation to study natural phenomena, complementing the more traditional experimental and theoretical approaches. Another broad discipline, it spans all the sciences with strong links to numerical analysis, computer science and software engineering. Our work covers the breadth of these disciplines from fundamental theory and algorithm development through to implementation in open source software. Our researchers have expertise in the following areas.
Approximation theory
Approximation theory is a key component of contemporary algorithms used in computational science and engineering.
Mathematical foundations of data science and AI
Data science refers to the study of theory, methods, algorithms, and applications focused around data, and is a highly interdisciplinary subject which relies on solid foundations of mathematical and statistical fundamentals.
Numerical linear algebra
Numerical linear algebra is at the heart of computational algorithms used in science and engineering, and in industry.
Scientific computing
Scientific computing is the study of the techniques that underpin disciplinespecific fields of computational science.
Uncertainty quantification
Uncertainty quantification is a modern interdisciplinary science that cuts across traditional research groups and combines statistics, numerical analysis and computational applied mathematics.
Research seminars on topics associated with numerical analysis and scientific computing take place regularly in the following series:
 Applied maths (informal)
We welcome applications for PhD study in all areas of mathematics in the life sciences. PhD enquiries related to this theme can be directed to Sean Holman .
To discover the PhD opportunities available in the Department of Mathematics, explore our Postgraduate research in mathematics .
Our staff, students and postgraduate researchers have access to a fantastic range of facilities across the University.
Your PhD journey
Exploring research as part of your postgraduate research journey? Our postgraduate research hub will help you plan your next steps.
The Department's recent publications in the University's database.
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Computational Science & Numerical Analysis
Computational science is a key area related to physical mathematics. The problems of interest in physical mathematics often require computations for their resolution. Conversely, the development of efficient computational algorithms often requires an understanding of the basic properties of the solutions to the equations to be solved numerically. For example, the development of methods for the solution of hyperbolic equations (e.g. shock capturing methods in, say, gasdynamics) has been characterized by a very close interaction between theoretical, computational, experimental scientists, and engineers.
Department Members in This Field
 Laurent Demanet Applied analysis, Scientific Computing
 Alan Edelman Parallel Computing, Numerical Linear Algebra, Random Matrices
 Steven Johnson Waves, PDEs, Scientific Computing
 Pablo Parrilo Optimization, Control Theory, Computational Algebraic Geometry, Applied Mathematics
 Gilbert Strang Numerical Analysis, Partial Differential Equations
 John Urschel Matrix Analysis, Numerical Linear Algebra, Spectral Graph Theory
Instructors & Postdocs
 Pengning Chao Scientific computing, Nanophotonics, Inverse problems, Fundamental limits
 Shi Chen Machine Learning, Gradient Flows and Optimization, Inverse Problems
 Ziang Chen applied analysis, applied probability, statistics, optimization, machine learning
 Nicholas Nelsen Scientific Machine Learning, Statistics, Inverse Problems
Researchers & Visitors
 Keaton Burns PDEs, Spectral Methods, Fluid Dynamics
 Raphaël Pestourie Surrogate Models; MLenhanced Optimization; Optical Computing.
Graduate Students*
 Rodrigo Arrieta Candia Numerical methods for PDEs, Numerical Analysis, Scientific Computing, Computational Electromagnetism
 Diego Chavez Physical Applied Math, Waves
 Mo Chen Optimization, Scientific Computing
 Charlie CowenBreen PDE, machine learning, computerassisted proofs
 Max Daniels Highdimensional statistics, optimization, sampling algorithms, machine learning
 Sarah Greer Imaging, inverse problems, signal processing
 Joonsoo Lee
 Evelyne Ringoot HighPerformance Computing
 Sarah Snider Physical Applied Mathematics, Computational Fluid Dynamics
 Songchen Tan computational science, numerical analysis, differentiable programming
*Only a partial list of graduate students
 Admission Policies
 Financial Support
 Ph.D. in Atmosphere Ocean Science
 M.S. at Graduate School of Arts & Science
 M.S. at Tandon School of Engineering
 Current Students
Ph.D. in Mathematics, Specializing in Applied Math
Table of contents, overview of applied mathematics at the courant institute.
 PhD Study in Applied Mathematics
 Applied math courses
Applied mathematics has long had a central role at the Courant Institute, and roughly half of all our PhD's in Mathematics are in some applied field. There are a large number of applied fields that are the subject of research. These include:
 Atmosphere and Ocean Science
 Biology, including biophysics, biological fluid dynamics, theoretical neuroscience, physiology, cellular biomechanics
 Computational Science, including computational fluid dynamics, adaptive mesh algorithms, analysisbased fast methods, computational electromagnetics, optimization, methods for stochastic systems.
 Data Science
 Financial Mathematics
 Fluid Dynamics, including geophysical flows, biophysical flows, fluidstructure interactions, complex fluids.
 Materials Science, including micromagnetics, surface growth, variational methods,
 Stochastic Processes, including statistical mechanics, MonteCarlo methods, rare events, molecular dynamics
PhD study in Applied Mathematics
PhD training in applied mathematics at Courant focuses on a broad and deep mathematical background, techniques of applied mathematics, computational methods, and specific application areas. Descriptions of several appliedmath graduate courses are given below.
Numerical analysis is the foundation of applied mathematics, and all PhD students in the field should take the Numerical Methods I and II classes in their first year, unless they have taken an equivalent twosemester PhDlevel graduate course in numerical computing/analysis at another institution. Afterwards, students can take a number of more advanced and specialized courses, some of which are detailed below. Important theoretical foundations for applied math are covered in the following courses: (1) Linear Algebra I and II, (2) Intro to PDEs, (3) Methods of Applied Math, and (4) Applied Stochastic Analysis. It is advised that students take these courses in their first year or two.
A list of the current research interests of individual faculty is available on the Math research page.
Courses in Applied Mathematics
The following list is for AY 2023/2024:

(MATHGA.2701) Methods Of Applied Math
Fall 2023, Oliver Buhler
Description: This is a firstyear course for all incoming PhD and Masters students interested in pursuing research in applied mathematics. It provides a concise and selfcontained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multiscale asymptotics, transform methods, geometric wave theory, and calculus of variations.
Prerequisites : Elementary linear algebra, ordinary differential equations; at least an undergraduate course on partial differential equations is strongly recommended.
(MATHGA.2704) Applied Stochastic Analysis
Spring 2024, Jonathan Weare
This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective. Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments. The target audience is PhD students in applied mathematics, who need to become familiar with the tools or use them in their research.
Prerequisites: Basic Probability (or equivalent masterslevel probability course), Linear Algebra (graduate course), and (beginning graduatelevel) knowledge of ODEs, PDEs, and analysis.
(MATHGA.2010/ CSCIGA.2420) Numerical Methods I
 Fall 2023, Benjamin Peherstorfer
Description: This course is part of a twocourse series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an inclass final. Topics covered in the class include floatingpoint arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, and nonlinear optimization, and iterative methods. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.
Prerequisites: A good background in linear algebra, and some experience with writing computer programs (in MATLAB, Python or another language).
(MATHGA.2020 / CSCIGA.2421) Numerical Methods II
Spring 2024, Aleksandar Donev
This course (3pts) will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments in MATLAB/Python will form an essential part of the course. The course will introduce students to numerical methods for (approximately in this order):
 The Fast Fourier Transform and pseudospectral methods for PDEs in periodic domains
 Ordinary differential equations, explicit and implicit RungeKutta and multistep methods, IMEX methods, exponential integrators, convergence and stability
 Finite difference/element, spectral, and integral equation methods for elliptic BVPs (Poisson)
 Finite difference/element methods for parabolic (diffusion/heat eq.) PDEs (diffusion/heat)
 Finite difference/volume methods for hyperbolic (advection and wave eqs.) PDEs (advection, wave if time permits).
Prerequisites
This course requires Numerical Methods I or equivalent graduate course in numerical analysis (as approved by instructor), preferably with a grade of B+ or higher.
( MATHGA.2011 / CSCIGA 2945) Computational Methods For PDE
Fall 2023, Aleksandar Donev & Georg Stadler
This course follows on Numerical Methods II and covers theoretical and practical aspects of advanced computational methods for the numerical solution of partial differential equations. The first part will focus on finite element methods (FEMs), and the second part on finite volume methods (FVMs) including discontinuous Galerkin (FE+FV) methods. In addition to setting up the numerical and functional analysis theory behind these methods, the course will also illustrate how these methods can be implemented and used in practice for solving partial differential equations in two and three dimensions. Example PDEs will include the Poisson equation, linear elasticity, advectiondiffusion(reaction) equations, the shallowwater equations, the incompressible NavierStokes equation, and others if time permits. Students will complete a final project that includes using, developing, and/or implementing stateoftheart solvers.
In the Fall of 2023, Georg Stadler will teach the first half of this course and cover FEMs, and Aleks Donev will teach in the second half of the course and cover FVMs.
A graduatelevel PDE course, Numerical Methods II (or equivalent, with approval of syllabus by instructor(s)), and programming experience.
 Elman, Silvester, and Wathen: Finite Elements and Fast Iterative Solvers , Oxford University Press, 2014.
 Farrell: Finite Element Methods for PDEs , lecture notes, 2021.
 Hundsdorfer & Verwer: Numerical Solution of TimeDependent AdvectionDiffusionReaction Equations , SpringerVerlag, 2003.
 Leveque: Finite Volume Methods for Hyperbolic Problems , Cambridge Press, 2002.

( MATHGA.2012 ) Immersed Boundary Method For FluidStructure Interaction
Not offered AY 23/24.
The immersed boundary (IB) method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry. It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide variety of problems in biofluid dynamics, such as wave propagation in the inner ear, blood clotting, swimming of creatures large and small, and the flight of insects. Nonbiological applications include sails, parachutes, flows of suspensions, and twofluid or multifluid problems. Topics to be covered include: mathematical formulation of fluidstructure interaction in Eulerian and Lagrangian variables, with interaction equations involving the Dirac delta function; discretization of the structure, fluid, and interaction equations, including energybased discretization of the structure equations, finitedifference discretization of the fluid equations, and IB delta functions with specified mathematical properties; a simple but effective method for adding mass to an immersed boundary; numerical simulation of rigid immersed structures or immersed structures with rigid parts; IB methods for immersed filaments with bend and twist; and a stochastic IB method for thermally fluctuating hydrodynamics within biological cells. Some recent developments to be discussed include stability analysis of the IB method and a FourierSpectral IB method with improved boundary resolution.
Course requirements include homework assignments and a computing project, but no exam. Students may collaborate on the homework and on the computing project, and are encouraged to present the results of their computing projects to the class.
Prerequisite: Familiarity with numerical methods and fluid dynamics.
(MATHGA.2012 / CSCIGA.2945) : High Performance Computing
Not offered AY 23/24
This class will be an introduction to the fundamentals of parallel scientific computing. We will establish a basic understanding of modern computer architectures (CPUs and accelerators, memory hierarchies, interconnects) and of parallel approaches to programming these machines (distributed vs. shared memory parallelism: MPI, OpenMP, OpenCL/CUDA). Issues such as load balancing, communication, and synchronization will be covered and illustrated in the context of parallel numerical algorithms. Since a prerequisite for good parallel performance is good serial performance, this aspect will also be addressed. Along the way you will be exposed to important tools for high performance computing such as debuggers, schedulers, visualization, and version control systems. This will be a handson class, with several parallel (and serial) computing assignments, in which you will explore material by yourself and try things out. There will be a larger final project at the end. You will learn some Unix in this course, if you don't know it already.
Prerequisites for the course are (serial) programming experience with C/C++ (I will use C in class) or Fortran, and some familiarity with numerical methods.
(MATHGA.2011) Monte Carlo Methods
Fall 2023, Jonathan Weare and Jonathan Goodman
Topics : The theory and practice of Monte Carlo methods. Random number generators and direct sampling methods, visualization and error bars. Variance reduction methods, including multilevel methods and importance sampling. Markov chain Monte Carlo (MCMC), detailed balance, nondegeneracy and convergence theorems. Advanced MCMC, including Langevin and MALA, Hamiltonian, and affine invariant ensemble samplers. Theory and estimation of autocorrelation functions for MCMC error bars. Rare event methods including nested sampling, milestoning, and transition path sampling. Multistep methods for integration including Wang Landau and related thermodynamic integration methods. Application to sampling problems in physical chemistry and statistical physics and to Bayesian statistics.
Required prerequisites:
 A good probability course at the level of Theory of Probability (undergrad) or Fundamentals of Probability (masters)
 Linear algebra: Factorizations (especially Cholesky), subspaces, solvability conditions, symmetric and nonsymmetric eigenvalue problem and applications
 Working knowledge of a programming language such as Python, Matlab, C++, Fortran, etc.
 Familiarity with numerical computing at the level of Scientific Computing (masters)
Desirable/suggested prerequisites:
 Numerical methods for ODE
 Applied Stochastic Analysis
 Familiarity with an application area, either basic statistical mechanics (Gibbs Boltzmann distribution), or Bayesian statistics
(MATHGA.2012 / CSCIGA.2945) Convex & Non Smooth Optimization
Spring 2024, Michael Overton
Convex optimization problems have many important properties, including a powerful duality theory and the property that any local minimum is also a global minimum. Nonsmooth optimization refers to minimization of functions that are not necessarily convex, usually locally Lipschitz, and typically not differentiable at their minimizers. Topics in convex optimization that will be covered include duality, CVX ("disciplined convex programming"), gradient and Newton methods, Nesterov's optimal gradient method, the alternating direction method of multipliers, the primal barrier method, primaldual interiorpoint methods for linear and semidefinite programs. Topics in nonsmooth optimization that will be covered include subgradients and subdifferentials, Clarke regularity, and algorithms, including gradient sampling and BFGS, for nonsmooth, nonconvex optimization. Homework will be assigned, both mathematical and computational. Students may submit a final project on a preapproved topic or take a written final exam.
Prerequisites: Undergraduate linear algebra and multivariable calculus
Q1: What is the difference between the Scientific Computing class and the Numerical Methods twosemester sequence?
The Scientific Computing class (MATHGA.2043, fall) is a onesemester masterslevel graduate class meant for graduate or advanced undergraduate students that wish to learn the basics of computational mathematics. This class requires a working knowledge of (abstract) linear algebra (at least at the masters level), some prior programming experience in Matlab, python+numpy, Julia, or a compiled programming language such as C++ or Fortran, and working knowledge of ODEs (e.g., an undergrad class in ODEs). It only briefly mentions numerical methods for PDEs at the very end, if time allows.
The Numerical Methods I (fall) and Numerical Methods II (spring) twosemester sequence is a Ph.D.level advanced class on numerical methods, meant for PhD students in the field of applied math, masters students in the SciComp program , or other masters or advanced undergraduate students that have already taken at least one class in numerical analysis/methods. It is intended that these two courses be taken one after the other, not in isolation . While it is possible to take just Numerical Methods I, it is instead strongly recommended to take the Scientific Computing class (fall) instead. Numerical Methods II requires part I, and at least an undergraduate class in ODEs, and also in PDEs. Students without a background in PDEs should not take Numerical Methods II; for exceptions contact Aleks Donev with a detailed justification.
The advanced topics class on Computational Methods for PDEs follows on and requires having taken NumMeth II or an equivalent graduatelevel course at another institution (contact Aleks Donev with a syllabus from that course for an evaluation), and can be thought of as Numerical Methods III.
Q2: How should I choose a first graduate course in numerical analysis/methods?
 If you are an undergraduate student interested in applied math graduate classes, you should take the undergraduate Numerical Analysis course (MATHUA.0252) first, or email the syllabus for the equivalent of a fullsemester equivalent class taken elsewhere to Aleks Donev for an evaluation.
 Take the Scientific Computing class (fall), or
 Take both Numerical Methods I (fall) and II (spring), see Q1 for details. This is required of masters students in the SciComp program .
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PhD Numerical Analysis
University of manchester, different course options.
 Key information
Course Summary
Tuition fees, entry requirements, similar courses at different universities, key information data source : idp connect, qualification type.
PhD/DPhil  Doctor of Philosophy
Subject areas
Numerical Analysis
Course type
Programme description
The Department of Mathematics has an outstanding research reputation. The research facilities include one of the finest libraries in the country, the John Rylands University Library. This library has recently made a very large commitment of resources to providing comprehensive online facilities for the free use of the University's research community. Postgraduate students in the Department benefit from direct access to all the Library electronic resources from their offices.
Many research seminars are held in the Department on a weekly basis and allow staff and research students to stay in touch with the latest developments in their fields. The Department is one of the lead partners in the MAGIC project and research students can attend any of the postgraduate courses offered by the MAGIC consortium.
Opportunities for PhD research are available in a wide range of topics in Numerical Analysis. For some of the available areas of possible PhD research please visit the Numerical Analysis research page or the Numerical Analysis group website . For more details about the specific projects, please contact the relevant individual members of staff or get in touch with the Numerical Analysis Postgraduate Admissions Tutor.
UK fees Course fees for UK students
For this course (per year)
International fees Course fees for EU and international students
Students would usually be expected to have a 1st or upper 2nd class Honours Bachelor degree and an MSc with Merit or Distinction (or an equivalent overseas qualification) in a mathematical subject. MSc with Merit or Distinction (or an equivalent overseas qualification) in a mathematical subject. In some research areas, a background in physics, engineering or computer science is also acceptable.
Applied Statistics MSc
Birkbeck, university of london, applied statistics postgraduate certificate, mathematics (graduate certificate), applied statistics and financial modelling msc, mathematics and statistics phd.
Quick links
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Research Areas
It is possible to apply mathematics to almost any field of human endeavor. Here are some of the fields we’re working on now.
Scientific Computing and Numerical Analysis
Researchers : Loyce Adams , Bernard Deconinck , Randy LeVeque , Ioana Dumitriu , Anne Greenbaum , James Riley
Many practical problems in science and engineering cannot be solved completely by analytical means. Research in the area of numerical analysis and scientific computation is concerned with the development and analysis of numerical algorithms, the implementation of these algorithms on modern computer architectures, and the use of numerical methods in conjunction with mathematical modeling to solve largescale practical problems. Major research areas in this department include computational fluid dynamics (CFD), interface and front tracking methods, iterative methods in numerical linear algebra, and algorithms for parallel computers.Current research topics in CFD include:
 high resolution methods for solving nonlinear conservation laws with shock wave solutions
 numerical methods for atmospheric flows, particularly cloud formation
 Cartesian grid methods for solving multidimensional problems in complicated geometries on uniform grids
 spectral methods for fluid stability problems
 front tracking methods for fluid flow problems with free surfaces or immersed interfaces in the context of porous media flow (ground water or oil reservoir simulation) and in physiological flows with elastic membranes.
 nonequilibrium flows in combustion and astrophysical simulation
 immersed interface methods for solidification or melting problems and seismic wave equations with discontinuous coefficients that arise in modeling the geological structure of the earth.
Another research focus is the development of methods for largescale scientific computations that are suited to implementation on parallel computer architectures. Current interests include:
 preconditioners for the iterative solution of large linear or nonlinear systems
 methods for the symmetric and nonsymmetric eigenvalue problems
 methods for general interface problems in complicated domains.
The actual implementation and testing of methods on parallel architectures is possible through collaboration with the Department of Computer Science, the Boeing Company, and the Pacific Northwest Labs.
Nonlinear Waves and Coherent Structures
Researchers : Bernard Deconinck , Nathan Kutz , Randy LeVeque
Most problems in applied mathematics are inherently nonlinear. The effects due to nonlinearities may become important under the right circumstances. The area of nonlinear waves and coherent structures considers how nonlinear effects influence problems involving wave propagation. Sometimes these effects are desirable and lead to new applications (modelocked lasers, optical solitons and nonlinear optics). Other times one has no choice but to consider their impact (water waves). The area of nonlinear waves encompasses a large collection of phenomena, such as the formation and propagation of shocks and solitary waves. The area received renewed interest starting in the 1960s with the development of soliton theory, which examines completely integrable systems and classes of their special solutions.
Mathematical Biology
Researchers : Mark Kot , Hong Qian , Eric SheaBrown , Elizabeth Halloran , Suresh Moolgavkar , Eli Shlizerman , Ivana Bozic
Mathematical biology is an increasingly large and wellestablished branch of applied mathematics. This growth reflects both the increasing importance of the biological and biomedical sciences and an appreciation for the mathematical subtleties and challenges that arise in the modelling of complex biological systems. Our interest, as a group, lies in understanding the spatial and temporal patterns that arise in dynamic biological systems. Our mathematical activities range from reactiondiffusion equations, to nonlinear and chaotic dynamics, to optimization. We employ a variety of tools and models to study problems that arise in development, epidemiology, ecology, neuroscience, resource management, and biomechanics; and we maintain active collaborations with a large number and variety of biologists and biomedical departments both in the University and elsewhere. For more information, please see the Mathematical Biology page .
Atmospheric Sciences and Climate Modeling
Researchers : Chris Bretherton , KaKit Tung , Dale Durran
Mathematical models play a crucial role in our understanding of the fluid dynamics of the atmosphere and oceans. Our interests include mathematical methods for studying the hydrodynamical instability of shear flows, transition from laminar flow to turbulence, applications of fractals to turbulence, twodimensional and quasigeostrophic turbulence theory and computation, and largescale nonlinear wave mechanics.We also develop and apply realistic coupled radiative chemicaldynamical models for studying stratospheric chemistry, and coupled radiativemicrophysicaldynamical models for studying the interaction of atmospheric turbulence and cloud systems These two topics are salient for understanding how man is changing the earth’s climate.Our work involves a strong interaction of computer modelling and classical applied analysis. This research group actively collaborates with scientists in the Atmospheric Science, Oceanography, and Geophysics department, and trains students in the emerging interdisciplinary area of earth system modeling, in addition to providing a traditional education in classical fluid dynamics.
Mathematical Methods
Researchers : Bernard Deconinck , Robert O'Malley , Jim Burke , Archis Ghate , John Sylvester , Gunther Uhlmann
The department maintains active research in fundamental methods of applied mathematics. These methods can be broadly applied to a vast number of problems in the engineering, physical and biological sciences. The particular strengths of the department of applied mathematics are in asymptotic and perturbation methods, applied analysis, optimization and control, and inverse problems.
Mathematical Finance
Researchers : Tim Leung , Matt Lorig , Doug Martin
The department’s growing financial math group is active in the areas of derivative pricing & hedging, algorithmic trading, portfolio optimization, insurance, risk measures, credit risk, and systemic risk. Research includes collaboration with students as well as partners from both academia and industry.
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Numerical Analysis
Numerical analysis research group.
Name  Title  phone  Office  

Associate Professor  Mathematical Sciences  General  Eng & Math Sciences E441  
Associate Professor; Applied Mathematics & Computer Science Undergraduate Coordinator  Mathematical Sciences  General  Eng & Math Sciences E453  
Professor  Mathematical Sciences  General  Eng & Math Sciences W411 
Numerical Analysis at UWM
The Numerical Analysis Group of the University of WisconsinMilwaukee Department of Mathematical Sciences offers many opportunities for students to enter the important and exciting field of research in numerical analysis and scientific computing. The cornerstone of the program is the PhD in Mathematics with specialty in Numerical Analysis (Applied Mathematics). It is also possible to receive a PhD in Industrial Mathematics while emphasizing scientific computing. There are five graduate research faculty and one senior lecturer, with a variety of fields of expertise and scholarly activities among the faculty. The research involves numerical analysis for integral equations, partial differential equations (finite difference, finite element, domain decomposition), and optimization.
Research Relationships
The Numerical Analysis Group is closely related to the Center Industrial Mathematics (CIM) and Atmospheric Science Groups, each of which is part of the Department of Mathematical Sciences. The CIM acts as a liaison between academic and corporate units, assists researchers with nondisclosure and patent issues, works to gather funding support, and sponsors conferences and workshops on industrial mathematics. The Atmospheric Science Group is distinguished with its mathematical approach within its field of discipline. Hence they create solid opportunities for interdisciplinary work for students of numerical analysis and scientific computation.
Student Opportunities
There are opportunities for teaching assistantships, research assistantships, and fellowships in numerical analysis, scientific computing, and industrial mathematics; if interested, please contact the Associate Chair for the Graduate Program .
The Department of Mathematical Sciences also offers an option within the PhD with industrial emphasis; students work on dissertations solving advanced mathematical problems with industrial sources. More information may be obtained at the Web site of the Center for Industrial Mathematics (CIM) .
Milwaukee is a very good location to carry out scientific computational research and industrial mathematics activities because of its excellent universities, easy access to many other universities (e.g., Chicago and Madison), as well as large variety of industry. UWM Numerical Analysis and Center for Industrial Mathematics faculty have published in areas of finite difference and finit element methods for ordinary and partial differential equations, computational aspects of biomathematics, computational aspects of control, numerical analysis and computational analysis for integral equations, domain decomposition for PDE, optimization, Computerized Tomography and Magnetic Resonance Imaging, Nonlinear PDE modeling and simulation (various types), statistics as well as computational statistics, and applications of artificial neural networks. UWM faculty and students have worked with industries in activities as diverse as airline scheduling, electrical power systems, engine performance modeling and simulation, finance, industrial controls, industrial printing, medical imaging (CT and MRI), paint production, refrigeration, telephone queuing systems, and travel industry data analysis.
 MyU : For Students, Faculty, and Staff
Numerical Analysis
+ more about numerical analysis, applications.
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 Weekly seminar series in data science and applied and computational mathematics
Douglas Arnold
McKnight Presidential Professor
[email protected] Numerical analysis, differential equations, mechanics, computational relativity
Jeffrey Calder
Associate Professor
[email protected] partial differential equations, numerical analysis, applied probability, machine learning, image processing and computer vision
Bernardo Cockburn
Distinguished McKnight University Professor
[email protected] numerical analysis
Jasmine Foo
Northrop Professor
[email protected] mathematical biology and applied mathematics
William Leeb
Associate Professor
[email protected] applied mathematics, computational harmonic analysis, signal and image processing, data analysis
Gilad Lerman
[email protected] computational harmonic analysis, analysis of large data sets and statistical learning, bioinformatics
Assistant Professor
[email protected] Mathematical foundations of machine learning and data sciences, applied probability and stochastic dynamics, applied analysis and PDEs, Bayesian and computational statistics, inverse problems and uncertainty quantification
Mitchell Luskin
[email protected] numerical analysis, scientific computing, applied mathematics, computational physics
Peter Olver
[email protected] Lie groups, differential equations, computer vision, applied mathematics, differential geometry, mathematical physics
[email protected] numerical analysis, scientific computing, applied math
Alex Watson
[email protected] Partial differential equations, mathematical physics, numerical analysis, computational physics, data science
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Discover more about postgraduate research
PhD Numerical Analysis / Overview
Year of entry: 2025
 View full page
The standard academic entry requirement for this PhD is an upper secondclass (2:1) honours degree in a discipline directly relevant to the PhD (or international equivalent) OR any uppersecond class (2:1) honours degree and a Master’s degree at merit in a discipline directly relevant to the PhD (or international equivalent).
Other combinations of qualifications and research or work experience may also be considered. Please contact the admissions team to check.
Full entry requirements
Apply online
In your application you’ll need to include:
 The name of this programme
 Your research project title (i.e. the advertised project name or proposed project name) or area of research
 Your proposed supervisor’s name
 If you already have funding or you wish to be considered for any of the available funding
 A supporting statement (see 'Advice to Applicants for what to include)
 Details of your previous university level study
 Names and contact details of your two referees.
Before applying we also recommend that you read the 'Advice to Applicants' section.
Programme options
Fulltime  Parttime  Fulltime distance learning  Parttime distance learning  

PhD  Y  Y  N  N 
Programme description
The Department of Mathematics has an outstanding research reputation. The research facilities include one of the finest libraries in the country, the John Rylands University Library. This library has recently made a very large commitment of resources to providing comprehensive online facilities for the free use of the University's research community. Postgraduate students in the Department benefit from direct access to all the Library electronic resources from their offices.
Many research seminars are held in the Department on a weekly basis and allow staff and research students to stay in touch with the latest developments in their fields. The Department is one of the lead partners in the MAGIC project and research students can attend any of the postgraduate courses offered by the MAGIC consortium.
Opportunities for PhD research are available in a wide range of topics in Mathematics . For more information, please see Numerical Analysis research page or the Numerical Analysis group website and find out more about specific projects . Please contact the relevant individual members of staff or get in touch with the Numerical Analysis Postgraduate Admissions Tutor .
Visit our Events and Opportunities page to find out about upcoming open days and webinars.
Fees for entry in 2025 have not yet been set. For reference, the fees for the academic year beginning September 2024 were as follows:
 PhD (fulltime) UK students (per annum): Band A £4,786; Band B £7,000; Band C £10,000; Band D £14,500; Band E £24,500 International, including EU, students (per annum): Band A £28,000; Band B £30,000; Band C £35,500; Band D £43,000; Band E £57,000
 PhD (parttime) UK students (per annum): Band A £2393; Band B £3,500; Band C £5,000; Band D £7,250; Band E 12,250 International, including EU, students (per annum): Band A £14,000; Band B £15,000; Band C £17,750; Band D £21,500; Band E £28,500
Further information for EU students can be found on our dedicated EU page.
The programme fee will vary depending on the cost of running the project. Fees quoted are fully inclusive and, therefore, you will not be required to pay any additional bench fees or administration costs.
All fees for entry will be subject to yearly review and incremental rises per annum are also likely over the duration of the course for Home students (fees are typically fixed for International students, for the course duration at the year of entry). For general fees information please visit the postgraduate fees page .
Always contact the Admissions team if you are unsure which fees apply to your project.
Scholarships/sponsorships
There are a range of scholarships, studentships and awards at university, faculty and department level to support both UK and overseas postgraduate researchers.
To be considered for many of our scholarships, you’ll need to be nominated by your proposed supervisor. Therefore, we’d highly recommend you discuss potential sources of funding with your supervisor first, so they can advise on your suitability and make sure you meet nomination deadlines.
For more information about our scholarships, visit our funding page to search for scholarships, studentships and awards you may be eligible for.
Contact details
Our internationallyrenowned expertise across the School of Natural Sciences informs research led teaching with strong collaboration across disciplines, unlocking new and exciting fields and translating science into reality. Our multidisciplinary learning and research activities advance the boundaries of science for the wider benefit of society, inspiring students to promote positive change through educating future leaders in the true fundamentals of science. Find out more about Science and Engineering at Manchester .
Programmes in related subject areas
Use the links below to view lists of programmes in related subject areas.
 Mathematics
Regulated by the Office for Students
The University of Manchester is regulated by the Office for Students (OfS). The OfS aims to help students succeed in Higher Education by ensuring they receive excellent information and guidance, get high quality education that prepares them for the future and by protecting their interests. More information can be found at the OfS website .
You can find regulations and policies relating to student life at The University of Manchester, including our Degree Regulations and Complaints Procedure, on our regulations website .
We have 40 Mathematics (numerical analysis) PhD Projects, Programmes & Scholarships
Mathematics
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Mathematics (numerical analysis) PhD Projects, Programmes & Scholarships
Connections between numerical analysis of differential equations and machine learning, phd research project.
PhD Research Projects are advertised opportunities to examine a predefined topic or answer a stated research question. Some projects may also provide scope for you to propose your own ideas and approaches.
Funded PhD Project (UK Students Only)
This research project has funding attached. It is only available to UK citizens or those who have been resident in the UK for a period of 3 years or more. Some projects, which are funded by charities or by the universities themselves may have more stringent restrictions.
PhD in Computer Arithmetic and Numerical Analysis
Competition funded phd project (students worldwide).
This project is in competition for funding with other projects. Usually the project which receives the best applicant will be successful. Unsuccessful projects may still go ahead as selffunded opportunities. Applications for the project are welcome from all suitably qualified candidates, but potential funding may be restricted to a limited set of nationalities. You should check the project and department details for more information.
Numerical Algorithms and Analysis for Deterministic and Stochastic Systems
Numerical algorithms for molecular systems and data science, projects in mathematical systems and control theory, selffunded phd students only.
This project does not have funding attached. You will need to have your own means of paying fees and living costs and / or seek separate funding from student finance, charities or trusts.
Next Generation Machine Learning for Data Analysis
Neural networks for complex dynamical systems, investigate cytoskeletal networks controlling cancer cell migration at the systems level., towards a clinical meg system: theoretical and practical insights into the use of optically pumped magnetometers, active particles in complex geometry environments, discovery, computational modelling and prediction of novel virucidal polymers against sarscov2, funded phd project (students worldwide).
This project has funding attached, subject to eligibility criteria. Applications for the project are welcome from all suitably qualified candidates, but its funding may be restricted to a limited set of nationalities. You should check the project and department details for more information.
Modelling and Computation for Radiation Chemistry at Interfaces
Phd mathematical sciences, funded phd programme (students worldwide).
Some or all of the PhD opportunities in this programme have funding attached. Applications for this programme are welcome from suitably qualified candidates worldwide. Funding may only be available to a limited set of nationalities and you should read the full programme details for further information.
China PhD Programme
A Chinese PhD usually takes 34 years and often involves following a formal teaching plan (set by your supervisor) as well as carrying out your own original research. Your PhD thesis will be publicly examined in front of a panel of expert. Some international programmes are offered in English, but others will be taught in Mandarin Chinese.
The University of Manchester  Department of Mathematics
Maths research programme.
PhD Research Programmes describe the opportunities for postgraduate research within a University department. You may often be asked to submit your own research project proposal as part of your application, although predefined research projects may also be available.
Machine Learning and Domain Decomposition methods for Fluid Dynamics
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Mathematical Modeling Doctor of Philosophy (Ph.D.) Degree
Request Info about graduate study Visit Apply
The mathematical modeling Ph.D. enables you to develop mathematical models to investigate, analyze, predict, and solve the behaviors of a range of fields from medicine, engineering, and business to physics and science.
Overview for Mathematical Modeling Ph.D.
Mathematical modeling is the process of developing mathematical descriptions, or models, of realworld systems. These models can be linear or nonlinear, discrete or continuous, deterministic or stochastic, and static or dynamic, and they enable investigating, analyzing, and predicting the behavior of systems in a wide variety of fields. Through extensive study and research, graduates of the mathematical modeling Ph.D. will have the expertise not only to use the tools of mathematical modeling in various application settings, but also to contribute in creative and innovative ways to the solution of complex interdisciplinary problems and to communicate effectively with domain experts in various fields.
Plan of Study
The degree requires at least 60 credit hours of course work and research. The curriculum consists of three required core courses, three required concentration foundation courses, a course in scientific computing and highperformance computing (HPC), three elective courses focused on the student’s chosen research concentration, and a doctoral dissertation. Elective courses are available from within the School of Mathematics and Statistics as well as from other graduate programs at RIT, which can provide applicationspecific courses of interest for particular research projects. A minimum of 30 credits hours of course work is required. In addition to courses, at least 30 credit hours of research, including the Graduate Research Seminar, and an interdisciplinary internship outside of RIT are required.
Students develop a plan of study in consultation with an application domain advisory committee. This committee consists of the program director, one of the concentration leads, and an expert from an application domain related to the student’s research interest. The committee ensures that all students have a roadmap for completing their degree based on their background and research interests. The plan of study may be revised as needed. Learn more about our mathematical modeling doctoral students and view a selection of mathematical modeling seminars hosted by the department.
Qualifying Examinations
All students must pass two qualifying examinations to determine whether they have sufficient knowledge of modeling principles, mathematics, and computational methods to conduct doctoral research. Students must pass the examinations in order to continue in the Ph.D. program.
The first exam is based on the Numerical Analysis I (MATH602) and Mathematical Modeling I, II (MATH622, 722). The second exam is based on the student's concentration foundation courses and additional material deemed appropriate by the committee and consists of a short research project.
Dissertation Research Advisor and Committee
A dissertation research advisor is selected from the program faculty based on the student's research interests, faculty research interest, and discussions with the program director. Once a student has chosen a dissertation advisor, the student, in consultation with the advisor, forms a dissertation committee consisting of at least four members, including the dissertation advisor. The committee includes the dissertation advisor, one other member of the mathematical modeling program faculty, and an external chair appointed by the dean of graduate education. The external chair must be a tenured member of the RIT faculty who is not a current member of the mathematical modeling program faculty. The fourth committee member must not be a member of the RIT faculty and may be a professional affiliated with industry or with another institution; the program director must approve this committee member.
The main duties of the dissertation committee are administering both the candidacy exam and final dissertation defense. In addition, the dissertation committee assists students in planning and conducting their dissertation research and provides guidance during the writing of the dissertation.
Admission to Candidacy
When a student has developed an indepth understanding of their dissertation research topic, the dissertation committee administers an examination to determine if the student will be admitted to candidacy for the doctoral degree. The purpose of the examination is to ensure that the student has the necessary background knowledge, command of the problem, and intellectual maturity to carry out the specific doctorallevel research project. The examination may include a review of the literature, preliminary research results, and proposed research directions for the completed dissertation. Requirements for the candidacy exam include both a written dissertation proposal and the presentation of an oral defense of the proposal. This examination must be completed at least one year before the student can graduate.
Dissertation Defense and Final Examination
The dissertation defense and final examination may be scheduled after the dissertation has been written and distributed to the dissertation committee and the committee has consented to administer the final examination. Copies of the dissertation must be distributed to all members of the dissertation committee at least four weeks prior to the final examination. The dissertation defense consists of an oral presentation of the dissertation research, which is open to the public. This public presentation must be scheduled and publicly advertised at least four weeks prior to the examination. After the presentation, questions will be fielded from the attending audience and the final examination, which consists of a private questioning of the candidate by the dissertation committee, will ensue. After the questioning, the dissertation committee immediately deliberates and thereafter notifies the candidate and the mathematical modeling graduate director of the result of the examination.
All students in the program must spend at least two consecutive semesters (summer excluded) as resident fulltime students to be eligible to receive the doctoral degree.
Maximum Time Limitations
University policy requires that doctoral programs be completed within seven years of the date of the student passing the qualifying exam. All candidates must maintain continuous enrollment during the research phase of the program. Such enrollment is not limited by the maximum number of research credits that apply to the degree.
National Labs Career Fair
Hosted by RIT’s Office of Career Services and Cooperative Education, the National Labs Career Fair is an annual event that brings representatives to campus from the United States’ federally funded research and development labs. These national labs focus on scientific discovery, clean energy development, national security, technology advancements, and more. Students are invited to attend the career fair to network with lab professionals, learn about opportunities, and interview for coops, internships, research positions, and fulltime employment.
Join us virtually and oncampus
Discover how graduate study at RIT can help further your career objectives.
Learn more and register for a graduate open house
The College of Science consistently receives research grant awards from organizations that include the National Science Foundation , National Institutes of Health , and NASA , which provide you with unique opportunities to conduct cuttingedge research with our faculty members.
Faculty in the School of Mathematics and Statistics conducts research on a broad variety of topics including:
 applied inverse problems and optimization
 applied statistics and data analytics
 biomedical mathematics
 discrete mathematics
 dynamical systems and fluid dynamics
 geometry, relativity, and gravitation
 mathematics of earth and environment systems
 multimessenger and multiwavelength astrophysics
Learn more by exploring the school’s mathematics research areas .
Michael Cromer
Basca Jadamba
Moumita Das
Carlos Lousto
Matthew Hoffman
Featured Work and Profiles
Ph.D. student explores fire through visual art and math modeling
Mathematical Modeling Ph.D. student, Jenna Sjunneson McDanold, explores fire through visual art and mathematical modeling.
Mathematical Modeling, Curtain Coating, and Glazed Donuts
Bridget Torsey (Mathematical Modeling)
In her research, Bridget Torsey, a Math Modeling Ph.D. student, developed a mathematical model that can optimize curtain coating processes used to cover donuts with glaze so they taste great.
Your Partners in Success: Meet Our Faculty, Dr. Wong
Dr. Tony Wong
Mathematics is a powerful tool for answering questions. From mitigating climate risks to splitting the dinner bill, Professor Wong shows students that math is more than just a prerequisite.
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Curriculum for 20242025 for Mathematical Modeling Ph.D.
Current Students: See Curriculum Requirements
Mathematical Modeling, Ph.D. degree, typical course sequence
Course  Sem. Cr. Hrs.  

MATH602  3  
MATH606  1  
MATH607  1  
MATH622  3  
MATH722  3  
6  
3  
MATH751  3  
MATH790  7  
3  
6  
MATH790  9  
MATH790  6  
MATH790  6  
Concentrations
Applied inverse problems.
Course  Sem. Cr. Hrs.  

MATH625  3  
MATH633  3  
MATH741  3 
Biomedical Mathematics
Course  Sem. Cr. Hrs.  

MATH631  3  
MATH702  3  
MATH761  3 
Discrete Mathematics
Course  Sem. Cr. Hrs.  

CSCI665  3  
MATH645  3  
MATH646  3 
Dynamical Systems and Fluid Dynamics
Course  Sem. Cr. Hrs.  

MATH631  3  
MATH741  3  
MATH831  3 
Geometry, Relativity and Gravitation
Course  Sem. Cr. Hrs.  

ASTP660  3  
ASTP861  3  
MATH702  3 
Students are also interested in
 Applied and Computational Mathematics MS
Admissions and Financial Aid
This program is available oncampus only.
Offered  Admit Term(s)  Application Deadline  STEM Designated 

Full‑time  Fall  January 15 priority deadline, rolling thereafter  Yes 
Fulltime study is 9+ semester credit hours. International students requiring a visa to study at the RIT Rochester campus must study full‑time.
Application Details
To be considered for admission to the Mathematical Modeling Ph.D. program, candidates must fulfill the following requirements:
 Learn tips to apply for a doctoral program and then complete a graduate application .
 Submit copies of official transcript(s) (in English) of all previously completed undergraduate and graduate course work, including any transfer credit earned.
 Hold a baccalaureate degree (or US equivalent) from an accredited university or college.
 A recommended minimum cumulative GPA of 3.0 (or equivalent).
 Satisfy prerequisite requirements and/or complete foundation courses prior to starting program coursework.
 Submit a current resume or curriculum vitae.
 Submit a statement of purpose for research which will allow the Admissions Committee to learn the most about you as a prospective researcher.
 Submit two letters of recommendation .
 Entrance exam requirements: None
 Writing samples are optional.
 Submit English language test scores (TOEFL, IELTS, PTE Academic), if required. Details are below.
English Language Test Scores
International applicants whose native language is not English must submit one of the following official English language test scores. Some international applicants may be considered for an English test requirement waiver .
TOEFL  IELTS  PTE Academic 

100  7.0  70 
International students below the minimum requirement may be considered for conditional admission. Each program requires balanced subscores when determining an applicant’s need for additional English language courses.
How to Apply Start or Manage Your Application
Cost and Financial Aid
An RIT graduate degree is an investment with lifelong returns. Ph.D. students typically receive full tuition and an RIT Graduate Assistantship that will consist of a research assistantship (stipend) or a teaching assistantship (salary).
Additional Information
Foundation courses.
Mathematical modeling encompasses a wide variety of scientific disciplines, and candidates from diverse backgrounds are encouraged to apply. If applicants have not taken the expected foundational course work, the program director may require the student to successfully complete foundational courses prior to matriculating into the Ph.D. program. Typical foundation course work includes calculus through multivariable and vector calculus, differential equations, linear algebra, probability and statistics, one course in computer programming, and at least one course in real analysis, numerical analysis, or upperlevel discrete mathematics.
2024 SIAM New YorkNew JerseyPennsylvania Section Meeting
Welcome to the Homepage of the research group Numerical Analysis and Uncertainty Quantification at University of Heidelberg
Postal address
Institute of Applied Mathematics and Interdisciplinary Center for Scientific Computing (IWR) Universität Heidelberg Im Neuenheimer Feld 205 69120 Heidelberg, Germany
Herta Fitzer NumOpt [at] uniheidelberg.de +49 6221 5414111 Room: 1 / 318
Many physical models from the natural sciences and from engineering involve sources of uncertainty that affect their outputs. An example are variations in the orientation of layers of carbon fibres occurring naturally in the manufacturing process of aircraft wings. As a consequence, the locations and types of possible defects and cracks are difficult to predict precisely. The goal of uncertainty quantification is to use mathematical and computational methods to account for such uncertainties, and to understand how they propagate through to model outputs.
The research of our group focuses on developing innovative numerical methods to efficiently quantify uncertainty. We apply these techniques to tackle datadriven, largescale problems that are typically modelled in the form of differential equations. Our methods strive for a balance between efficiency, a rigorous mathematical foundation and realistic model problems. As a brief overview, some of the techniques we develop are based on Monte Carlo/quasiMonte Carlo sampling, stochastic collocation, sophisticated hierarchical/multilevel strategies and dimension reduction through lowrank tensor approximations. The applications we study range from groundwater flow, to nuclear physics and carbon fibre composites in manufacturing.
When studying real world processes mathematically, essentially all problems can be roughly put into two classes: Forward and inverse problems. A forward problem concerns the calculation of the state of a physical system, given all the necessary parameters, as well as boundary and initial conditions. Inverse problems, on the other hand, are concerned with computing parameters given observations of the state of the system. Good estimates of these parameters give us an insight into hidden quantities that typically cannot be observed directly and are otherwise very difficult to grasp. Consequently, inverse problems are among the most important in mathematical applications and uncertainty quantification plays a crucial role here.
A very popular approach to address inverse problems is Bayesian inference, a subbranch of statistics and data science. It facilitates the quantification of uncertainties regarding the model and its parameters. This resolves the inherent illposedness of inverse problems and has proven to be exceedingly fruitful in many applications. A particular focus in our group is the design and analysis of efficient numerical techniques for high and infinitedimensional Bayesian inverse problems, especially those constrained by differential equations. We look at the effect of choosing highlevel priors, develop efficient multilevel algorithms and surrogates, incorporate novel ideas from numerical analysis into this setting and explore links to machine learning.
Natural or engineered materials often contain two or more key constituents, arranged in a heterogeneous structure varying at different scales. Such materials are desirable because their macroscopic properties can be superior to the properties of the individual constituents. It is even possible to explicitly design them for a particular purpose by changing the composition of the constituents. An example are carbon fibre composites for lightweight structures and vehicles. The mathematical modelling of such heterogeneous or composite materials naturally leads to partial differential equations (PDEs) with highly oscillating coefficients. Direct numerical solution of such problems with traditional methods, such as finite elements is computationally expensive. Just to compute the correct qualitative behaviour, the mesh resolution would need to be sufficiently high to capture all the fine scale variation.
In our group, we study and develop multiscale numerical methods that do not suffer from this drawback. We are particularly interested in systems without periodic structure or scale separation and in problems with a high contrast in the constituent material properties. Such problems require customized approximation spaces, computable via localised boundary value or eigenproblems. There are also strong links to model order reduction and domain decomposition methods. Examples include multiscale finite elements, generalised multiscale finite elements, or the localizable orthogonal decomposition method. Target applications are again subsurface flow and carbon fibre composites, but also biological materials such as bone or cells.
Scientific computing and numerical simulation are playing an ever more important role in science and technology. Hardly any new developments, e.g., in engineering or the geosciences, take place without careful mathematical modelling, analysis and optimisation. More and more complicated systems are being tackled, in particular in the life sciences or in the context of climate change. This requires continued research into efficient and robust numerical methods, especially in the context of heterogeneous or random media, and in their careful and rigorous numerical analysis. Increased efficiency requires a redesign of traditional algorithms to harness the power of modern manycore computing architectures (hardwareaware scientific computing), while datadriven (predictive) scientific computing poses new challenges for the robustness of existing methods.
Here, the focus of our group encompasses
 traditional topics, such as efficient preconditioning and discretisation methods for heterogeneous and anisotropic PDEs, or efficient algorithms for largescale eigenproblems, as well as
 more modern topics that arise naturally in the context of uncertainty quantification or Bayesian inference, such as highdimensional approximation and quadrature, including Monte Carlo, quasiMonte Carlo, sparse grid, lowrank tensor approximation or deep learning.
In terms of novel software, we contribute in particular to DUNE (the Distributed and Unified Numerical Environment) and to MUQ (the MIT Uncertainty Quantification Library).
Recent PhD Theses  Applied Mathematics
 2007  2008  2009  2010  2011  2012  2013  2014  2015  2016  2017  2018  2019  2022  2023  2024
PhD Theses 2024
Author  Title 

Hank Chen  
Stephanie Abo  
Ala' Alalabi  
Liang Chen 

Alexey Smirnov  
Mengyao Zhang  
Yuan Wang  
Chuanzheng Wang 
PhD Theses 2023
Author  Title  

Donovan Allum 
 
Maxwell Fitzsimmons 
 
MariaEftychia Papageorgiou 
 
Esha Saha 
 
Zhanlue Liang 
 
Dorsa Mohammadrezaei  
Aditya Jain  
Cameron Meaney  
Panagiotis Tsimiklis  
Julian SmithRoberge  
Vahid Nourian 
PhD Theses 2022
Author  Title  

James Petrie  
Yiming Meng 
 
Mana Donganont  
Russell Milne 
 
Alison Cheeseman 
 
Qiaoyin Pan  
Krishna Dutt  
Andrew Grace  
Keegan Kirk  
Brydon Eastman  
Benjamin Lovitz 
PhD Theses 2019
Author  Title  

Yangang Chen 
 
Kamran Akbari  
Lindsey Daniels  
Kevin E.M. Church  
Chengzhu Xu 
PhD Theses 2018
Author  Title  

Julian Rennert  
Andrew Giuliani  
Ian Hincks  
Daniel Puzzuoli  
David Deepwell  
Tahmina Akhter 

PhD Theses 2017
Author  Title  

Cong Wu 
 
Monjur Morshed 
 
Alexander James Maxwell Howse 
 
Zhen Wang 
 
Kexue Zhang 
 
Mikhail Panine 
 
Keegan Keplinger 
 
Subasha Wickramarachchi 
 
Jared Penney 
 
Sepideh Afshar 
 
Ali Mahdipour Shirayeh 

PhD Theses 2016
Author  Title  

Giuseppe Sellaroli 
 
Robert H. Jonsson 
 
John Lang 
 
John Yawney 
 
Kristopher Rowe 
 
Manda Winlaw 
 
Anton Baglaenko 

PhD Theses 2015
Author  Title  

Taghreed Sugati 
 
Wilten Nicola 
 
Herbert Tang 
 
Noel Chalmers  
Puneet Sharma 
 
Daniel Otero 
 
Sina Khani  
Ilona Anna KowalikUrbaniak 
PhD Theses 2014
Author  Title  

Michael Dunphy  
Peter Stechlinski  
Colin Phipps  
Andree Susanto  
Derek Steinmoeller  
Venkata Manem  
Alex Shum 
PhD Theses 2013
Author  Title  

József Vass  
Andrijana Burazin  
Nancy Soontiens  
Amenda Chow  
Rasha Al Jamal  
Wentao Liu   
Minghua Lin  
Killian Miller 
PhD Theses 2012
Author  Title  

Rahul Rahul  
Ruibin Qin  
Dominique Brunet  
Yasunori Aoki  
Easwar Magesan  
Christopher Ferrie  
Dhanaraja Kasinathan  
Wai Man NG  
Matthew Johnston 
PhD Theses 2011
Author  Title  

Raluca Jessop  
Yufang Hao  
Mohamad Alwan  
Yanwei Wang  
Christopher Subich  
Timothy Rees  
Volodymyr Gerasik 

PhD Theses 2010
Author  Title  

Jun Liu 
 
Kathleen Wilkie 
 
Sean Speziale 
 
Nataliya Portman 

PhD Theses 2009
Author  Title  

Rudy Gunawan  
Gibin George Powathil  
Matthew Calder 
PhD Theses 2008
Author  Title  

Gregory Mayer  
Cedric Beny  
Lijun Wang  
Kahrizsangi Ebrahimi  
Robert Martin 
PhD Theses 2007
Author  Title  

Shannon Kennedy  
Alexander Korobov  
Qing Wang  
Duncan Mowbray  
Donald Campbell 
Graduate School
Applied and Computational Math
General information, program offerings:, department for program:, director of graduate studies:, graduate program administrator:.
The Program in Applied and Computational Mathematics offers a select group of highly qualified students the opportunity to obtain a thorough knowledge of branches of mathematics indispensable to science and engineering applications, including numerical analysis and other computational methods.
Program Offerings
Program offering: ph.d..
Students enroll in courses based on research topics that they choose in consultation with faculty. Typically, students take regular or reading courses with their advisers in the three topic areas of their choice, completing the regular exams and course work for these courses.
Students must choose three areas out of a list of the following six topic areas in which to take courses and be examined:
 Asymptotics, analysis, numerical analysis, and signal processing;
 Discrete mathematics, combinatorics, algorithms, computational geometry, and graphics;
 Mechanics and field theories (including computational physics/chemistry/biology);
 Optimization (including linear and nonlinear programming and control theory);
 Partial differential equations and ordinary differential equations (including dynamical systems);
 Stochastic modeling, probability, statistics, and information theory
Additional topics may be considered with prior approval by the director of graduate studies.
The student should choose specific topics by the end of October. In consultation with the student, the director of graduate studies appoints a set of advisers from among the faculty and associated faculty. The adviser in each topic meets regularly with the student, monitors progress, and assigns additional reading material. Advisers are usually program faculty or associated faculty, but faculty members from other departments may serve as advisers with approval.
Additional pregenerals requirements
At the end of the first year, students will take a preliminary exam, consisting of a joint interview by their three firstyear topic advisers. Each student should decide with their firstyear advisers which courses are relevant for the examination areas.
Students should assess their level of preparation for the preliminary examination by reviewing homework and examinations from the previous year’s work. Students who fail the preliminary examination may take the examination a second time with the support of the firstyear advisers.
General exam
Before being admitted to the third year of study, students must pass the general examination. The general examination, or generals, is designed as a sequence of interviews with assigned professors covering three applied mathematics areas. The generals culminate in a seminar on a research topic, usually delivered toward the end of the fourth term. A student who completes all program requirements (coursework, preliminary exams, with no incompletes) but fails the general examination may take it a second time. Students who fail the general examination a second time will have their degree candidacy terminated.
Qualifying for the M.A.
The Master of Arts degree is normally an incidental degree on the way to full Ph.D. candidacy but may also be awarded to students who, for various reasons, leave the Ph.D. program. Students who have successfully completed all courses undertaken during their graduate study, have satisfactorily resolved all incompletes (if any), and have passed the preliminary exam may be awarded an M.A. degree. Upon learning the program’s determination of their candidacy to receive the M.A., students apply for the master's degree online through the advanced degree application system.
Dissertation and FPO
The doctoral dissertation must consist of either a mathematical contribution to some field of science or engineering or the development or analysis of mathematical or computational methods useful for, inspired by, or relevant to science or engineering.
The Ph.D. is awarded after the candidate’s doctoral dissertation has been accepted and the final public oral examination sustained.
 Amit Singer
Director of Undergraduate Program
 Paul Seymour
Executive Committee
 Noga M. Alon, Mathematics
 René A. Carmona, Oper Res and Financial Eng
 Emily Ann Carter, Mechanical & Aerospace Eng
 Maria Chudnovsky, Mathematics
 Peter Constantin, Mathematics
 Amit Singer, Mathematics
 Howard A. Stone, Mechanical & Aerospace Eng
 Romain Teyssier, Astrophysical Sciences
 Jeroen Tromp, Geosciences
 Ramon van Handel, Oper Res and Financial Eng
Associated Faculty
 Ryan P. Adams, Computer Science
 Amir Ali Ahmadi, Oper Res and Financial Eng
 Michael Aizenman, Physics
 Yacine AïtSahalia, Economics
 William Bialek, Physics
 Mark Braverman, Computer Science
 Carlos D. Brody, Princeton Neuroscience Inst
 Adam S. Burrows, Astrophysical Sciences
 Roberto Car, Chemistry
 Bernard Chazelle, Computer Science
 Jianqing Fan, Oper Res and Financial Eng
 Jason W. Fleischer, Electrical & Comp Engineering
 Mikko P. Haataja, Mechanical & Aerospace Eng
 Gregory W. Hammett, PPPL Theory
 Isaac M. Held, Atmospheric & Oceanic Sciences
 Sergiu Klainerman, Mathematics
 Naomi E. Leonard, Mechanical & Aerospace Eng
 Simon A. Levin, Ecology & Evolutionary Biology
 Luigi Martinelli, Mechanical & Aerospace Eng
 William A. Massey, Oper Res and Financial Eng
 Assaf Naor, Mathematics
 Jonathan W. Pillow, Princeton Neuroscience Inst
 H. Vincent Poor, Electrical & Comp Engineering
 Frans Pretorius, Physics
 Herschel A. Rabitz, Chemistry
 Peter J. Ramadge, Electrical & Comp Engineering
 Jennifer Rexford, Computer Science
 Clarence W. Rowley, Mechanical & Aerospace Eng
 Szymon M. Rusinkiewicz, Computer Science
 Frederik J. Simons, Geosciences
 Jaswinder P. Singh, Computer Science
 Ronnie Sircar, Oper Res and Financial Eng
 Mete Soner, Oper Res and Financial Eng
 John D. Storey, Integrative Genomics
 Sankaran Sundaresan, Chemical and Biological Eng
 Ludovic Tangpi, Oper Res and Financial Eng
 Robert E. Tarjan, Computer Science
 Corina E. Tarnita, Ecology & Evolutionary Biology
 Salvatore Torquato, Chemistry
 Olga G. Troyanskaya, Computer Science
 Matt Weinberg, Computer Science
 Noga M. Alon
 Maria Chudnovsky
 Peter Constantin
 Romain Teyssier
 Jeroen Tromp
Associate Professor
 Ramon van Handel
 Henry F. Schreiner
Visiting Professor
For a full list of faculty members and fellows please visit the department or program website.
Permanent Courses
Courses listed below are graduatelevel courses that have been approved by the program’s faculty as well as the Curriculum Subcommittee of the Faculty Committee on the Graduate School as permanent course offerings. Permanent courses may be offered by the department or program on an ongoing basis, depending on curricular needs, scheduling requirements, and student interest. Not listed below are undergraduate courses and onetimeonly graduate courses, which may be found for a specific term through the Registrar’s website. Also not listed are graduatelevel independent reading and research courses, which may be approved by the Graduate School for individual students.
AOS 576  Current Topics in Dynamic Meteorology (also APC 576)
Apc 503  analytical techniques in differential equations (also ast 557), apc 523  numerical algorithms for scientific computing (also ast 523/cse 523/mae 507), apc 524  software engineering for scientific computing (also ast 506/cse 524/mae 506), apc 599  summer extramural research project, ast 559  turbulence and nonlinear processes in fluids and plasmas (also apc 539), cbe 502  mathematical methods of engineering analysis ii (also apc 502), cbe 554  topics in computational nonlinear dynamics (also apc 544), mae 501  mathematical methods of engineering analysis i (also apc 501/cbe 509), mae 502  mathematical methods of engineering analysis ii (also apc 506), mae 541  applied dynamical systems (also apc 571), mat 522  introduction to pde (also apc 522), mat 572  topics in combinatorial optimization (also apc 572), mat 585  mathematical analysis of massive data sets (also apc 520), mat 586  computational methods in cryoelectron microscopy (also apc 511/mol 511/qcb 513), mse 515  random heterogeneous materials (also apc 515/chm 559), orf 550  topics in probability (also apc 550).
Numerical Analysis Research Topics Ideas [MS PhD]
List of Research Topics and Ideas of Numerical Analysis for MS and Ph.D. Thesis.
 A comprehensive numerical analysis of heat and mass transfer phenomenons during cavitation sonoprocess
 Numerical analysis of mechanical behaviour of lattice and porous structures
 ATHENA: Advanced Techniques for High dimensional parameter spaces to Enhance Numerical Analysis
 Experimental and numerical analysis on coupled hygrothermochemomechanical effect in earlyage concrete
 Numerical analysis of natural rubber bearing equipped with steel and shape memory alloys dampers
 A numerical analysis of the effects of nanofluid and porous media utilization on the performance of parabolic trough solar collectors
 Numerical analysis of crossflow plate type indirect evaporative cooler: Modeling and parametric analysis
 Numerical analysis of the effect of fire source configuration on firewind enhancement
 Numerical analysis
 An influence mechanism of shale barrier on heavy oil recovery using SAGD based on theoretical and numerical analysis
 Numerical analysis of the stability of arched sill mats made of cemented backfill
 Theoretical and numerical analysis of regular oneside oscillations in a single pendulum system driven by a magnetic field
 Numerical analysis of performance and emission behavior of CI engine fueled with microalgae biodiesel blend
 Numerical analysis of ultrasonic multiple scattering for fine dust number density estimation
 Numerical analysis of fractional Volterra integral equations via Bernstein approximation method
 Numerical analysis of a highly sensitive surface plasmon resonance sensor for sarscov2 detection
 Numerical analysis of enhanced conductive deep borehole heat exchangers
 Numerical analysis of the robustness of clinching process considering the preforming of the parts
 Experimental study and numerical analysis on seismic performance of FRP confined highstrength rectangular concretefilled steel tube columns
 Numerical analysis of twinprecipitate interactions in magnesium alloys
 A twostep procedure for the numerical analysis of curved masonry structures
 Numerical analysis of subsurface deformation induced by groundwater level changes in the Bangkok aquifer system
 Numerical analysis and applications of explicit high order maximum principle preserving integrating factor RungeKutta schemes for AllenCahn equation
 Numerical analysis of the cyclic mechanical damage of Liion battery electrode and experimental validation
 Experimental and numerical analysis of circular concretefilled double steel tubular stub columns with inner square hollow section
 Numerical analysis of sand erosion for a pelton turbine injector at high concentration
 Deviations in yield and ultimate tensile strength estimation with the Small Punch Test: Numerical analysis of prestraining and Bauschinger effect influence
 Numerical analysis applied to the study of soil stress and compaction due to mechanised sugarcane harvest
 Numerical analysis of combined electroosmoticpressure driven flow of a viscoelastic fluid over high zeta potential modulated surfaces
 Numerical analysis of the flowinduced vibrations in the laminar wake behind a blunt body with rear flexible cavities
 Regular perturbation solution of Couette flow (nonNewtonian) between two parallel porous plates: a numerical analysis with irreversibility
 Numerical Analysis of the Coupling between Heat Transfer and Deformation in Rotary Air Preheater
 Numerical analysis of tinyfocalspot generation by focusing linearly, circularly, and radially polarized beams through a micro/nanoparticle
 An investigation on viscoelastic characteristics of 3Dprinted FDM components using RVE numerical analysis
 Experimental and static numerical analysis on bumper beam to be proposed for Indian passenger car
 Numerical analysis of special concentric braced frames using experimentallyvalidated fatigue and fracture model under short and long duration earthquakes
 Theoretical and numerical analysis of the evaporation of monoand multicomponent single fuel droplets
 Investigation of numerical analysis and seismic performance of underground loess cave with traditional dwellings
 Unsteadystate numerical analysis of advanced Savonius wind turbine
 Numerical analysis of dynamic compaction using FEMSPH coupling method
 Numerical Analysis of Macrofiber Composite Structures
 Numerical analysis of a monotube mixed mode magnetorheological damper by using a new rheological approach in CFD
 Laboratory and numerical analysis of geogrid encased stone columns
 Numerical analysis of thermalhydraulic characteristics of steamair condensation in vertical sinusoidal corrugated tubes
 Numerical analysis of dynamic characteristics of inarm type hydropneumatic suspension unit
 Numerical Analysis of the Hydrogen Dispersion Behavior in Different Directions in a Naturally Ventilated Space
 Material nonlinearity in the numerical analysis of SLJ bonded with ductile adhesives: A meshless approach
 Numerical analysis of the shear behavior of FRPstrengthened continuous RC beams having web openings
 Numerical Analysis of Phosphorus Concentration Distribution in a Silicon Crystal during Directional Solidification Process
 An improved numerical analysis of the transient oil decongealing process in a heat exchanger under low temperature conditions
 Numerical Analysis and Prediction of FLD For Al Alloy6063
 Improvement of ammonia mixing in an industrial scale selective catalytic reduction DeNOx system of a coalfired power plant: A numerical analysis
 Experimental and Numerical Analysis of Water Hammer Phenomenon in Pipeline with Fiber Optic Cable
 Numerical analysis of the effect of air pressure and oil flow rate on droplet size and tool temperature in MQL machining
 Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm
 Numerical linear algebra and optimization
 Numerical Analysis of Serrated Chip Formation Mechanism with JohnsonCook Parameters in MicroCutting of Ti6Al4V
 Numerical Analysis of the Mechanical Response of Anchored Wire Meshes
 Numerical Analysis for the Assessment of Factors Influencing the Breakdown of Swirl Flow in a Cylinder Driven by a Rotating End Wall
 Numerical analysis of the ultimate compressibility of concrete with indirect reinforcement for plotting a stressstrain diagram
 Numerical analysis and prediction of Aerooptical effects
 Numerical analysis for stochastic timespace fractional diffusion equation driven by fractional Gaussion noise
 Numerical analysis and performance improvement of nanostructured Cu2O/TiO2pn heterojunction solar cells using SCAPS
 Numerical Analysis of Mechanical Characteristics of Joint Structure of Steel Pipe Sheet Pile Foundation
 Numerical Analysis of Rotor Dynamics of Dredge Pump Shafting
 A numerical analysis of convection heat transfer and friction factor for oscillating corrugated channel flows
 Numerical Analysis on Aerodynamic Characteristics of Slender Body with Asymmetric Double Protuberance
 Mixed Finite ElementSecond Order Upwind Fractional Step Difference Scheme of Darcy–Forchheimer Miscible Displacement and Its Numerical Analysis
 Numerical Analysis of JAXA Standard Model in High Lift Configuration
 Numerical Analysis of Shock Speed Attenuation in Expansion Tube
 Experimental and Numerical Analysis of the Mechanical Properties of a Pretreated Shape Memory Alloy Wire in a SelfCentering Steel Brace
 Numerical Analysis of Shallow Foundations Considering Hydraulic Hysteresis and Deformation Dependent SoilWater Retention
 Experimental and Numerical Analysis of Laterally Loaded Pile Subjected to Earthquake Loading
 Numerical analysis of static behavior of caissontype quay wall deepened by grouting rubblemound
 Experimental and Numerical Analysis of Steel BeamColumn Connections
 Numerical Analysis of Shear and Particle Crushing Characteristics in Ring Shear System Using the PFC2D. Materials 2021, 14, 229
 Numerical Analysis on a Selection of Horn Material for the Design of Cylindrical Horn in Ultrasonic Machining
 A Simplified Francis Turbine for Micro Hydro Application: Design and Numerical Analysis
 Numerical Analysis of Nonlinear Dynamics Based on SpinVCSELs with Optical Feedback. Photonics 2021, 8, 10
 Numerical analysis of the flow behavior in the throat section of an experimental conical nozzle
 Numerical Analysis OF PCM Within a Square Enclosure Having Different Wall Heating Conditions
 Numerical Analysis of Dynamic Hysteresis in Tape Springs for Space Applications
 Numerical Analysis of Autofrettaged HighPressure Aluminium Cylinder
 Numerical analysis of steel columns subject to eccentric loadings
 Numerical Analysis of Flow and Radiative Transfer with Excitation Nonequilibrium in an Arcjet Facility
 Numerical Analysis of Scramjet Intake with Spoonshaped Isolator
 Numerical Analysis on Gas Turbine Blade of a NickelBased Alloy with Titanium Alloy
 Experimental and Numerical Analysis on the Mechanical Behaviour of Box Beam ColdFormed Steel Builtup CSections
 Numerical Analysis of Segmental Tunnel Linings employing a Hybrid Modeling Approach
 Numerical Analysis of Longspan Floor Vibration Due to Crowd Synchronized Walking
 Numerical analysis of the carboncontaining pellet direct reduction process with central heat transfer enhancement
 Numerical analysis of coupled systems of ODEs and applications to enzymatic competitive inhibition by product
 Mathematical model of SIR epidemic system (COVID19) with fractional derivative: stability and numerical analysis
 Numerical analysis of Vertical Axis Wind Turbine performance at varied AngleofAttack
 Numerical analysis of fretting wear in lateral contact of sphere/sphere
 Hybrid Algorithms for Numerical Solution of Optoelectronics Applications
 Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games I: The Ergodic Case
 Numerical–Experimental Analysis of Polyethylene Pipe Deformation at Different Load Values
 Bending behavior of octettruss lattice structures: Modelling options, numerical characterization and experimental validation
 Numerical Simulation of Photovoltaic Cell
 Numerical Solving for Nonlinear Problems Using Iterative Techniques
 An introduction to numerical methods and analysis
 On Solving Nonlinear Equation Via Numerical Analysis for Photovoltaic Cell (HMDM)
 Application of Numerical Analysis for Solving Nonlinear Equation (AHMDM)
 On the Globalization of ASPIN Employing TrustRegion Control Strategies–Convergence Analysis and Numerical Examples
 Numerical modeling of two microwave sensors for biomedical applications
 Numerical modeling of solidliquid phase change under the influence an external electric field
 The damagefailure criteria for numerical stability analysis of underground excavations: A review
 Numerical and chemical kinetic analysis to evaluate the effect of steam dilution and pressure on combustion of ndodecane in a swirling flow environment
 Numerical simulation and stability analysis for the fractionalorder dynamics of COVID19
 Scenarios in the experimental response of a vibroimpact singledegreeoffreedom system and numerical simulations
 Experimental and Numerical Investigation of OneDimensional Infiltration into Unsaturated Soil
 Numerical Modeling of Heat and Mass Transfer during Cryopreservation Using Interval Analysis
 Numerical techniques for infrared spectra analysis of organic and inorganic volatile compounds for biomedical applications
 TwoDimensional Numerical Model for Stability Analysis of Tunnel Face Based on Particle Flow Code
 Numerical simulation and CFDbased correlations for artificially roughened solar air heater
 Numerical study on the thermohydraulic performance analysis of fly ash nanofluid
 Numerical Modeling and Stability Analysis of Surrounding Rock of Yuanjue Cave
 The theoretical performance analysis and numerical simulation of the cylindrical vane pump
 Investigation of novel multilayer sandwich panels under quasistatic indentation loading using experimental and numerical analyses
 Numerical approximations for a fully fractional Allen–Cahn equation
 Quasistatic numerical modeling of miniature RF circuits based on lumped equivalent circuits
 A comparative Analysis of PV Cell Mathematical Model
 Experimental, analytical and numerical performance of RC beams with Vshaped reinforcement
 Shape and topology optimization. to appear in Handbook of Numerical Analysis, 22
 Geotechnical Properties of Stabilized Sedimentary Formation for Numerical Analysis , 14 Chapters of Research Work
 Numerical analysis of a distributed control problem with a stochastic parabolic equation
 Numerical Analysis of Reinforced Exterior Concrete BeamColumn Joints Retrofitted Using FRP under Cyclic Loads
 Experimental and Numerical Analysis of Energy Absorption of Hollow and Foam filled thickwall Aluminum Tubes Considering Different Damage Models
 The dynamical model for COVID19 with asymptotic analysis and numerical implementations
 Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods
 A novel approach for the numerical approximation to the solution of singularly perturbed differentialdifference equations with small shifts
 Hybrid bistable composite laminates for structural assemblies: A numerical and experimental study
 The relative salience of numerical and nonnumerical dimensions shifts over development: A reanalysis of
 Numerical analyses on geogridreinforced cushion in pilesupported composite foundation
 On the numerical solution of stochastic oscillators driven by timevarying and random forces
 A New Blast Absorbing Sandwich Panel with Unconnected Corrugated Layers—Numerical Study
 Combined numerical and experimental study on the use of Gurney flaps for the performance enhancement of NACA0021 airfoil in static and dynamic conditions
 Supporting Data: Numerical Simulation of Sleeve Fracturing for InSitu Stress Measurement using Cohesive Elements
 Numerical modeling of postflood water flow in pavement structures
 Fatigue Analysis of Actuators with Teflon Impregnated Coating—Challenges in Numerical Simulation
 Seismic assessment of wind turbines: How crucial is rotornacelleassembly numerical modeling?
 A Numerical Study on Interference Effects of Closely Spaced Strip Footings on Cohesionless Soils
 Numerical Investigation of a Phase Change Material Including Natural Convection Effects
 CSCI, MATH 6820 Numerical Solution of Ordinary Differential Equations Lecture Notes: Spring 1999
 A New Technique for Solar Cell Parameters Estimation of The SingleDiode Model
 Analysis and numerical simulation of fractional Biswas–Milovic model
 Thermal and Hydrodynamic Phenomena in the Stagnation Zone—Impact of the Inlet Turbulence Characteristics on the Numerical Analyses
 ElastoPlastic Numerical Analyses for Predicting CaveIns of Tunnels and Caverns
 … pass location on thermofluidic characteristic on the novel aircooled branched wavy minichannel heat sink: A comprehensive numerical and experimental analysis
 Numerical and experimental analysis of the structural performance of AM components built by fused filament fabrication
 Experimental and numerical simulation of the piston engine fueled with alternative fuel blends: CFD approach
 Comparison between experimental digital image processing and numerical methods for stress analysis in dental implants with different restorative materials
 Numerical investigation for heat transfer enhancement of nanofluid in the solar flat plate collector with insertion of multichannel twisted tape
 A numerical study on energy absorption capability of lateral corrugated composite tube under axial crushing
 Parameters Estimation of Photovoltaic Model Using Nonlinear Algorithms
 The Influence of the Mesh Element Size on Critical Bending Speeds of a Rotor in the Finite Element Analysis
 Numerical simulations of Pelton turbine flow to predict large head variation influence
 Use of Vertical and Inclined Walls to Mitigate the Interaction of Reverse Faulting and Shallow Foundations: Centrifuge Tests and Numerical Simulation
 Numerical simulation and experimental study on the drilling process of 7075t6 aerospace aluminum alloy
 A numerical study of field strength and clay morphology impact on NMR transverse relaxation in sandstones
 NUMERICAL SIMULATIONS FOR SHALLOW WATER FLOWS OVER ERODIBLE BEDS BY CENTRAL DG METHODS.
 Numerical modeling and parametric study of piled rafts foundations
 Direct numerical simulation of electroconvection with thin Debye layer matching canonical experiments
 A novel numerical approach to timefractional parabolic equations with nonsmooth solutions
 Experimental and numerical investigation of the thermomechanical behaviour of an energy sheet pile wall
 Experimental and numerical investigation on dynamic responses of the umbrella membrane structure excited by heavy rainfall
 A novel Covid19 model with fractional differential operators with singular and nonsingular kernels: Analysis and numerical scheme based on Newton …
 Numerical investigation of sloshing in tank with horivert baffles under resonant excitation using CFD code
 Parametric investigations of magnetic nanoparticles hyperthermia in ferrofluid using finite element analysis
 Experimental and numerical investigations on seismic performance of RC bridge piers considering buckling and lowcycle fatigue of highstrength steel bars
 Numerical assessment of effective width in steelconcrete composite box girder bridges
 Structurepreserving, energy stable numerical schemes for a liquid thin film coarsening model
 Stability analysis of thinwalled composite plate in unsymmetrical configuration subjected to axial load
 Numerical Simulation of Swelling in Tunnels
 Numerical study on the behaviour of AL2O3/water nanofluid at multiple flows and concentration
 Weighted finite element method for elasticity problem with a crack
 The impact of a multilevel protection column on the propagation of a water wave and pressure distribution during a dam break: Numerical simulation
 Analysis and numerical simulation of novel coronavirus (COVID19) model with MittagLeffler Kernel
 Numerical study of mist film cooling on a flat plate with various numbers of deposition
 Effect of Reinforcing Steel on the Impact Resistance of Reinforced Concrete Panel Subjected to HardProjectile Impact
 Comparative numerical and experimental analysis of thermal and hydraulic performance of improved plate fin heat sinks
 Study of the Effect of Welding Current on Heat Transfer and Melt Pool Geometry on Mild Steel Specimen Through Finite Element Analysis
 Sensitivity analysis of design parameters for erythritol melting in a horizontal shell and multifinned tube system: Numerical investigation
 Numerical investigation on a dual loop EGR optimization of a light duty diesel engine based on water condensation analysis
 Ducted Fuel Injection: Experimental and numerical investigation on fuel spray characteristics, air/fuel mixing and soot mitigation potential
 Numerical simulation and experimental investigation on laser beam welding of Inconel 625
 Numerical limit analysisbased modelling of masonry structures subjected to large displacements
 Numerical performance of blindbolted demountable square CFST Kjoints
 Analysis and development of novel datadriven drag models based on direct numerical simulations of fluidized beds
 Experimental and numerical fretting fatigue using a new test fixture
 Numerical treatment for dynamics of second law analysis and magnetic induction effects on ciliary induced peristaltic transport of hybrid nanomaterial
 Experimental and numerical investigation of LowVelocity impact on steel wire reinforced foam Core/Composite skin sandwich panels
 Laboratory and numerical experimentation for masonry in compression
 A radiatively induced neutrino mass model with hidden local U (1) and LFV processes l i? l j ?, µ? eZ’ and µe? ee
 Influence of lowering groundwater level on the behavior of pile in soft soil
 Crushing of SingleWalled Corrugated Board during Converting: Experimental and Numerical Study
 Numerical optimization of key design parameters of a thermoelectric microfluidic sensor for ultrasensitive detection of biochemical analytes
 Numerical modeling and analysis of the effect of pressure on the performance of an alkaline water electrolysis system
 Numerical Simulation of the Daikai Station Subway Structure Collapse due to Sudden Uplift during Earthquake
 Advanced Numerical Modelling of Geogrids and Steel Wire Meshes
 Explicit Numerical Model of Solar Cells to Determine Current and Voltage
 Numerical Dynamic Programming for Continuous States
 A NumericalAnalytical Method for Dynamic Analysis of Piles in Nonhomogeneous Transversely Isotropic Media
 Failure mode analysis of postseismic rockfall in shattered mountains exemplified by detailed investigation and numerical modelling
 Estimating water balance components in irrigated agriculture using a combined approach of soil moisture and energy balance monitoring, and numerical modelling
 Discovery of dynamics using linear multistep methods
 A numerical study of arsenic contamination at the Bagnoli bay seabed by a semianthropogenic source. Analysis of current regime
 Model uncertainty in nonlinear numerical analyses of slender reinforced concrete members
 Optimalorder error estimates of finite element approximations to variableorder timefractional diffusion equations without regularity assumptions of the true solutions
 Seismic Response of Embankment on Soft Clay Based on Shaking Table Test
 Numerical Evaluation of Bearing Capacity of StepTapered Piles Using PY Curves Analysis
 Mitigation of Multipactor Effect in High Power Microwave Systems for SPS
 StructurePreserving Numerical Methods for Nonlinear Fokker–Planck Equations with Nonlocal Interactions by an Energetic Variational Approach
 Numerical Investigation of ThreeDimensional and Vortical Flow Phenomena to Enhance the Power Performance of a Wind Turbine Blade
 Structural Analysis of a StressLaminatedTimber Bridge Deck Using Hardwood
 Experimental and Numerical Study on the Pullout Resistance of a Single and Group of Granular Pile Anchor (GPA) in Soft Soils
 Nonlinear Vibrations of a Composite Beam in Large Displacements: Analytical, Numerical, and Experimental Approaches
 Numerical Investigation of Flow Inside the Collector of a Solar Chimney Power Plant
 An integrated approach for the numerical modeling of severely damaged historic structures: Application to a masonry bridge
 Experimental and Numerical Studies on the Structural Performance of a Double Composite Wall
 The new operational matrix of integration for the numerical solution of integrodifferential equations via Hermite wavelet
 Numerical Modelling of Heat Transfer
 A Narrowstencil Finite Difference Method for Approximating Viscosity Solutions of Hamilton–Jacobi–Bellman Equations
 Experimental and numerical investigation on flow characteristics of interwrapper channel in LMFBR
 Numerical Investigation on Seismic Performance of a PiledRaft Foundation with GridForm DMWs
 Global sensitivity analysis on numerical solver parameters of ParticleInCell models in particle accelerator systems
 Unified analysis of discontinuous Galerkin and C0interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations
 A velocityvorticity method for highly viscous 3D flows with application to digital rock physics
 Numerical modeling of strain localization
 Failure process analysis of frustum windows for deepsea manned cabin
 Numerical Modeling of the Seismically Induced Uplift Behavior of Twin Tunnels
 A new numerical method to solve pantograph delay differential equations with convergence analysis
 Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems
 Numerical Assessment of Laterally Loaded Pile Group Efficiency
 Numerical evaluation of highintensity focused ultrasoundinduced thermal lesions in atherosclerotic plaques [J]
 Verification of numerical models for seismic fluidstructure interaction analysis of internal components in liquidfilled advanced reactors
 Effect of Domain Size in the Modelled Response of ThermallyActivated Piles
 Analysis of calendering process of nonisothermal flow of nonNewtonian fluid: A perturbative and numerical study
 Pogo analysis for a clustered rocket engine by sophisticated branchpipe modeling
 Numerical simulation of compressive failure of carbonfiberreinforced plastic laminates with various hole shapes
 IBIM based approach for geotechnical and numerical modelling of a conventional tunnel excavation
 Performance analysis of a gassolid thermochemical energy storage using numerical and experimental methods
 CFD analysis for airflow distribution of a conventional building plan for different wind directions
 A numerical model to simulate the dynamic performance of Breathing Walls
 Adaptive Control of a StringPlate Coupled System
 Remarks on numerical simulation of the LEAPAsia2019 centrifuge tests
 Numerical and Experimental Studies of Ballistic Compression Process in a Soft Recovery System
 Numerical study on the behaviour of vertical bolted joints for precast concrete wallbased lowrise buildings
 Experimental and numerical investigation of hybrid laser arc welding process and the influence of welding sequence on the manufacture of stiffened flat panels
 Numerical investigation of wideband Lshaped metasurface based solar absorber for visible and ultraviolet region
 Experimental Testing Conducted in the Course of the GIPIPE Project and Their Numerical Simulation
 Williams expansion utilized for assessment of crack behaviour under mixedmode loading in alkaliactivated finegrained composite
 Numerical modelling and optimization of vertical axis wind turbine pairs: A scale up approach
 Numerical simulation of strain localization based on Cosserat continuum theory and isogeometric analysis
 A Novel PredictorCorrector Hally Technique for Determining The Parameters for Nonlinear Solar Cell Equation
 Numerical Investigation on the Behavior of Reinforced Lightweight Concrete Beams at Elevated Temperature
 Optimisation on thermoforming of biodegradable poly (lactic acid)(PLA) by numerical modelling
 Norm Error Estimates for HDG Methods Applied to the Poisson Equation with an Application to the Dirichlet Boundary Control Problem
 A numerical study on the effects of nanoparticles and stair fins on performance improvement of phase change thermal energy storages
 The Numerical and Experimental Investigation of Particle Size Distribution Produced by an Electrical Discharge Process
 Numerical and Experimental Evaluation on the Behaviour of ColdFormed Steel Box Struts and Prediction of Experimental Results Using Artificial Neural Networks
 Runtime Abstract Interpretation for Numerical Accuracy and Robustness
 Performance Analysis of a Proposed Design of Diverterless Supersonic Inlet at Various Flight Conditions
 Novel Approach for Optical Characterization of Thrust Collar Lubricated Area: Experimental and Numerical Results
 Numerical Study on Solidification of Phase Change Materials Embedded with Metal Foam
 Experimental and numerical investigation of lashing bridge and container stack dynamics using a scaled model test
 Contour integrals of analytic functions given on a grid in the complex plane
 Dispersion and light loss reduction in photonic crystal fibers using nanoparticles optimization
 Extraction and numerical simulation of gas–water flow in low permeability coal reservoirs based on a pore network model
 Numerical modeling of the tension stiffening in reinforced concrete members via discontinuum models
 Numerical Simulations of Centrifugal Experiments on Seismic Behavior of Shallow Overburden Tunnels with Preground Improvement
 A new numerical method to solve the CaputoFabrizio fractional differential equation: error analysis and stability
 Regularization of algorithms for estimation of errors of differential equations approximate solutions
 Study of the Earth Pressure of the Finite Soil Layer
 Hailstoneinduced dynamic responses of pretensioned umbrella membrane structure
 Bearing Characteristics of Surrounding Rock of Deep Mining Roadway with Full and End Bolt Anchorages: A Comparative Numerical and Experimental Study
 A Numerical Model of a Perforated Microcantilever Covered with Cardiomyocytes to Improve the Performance of the Microcantilever Sensor
 Numerical investigation and experiment on pressure pulsation characteristics in a full tubular pump
 Analysis of critical imposed load of plate using variational calculus
 Numerical study on the nonlinear behavior of fullscale timber framed joints
 Numerical scrutinization of DarcyForchheimer relation in convective magnetohydrodynamic nanofluid flow bounded by nonlinear stretching surface in the perspective …
 Experimental and Numerical Study of Air Vessel in Quasi2D Transient Flow Analysis
 Storm Tide and Wave Simulations and Assessment
 On analytical and numerical solutions of inverse problems of the mechanics of composites
 Adjointbased sensitivity analysis for a numerical storm surge model
 Evaluation and Determination of the Parameters of a Photovoltaic Cell by an Iterative Method
 Towards stratigraphicthermomechanical numerical modelling: Integrated analysis of asymmetric extensional basins
 Numerical Dynamic Analysis of Gearbox Behaviour
 Numerical investigation of splitter plate effect on bluff body using finite volume method
 The performance of TBM disc cutter in soft strata: A numerical simulation using the threedimensional RBDDEM coupled method
 Simplified calculation method for supplemental viscous dampers of cablestayed bridges under nearfault ground motions
 Numerical approach to chaotic pattern formation in diffusive predator–prey system with Caputo fractional operator
 Numerical and experimental investigation of Yshaped micromixers with mixing units based on cantor fractal structure for biodiesel applications
 Influence of SoilStructure Interaction on Seismic Response of MultiStorey Buildings
 A twophase numerical model of well test analysis to characterize formation damage in nearwell regions of injection wells
 Using prefailure and postfailure remote sensing data to constrain the threedimensional numerical model of a large rock slope failure
 On the modified circular restricted threebody problem with variable mass
 Experimental and Numerical Study on the Mechanical Behavior of Composite Steel Structure under Explosion Load
 Dynamics of detonation transmission and propagation in a curved chamber: a numerical and experimental analysis
 Determination of PV Model Parameters Using Bisection and Secant Methods
 Thermal Buffering Performance of a Propyl Palmitate/Expanded PerliteBased FormStable Composite: Experiment and Numerical Modeling in a Building Model
 A numerical solution of the wave–body interactions for a freely floating vertical cylinder in different water depths using OpenFOAM
 The Influence of the Window Width on FRA Assessment with Numerical Indices
 Numerical modeling of axially loaded circular concretefilled doubleskin steel tubular short columns incorporating a new concrete confinement model
 Structural Safety and Stability of the Bridge on the Paraopeba River in Moeda, Minas Gerais, Brazil: Case Study
 Modeling and analysis of the coupling in discrete fracture matrix models
 An inclusive analysis of inductive dielectric and/or metallic discontinuities in a rectangular waveguide
 Numerical study on blast responses of rubberized concrete slabs using the Karagozian and Case concrete model
 A decision tree lifted domain for analyzing program families with numerical features
 Numerical simulation of nonlinear thermal radiation on the 3D flow of a couple stress Casson nanofluid due to a stretching sheet
 Experimental and numerical investigation on compressive and flexural behavior of structural steel tubular beams strengthened with AFRP composites
 Horizontal Displacement of Urban Deep Excavated Walls Supported by Multistrands Anchors, Steel Piles, and In Situ Concrete Piles: Case Study
 A study on transmission dynamics of the emerging Candida Auris infections in Intensive Care Units: Optimal control analysis and numerical computations
 Numerical and Experimental Response of FreeStanding Art Objects Subjected to Ground Motion
 A combined convection carreau–yasuda nanofluid model over a convective heated surface near a stagnation point: a numerical study
 Numerical study on magnetoacoustic thermal characteristics of microspeaker for mobile phones
 Analysis of the triggering mechanism of landslide in the village Podi, Montenegro
 Numerical prediction of temperature distribution and residual stresses on plasma arc welded thin titanium sheets
 Significance of Interface Modeling in the Analysis of Laterally Loaded Deep Foundations
 Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion
 Study of Dynamic Characteristics of Circular Liquid Storage Tanks Using Acoustic Principles
 Numerical investigation on flow distortion in a vertical inline pump
 A study on rock mass crack propagation and coalescence simulation based on improved numerical manifold method (NMM)
 Numerical investigation of the recent Chenhecun landslide (Gansu, China) using the discrete element method
 Closure to “Experimental Evaluation and Numerical Modeling of WideFlange Steel Columns Subjected to Constant and Variable Axial Load Coupled with Lateral Drift …
 A coupled numerical approach to simulate the effect of earthquake frequency content on seismic behavior of submarine tunnel
 Numerical simulation and experimental analysis on the performance of a modified shower cooling tower
 A coupled 3D wear and fatigue numerical procedure: Application to fretting problems in ultrahigh strength steel wires
 Finding the Sequence of Largest Small nPolygons by Numerical Optimization
 Application of stressstrength interference theory in butt welds
 Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates
 Bearing Capacity of Soft Clays Improved by Stone Columns: A Parametric Analysis
 On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators
 Numerical investigation of the fatigue performance of elastic rail clips considering rail corrugation and dynamic axle load
 Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels
 Numerical modeling to determine the limits on photovoltaic capacity when operating in a microgrid with solidoxide fuel cell triple combinedcycle plants
 Spacetime finite element discretization of parabolic optimal control problems with energy regularization
 Linear stability analysis of heterogeneous traffic flow considering degradations of connected automated vehicles and reaction time
 Stability of Galerkin discretizations of a mixed space–time variational formulation of parabolic evolution equations
 Convergence of a finitevolume scheme for a degeneratesingular crossdiffusion system for biofilms
 Numerical reconstruction of paleolithic fires in the ChauvetPont d’Arc Cave (Ardèche, France)
 Numerical and Experimental Analysis of Hardening Distortions of Drawpieces Produced in Hot Stamping Process. Metals 2021, 11, 457
 ThreeDimensional Modeling of Laterally Loaded Pile Embedded in Unsaturated Sandy Soil
 Influence of Broken Twisted Tape on Heat Transfer Performance in Novel Axial Corrugated Tubes: Experimental and Numerical Study
 Numerical study of oblique droplet impact on a liquid film
 Experimental and numerical modelling of group of geosyntheticencased stone columns
 Lowvelocity drop weight impact behavior of Twaron® fabric investigated using experimental and numerical simulations
 Numerical simulation to determine the fracture aperture in a typical basin of China
 Experimental and numerical thermal analysis for direct microwave heating of silicon carbide
 A Thermal Deformation Test Technique Robust to Change in Room Temperature Using Multiple Periodic Heating
 Applicability assessment of a stentretriever thrombectomy finiteelement model
 Plastic Hinge Model for PerformanceBased Design of BeamColumn Joints
 Thermoelastic Investigation of CarbonFiberReinforced Composites Using a DropWeight Impact Test
 Numerical Modeling and Optimization of an Air Handling Unit
 The effect of a hotwire in the tandem GMAW process ascertained by developing a multiphysics simulation model
 Numerical Study on CO2 Injection in Indian Geothermal Reservoirs Using COMSOL Multiphysics 5.2a
 Thermal–electrical–structural performances of hot heat exchanger with different internal fins of thermoelectric generator for low power generation application
 Numerical investigation on subcooled pool film boiling of liquid hydrogen in different gravities
 Dimensional Analysis of Shelllike Buckling in IsoTruss® Structures using Numerical Methods
 Study of Tilt on Adjacent Strip Footings
 Experimental and numerical investigation in the series arrangement square cyclone separator
 Strong error analysis for stochastic gradient descent optimization algorithms
 Numerical investigation of timefractional equivalent width equations that describe hydromagnetic waves
 Reliability of Chaotic Vibrations of EulerBernoulli Beams with Clearance
 MAP123EPF: A mechanisticbased datadriven approach for numerical elastoplastic modeling at finite strain
 On the Construction and Analysis of Finite Volume Element Schemes with Optimal L 2 Convergence Rate
 Numerical Study of SingleHole and MultiHoles Orifice Flow Parameters
 Theoretical, Numerical, and Experimental Study on an Unsteady Venturi Flowmeter for Incompressible Flows
 Numerical Modelling and Thermohydraulic Analysis of Circular Pipe Having Internal Vortex Generators
 Variational analysis of the discontinuous Galerkin timestepping method for parabolic equations
 New Exact Operational Shifted Pell Matrices and Their Application in Astrophysics
 Laboratory and Numerical Studies on the Performance of Geocell Reinforced Base Layer Overlying Soft Subgrade
 EXPERIMENTAL AND NUMERICAL STUDY OF INSERTING AN INTERNAL HOLLOW CORE TO FINNED HELICAL COIL TUBESHELL HEAT EXCHANGER
 A smoothed iFEM approach for efficient shapesensing applications: Numerical and experimental validation on composite structures
 New analysis and application of fractional order Schrödinger equation using with Atangana–Batogna numerical scheme
 Experimental and Numerical Study of LowVelocity Impact and Tensile after Impact for CFRP Laminates SingleLap Joints Adhesively Bonded Structure
 Basalt Fibre Reinforcement of Bent Heterogeneous Glued Laminated Beams
 Midinfrared supercontinuum generation in a lowloss germaniumonsilicon waveguide
 Mapping sequence to feature vector using numerical representation of codons targeted to amino acids for alignmentfree sequence analysis
 UltraLean Gaseous Flames in Terrestrial Gravity Conditions. Fluids 2021, 6, 21
 Finite Element Analysis of Ballistic Impact on Monolithic and Multilayered Target Plate with and Without Air Gap
 Quantitative Acoustic Emissions Source Mechanisms Analysis of Soft and Competent Rocks through MicromechanicsSeismicity Coupled Modeling
 Numerical investigation of free convection through a horizontal openended axisymmetric cavity
 Spectral monic chebyshev approximation for higher order differential equations
 Dynamic Characteristics of Woven Flax/Epoxy Laminated Composite Plate. Polymers 2021, 13, 209
 Numerical Solution of Interval VolterraFredholmHammerstein Integral Equations via Interval Legendre Wavelets? Method?
 Numerical and Experimental Investigation of Rectangular LiquidContaining Structures under Seismic Excitation
 Natural Convection from Two Cylinders in an Enclosure with Sinusoidal Bottom Wall: A Numerical Study
 Numerical investigation on a multichannel micro combustor fueled with hydrogen for a microthermophotovoltaic system
 Mesoscopic models and numerical simulations of the temperature field and hydration degree in earlyage concrete
 Blown Wing Aerodynamic Coefficient Predictions Using Traditional Machine Learning and Data Science Approaches
 Study on pure annihilation type B to V gamma decays
 Resistor–capacitor modeling of the cell membrane: A multiphysics analysis
 Two Step and NewtonRaphson Algorithms in the Extraction for the Parameters of Solar Cell
 A two loop induced neutrino mass model with modular A4 symmetry
 Some Step Iterative Method for Finding Roots of a Nonlinear Equation
 Analysis on Movement of Conducting Particle by Varying the Particle Dimensions in Gas Insulated Busduct Using Numerical Methods
 A numerical method for magnetohygrothermal dynamic stability analysis of defective quadrilateral graphene sheets using higher order nonlocal strain gradient …
 Numerical Investigation of Heat Transfer in Aircraft Engine Blade Using k? and SST k ? Model
 Advanced complex analysis of the thermal softening of nitrided layers in tools during hot die forging
 Prediction of Restrained Expansion and Shrinkage Strains of Reinforced Concrete Specimens by Using Finite Element Analysis
 Analytical and numerical study of the influence of different support types in the nonlinear vibrations of beams
 Optimalrate finiteelement solution of Dirichlet problems in curved domains with straightedged tetrahedra
 Stability Analysis of Hill Slopes in The Nilgiris, Tamil Nadu
 An experimental and numerical investigation of novel solution for energy management enhancement in data centers using underfloor plenum porous obstructions
 GaussLegendre Features for Gaussian Process Regression
 Can the Hyperfine Mass Splitting Formula in Heavy Quarkonia be Applied to the B c System?
 Numerical solution of the multiterm timefractional diffusion equation based on reproducing kernel theory
 Numerical Investigation of Fabricated MWCNTs/Polystyrene Nanofibrous Membrane for DCMD. Polymers 2021, 13, 160
 [PS][PS] 1 Numerical Integration with a Singular Integrand
 Highly sensitive nonlinear photonic crystal fiber based sensor for chemical sensing applications
 Numerical investigation of a central fuel property hypothesis under boosted sparkignition conditions
 Numerical Simulation Analysis of Cutter Head of Large Diameter Metro Shield with Soft Upper and Hard Bottom
 Numerical modelling of a segmented annular seal with enhanced lift effects
 Surface tensionmediated trapping and propulsion of small objects at liquid interfaces by using linespot lasers
 Numerical Investigation of an Efficient Blade Design for a Flow Driven Horizontal Axis Marine Current Turbine
 Numerical and analytical investigations for solving the inverse tempered fractional diffusion equation via interpolating elementfree Galerkin (IEFG) method
 Numerical and experimental investigation on the forward osmosis (FO) process for the operational conditions and spacer configuration optimization in microalgae …
 Study on effects of wind velocity on thermal efficiency and heat dynamics of pavement solar collectors: An experimental and numerical study
 Experimental and numerical investigations of steel fiber reinforced concrete dappedend purlins
 Numerical investigation of MaxwellVlasov equations. Part I: basic physics and algoritms
 Equivalence of pth moment stability between stochastic differential delay equations and their numerical methods
 Performance Study of Cross Flow Hybrid Hydrokinetic Turbine
 A problembased learning proposal to teach numerical and analytical nonlinear root searching methods
 Experimental and numerical investigation on the complex behaviour of the localised seismic response in a multistorey planasymmetric structure
 Numerical investigation of nanoparticle deposition location and pattern on a sharpbent tube wall
 Numerical study of rockfall impact on bridge piers and its effect on the safe operation of highspeed trains
 Numerical investigation of distributedorder fractional optimal control problems via Bernstein wavelets
 Fast parallel solution of fully implicit RungeKutta and discontinuous Galerkin in time for numerical PDEs, Part II: nonlinearities and DAEs
 Optimization of the Clinical Setting Using Numerical Simulations of the Electromagnetic Field in an Obese Patient Model for Deep Regional Hyperthermia of an 8 MHz …
 Vibrations of multilayer composite viscoelastic curved pipe under internal pressure
 Numerical and experimental investigation of tool geometry effect on residual stresses in orthogonal machining of Inconel 718
 Chaotic or hyperchaotic oscillator? Numerical solution, circuit design, MATLAB HDLcoder implementation, VHDL code, security analysis, and FPGA realization
 Numerical simulations of a Cu–water nanofluidbased parabolictrough solar collector
 Numerical Investigation of Fabricated MWCNTs/Polystyrene Nanofibrous Membrane for DCMD
 The incentive effects of different government subsidy policies on green buildings
 Numerical study of the effects of CO2 gas in geothermal water on the fluidflow characteristics in production wells
 Numerical evaluation of builtin temperature distribution effects on stress development in concrete pavements
 Numerical simulations of perforated plate liners: Analysis of the viscothermal dissipation mechanisms
 Effects of variable transport properties on heat and mass transfer in MHD bioconvective nanofluid rheology with gyrotactic microorganisms: numerical approach
 Evaluation of modified CamClay constitutive model in FLAC and its development by FISH programming
 Comprehensive 3D numerical study on interaction between structure and dipslip faulting
 Analysis of different combustion chamber geometries using hydrogen/diesel fuel in a diesel engine
 Analysis of Influencing Factors of the Ice Shedding Vibration on Iced Transmission Lines
 A numerical approach for 2D Sutterby fluidflow bounded at a stagnation point with an inclined magnetic field and thermal radiation impacts
 Analysis of a twogrid method for semiconductor device problem
 The behavior of concreteencased steel composite columnbeam joints under cyclic loading
 Variability and plant communities’ diversity of acidophilous dwarfheath mountain tundra (the class LoiseleurioVaccinietea) in Romanian Carpathians
 A numerical study on rutting behaviour of direct coal liquefaction residue modified asphalt mixture
 Convergence analysis of a secondorder semiimplicit projection method for LandauLifshitz equation
 Design and Analysis of an AirPurifier Using Cyclone Separator for Industries
 Numerical investigation of a novel singlepass allglass receiver for parabolic trough collector
 Analysis and optimization of the performance parameters of non cooperative target location detection system
 Improving accuracy of the fifthorder WENO scheme by using the exponential approximation space
 How to Transform BMatrix Chains into Markov Chains and Vice Versa
 A series representation of the discrete fractional Laplace operator of arbitrary order
 Energy and exergy analysis of solar photovoltaic thermal system: experimental and numerical verification
 Optimal Control of Acoustic Radiation Power for a TripleWalled Structure
 Strong convergence analysis for Volterra integrodifferential equations with fractional Brownian motions
 Numerical Modeling of WaterVapor Migration and Phase Transformation in Unsaturated Freezing Soils
 Experimental and Numerical Study on Cooling System Waste Heat Recovery for Engine Air Preheating by Ternary Hybrid Nanofluid
 Synchronization characteristics of an array of coupled MEMS limit cycle oscillators
 Field measurements for identification of modal parameters for highrise buildings under construction or in use
 Comparative thermal hydraulic performance analysis on helical screw insert in tube with different number of strips in transition flow regime
 A neural network constitutive model for hyperelasticity based on molecular dynamics simulations
 Numerical Procedures for Simulation of Wave Propagation in Plates
 Optimization of pipeandspike discharge electrode shape for improving electrostatic precipitator collection efficiency
 Hypocoercivitycompatible finite element methods for the longtime computation of Kolmogorov’s equation
 Proposal of a stagebystage design method and its application on a multistage multiphase pump based on numerical simulations
 … of the 3rd Dimension within the Representative Volume Element (RVE) on Damage Initiation and Propagation during FullPhase Numerical Simulations of Single and …
 Numerical and Experimental Study on Combined Seals With the Consideration of Stretching Effects
 Thermal Upheaval Buckling of Buried Pipelines: Experimental Behavior and Numerical Modeling
 A numerical study on the effects of spatial and temporal discretization in cardiac electrophysiology
 On a multiscale a posteriori error estimator for the stokes and Brinkman equations
 Flow analyses of diffuser augmented wind turbines
 Parameters Extraction of a SingleDiode Model of Photovoltaic Cell Using False Position Iterative Method
 Vibration Behavior Analysis of Anchor Cables in Submerged Floating Tunnel and Its Application
 Numerical solutions of hypersingular integral equations for interface circular crack under axisymmetric loadings
 Evaluation of the soil–pile interface properties in the lateral direction for seismic analysis in sand
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Admission Open – batch#11
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Mathematics PhD theses
A selection of Mathematics PhD thesis titles is listed below, some of which are available online:
2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991
Reham Alahmadi  Asymptotic Study of Toeplitz Determinants with FisherHartwig Symbols and Their DoubleScaling Limits
Anne Sophie Rojahn – Localised adaptive Particle Filters for large scale operational NWP model
Melanie Kobras – Low order models of storm track variability
Ed Clark – Vectorial Variational Problems in L∞ and Applications to Data Assimilation
Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes
Chiara Cecilia Maiocchi – Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems
Samuel R Harrison – Stalactite Inspired Thin Film Flow
Elena Saggioro – Causal network approaches for the study of subseasonal to seasonal variability and predictability
Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions
Jennifer E. Israelsson – The spatial statistical distribution for multiple rainfall intensities over Ghana
Giulia Carigi – Ergodic properties and response theory for a stochastic twolayer model of geophysical fluid dynamics
André Macedo – Localglobal principles for norms
Tsz Yan Leung – Weather Predictability: Some Theoretical Considerations
Jehan Alswaihli – Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations
Jemima M Tabeart – On the treatment of correlated observation errors in data assimilation
Chris Davies – Computer Simulation Studies of Dynamics and SelfAssembly Behaviour of Charged Polymer Systems
Birzhan Ayanbayev – Some Problems in Vectorial Calculus of Variations in L∞
Penpark Sirimark – Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation
Adam Barker – Path Properties of Levy Processes
Hasen Mekki Öztürk – Spectra of Indefinite Linear Operator Pencils
Carlo Cafaro – Information gain that convectivescale models bring to probabilistic weather forecasts
Nicola Thorn – The boundedness and spectral properties of multiplicative Toeplitz operators
James Jackaman – Finite element methods as geometric structure preserving algorithms
Changqiong Wang  Applications of Monte Carlo Methods in Studying Polymer Dynamics
Jack Kirk  The molecular dynamics and rheology of polymer melts near the flat surface
Hussien Ali Hussien Abugirda  Linear and Nonlinear NonDivergence Elliptic Systems of Partial Differential Equations
Andrew Gibbs  Numerical methods for high frequency scattering by multiple obstacles (PDF2.63MB)
Mohammad Al Azah  Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF913KB)
Katarzyna (Kasia) Kozlowska  RiemannHilbert Problems and their applications in mathematical physics (PDF1.16MB)
Anna Watkins  A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF2.46MB)
Niall Arthurs  An Investigation of Conservative MovingMesh Methods for Conservation Laws (PDF1.1MB)
Samuel Groth  Numerical and asymptotic methods for scattering by penetrable obstacles (PDF6.29MB)
Katherine E. Howes  Accounting for Model Error in FourDimensional Variational Data Assimilation (PDF2.69MB)
Jian Zhu  Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF1.69MB)
Tommy Liu  Stochastic Resonance for a Model with Two Pathways (PDF11.4MB)
Matthew Paul Edgington  Mathematical modelling of bacterial chemotaxis signalling pathways (PDF9.04MB)
Anne Reinarz  Sparse spacetime boundary element methods for the heat equation (PDF1.39MB)
Adam ElSaid  Conditioning of the WeakConstraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF2.64MB)
Nicholas Bird  A MovingMesh Method for High Order Nonlinear Diffusion (PDF1.30MB)
Charlotta Jasmine Howarth  New generation finite element methods for forward seismic modelling (PDF5,52MB)
Aldo Rota  From the classical moment problem to the realizability problem on basic semialgebraic sets of generalized functions (PDF1.0MB)
Sarah Lianne Cole  Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF2.84MB)
Alexander J. F. Moodey  Instability and Regularization for Data Assimilation (PDF1.32MB)
Dale Partridge  Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF3.19MB)
Joanne A. Waller  Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF6.75MB)
Faez Ali ALMaamori  Theory and Examples of Generalised Prime Systems (PDF503KB)
Mark Parsons  Mathematical Modelling of Evolving Networks
Natalie L.H. Lowery  Classification methods for an illposed reconstruction with an application to fuel cell monitoring
David Gilbert  Analysis of largescale atmospheric flows
Peter Spence  Free and Moving Boundary Problems in Ion Beam Dynamics (PDF5MB)
Timothy S. Palmer  Modelling a single polymer entanglement (PDF5.02MB)
Mohamad Shukor Talib  Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF2.49MB)
Cassandra A.J. Moran  Wave scattering by harbours and offshore structures
Ashley Twigger  Boundary element methods for high frequency scattering
David A. Smith  Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF1.05MB)
Stephen A. Haben  Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF3.51MB)
Jing Cao  Molecular dynamics study of polymer melts (PDF3.98MB)
Bonhi Bhattacharya  Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF4.06MB)
Tamsin E. Lee  Modelling timedependent partial differential equations using a moving mesh approach based on conservation (PDF2.17MB)
Polly J. Smith  Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF3Mb)
Corinna Burkard  Threedimensional Scattering Problems with applications to Optical Security Devices (PDF1.85Mb)
Laura M. Stewart  Correlated observation errors in data assimilation (PDF4.07MB)
R.D. Giddings  Mesh Movement via Optimal Transportation (PDF29.1MbB)
G.M. Baxter  4DVar for high resolution, nested models with a range of scales (PDF1.06MB)
C. Spencer  A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.
P. Jelfs  A Cproperty satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF11.7MB)
L. Bennetts  Wave scattering by ice sheets of varying thickness
M. Preston  Boundary Integral Equations method for 3D water waves
J. Percival  Displacement Assimilation for Ocean Models (PDF  7.70MB)
D. Katz  The Application of PVbased Control Variable Transformations in Variational Data Assimilation (PDF 1.75MB)
S. Pimentel  Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF5.9MB)
J.M. Morrell  A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF7.7MB)
L. Watkinson  Four dimensional variational data assimilation for Hamiltonian problems
M. Hunt  Unique extension of atomic functionals of JB*Triples
D. Chilton  An alternative approach to the analysis of twopoint boundary value problems for linear evolutionary PDEs and applications
T.H.A. Frame  Methods of targeting observations for the improvement of weather forecast skill
C. Hughes  On the topographical scattering and neartrapping of water waves
B.V. Wells  A moving mesh finite element method for the numerical solution of partial differential equations and systems
D.A. Bailey  A ghost fluid, finite volume continuous rezone/remap Eulerian method for timedependent compressible Euler flows
M. Henderson  Extending the edgecolouring of graphs
K. Allen  The propagation of large scale sediment structures in closed channels
D. Cariolaro  The 1Factorization problem and same related conjectures
A.C.P. Steptoe  Extreme functionals and StoneWeierstrass theory of inner ideals in JB*Triples
D.E. Brown  Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling
S.J. Fletcher  High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction
C. Johnson  Information Content of Observations in Variational Data Assimilation
M.A. Wakefield  Bounds on Quantities of Physical Interest
M. Johnson  Some problems on graphs and designs
A.C. Lemos  Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts
R.K. Lashley  Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems
J.V. Morgan  Numerical Methods for Macroscopic Traffic Models
M.A. Wlasak  The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling
M. Martin  Data Assimilation in Ocean circulation models with systematic errors
K.W. Blake  Moving Mesh Methods for NonLinear Parabolic Partial Differential Equations
J. Hudson  Numerical Techniques for Morphodynamic Modelling
A.S. Lawless  Development of linear models for data assimilation in numerical weather prediction .
C.J.Smith  The semi lagrangian method in atmospheric modelling
T.C. Johnson  Implicit Numerical Schemes for Transcritical Shallow Water Flow
M.J. Hoyle  Some Approximations to Water Wave Motion over Topography.
P. Samuels  An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.
M.J. Martin  Data Assimulation in Ocean Circulation with Systematic Errors
P. Sims  Interface Tracking using Lagrangian Eulerian Methods.
P. Macabe  The Mathematical Analysis of a Class of Singular ReactionDiffusion Systems.
B. Sheppard  On Generalisations of the StoneWeisstrass Theorem to Jordan Structures.
S. Leary  Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.
I. Sciriha  On Some Aspects of Graph Spectra.
P.A. Burton  Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.
J.F. Goodwin  Developing a practical approach to water wave scattering problems.
N.R.T. Biggs  Integral equation embedding methods in wavediffraction methods.
L.P. Gibson  Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.
A.K. Griffith  Data assimilation for numerical weather prediction using control theory. .
J. Bryans  Denotational semantic models for realtime LOTOS.
I. MacDonald  Analysis and computation of steady open channel flow .
A. Morton  Higher order Godunov IMPES compositional modelling of oil reservoirs.
S.M. Allen  Extended edgecolourings of graphs.
M.E. Hubbard  Multidimensional upwinding and grid adaptation for conservation laws.
C.J. Chikunji  On the classification of finite rings.
S.J.G. Bell  Numerical techniques for smooth transformation and regularisation of timevarying linear descriptor systems.
D.J. Staziker  Water wave scattering by undulating bed topography .
K.J. Neylon  Nonsymmetric methods in the modelling of contaminant transport in porous media. .
D.M. Littleboy  Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .
M.P. Dainton  Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .
M.H. Mawson  The shallowwater semigeostrophic equations on the sphere. .
S.M. Stringer  The use of robust observers in the simulation of gas supply networks .
S.L. Wakelin  Variational principles and the finite element method for channel flows. .
E.M. Dicks  Higher order Godunov blackoil simulations for compressible flow in porous media .
C.P. Reeves  Moving finite elements and overturning solutions .
A.J. Malcolm  Data dependent triangular grid generation. .
Department of Mathematics and Computer Science
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 Numerical Analysis of Partial Differential Equations
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 Mathematics
Podlesny, Joscha: Multiscale modelling and simulation of deformation accumulation in fault networks
Kahnt, max: numerical approximation of nonisothermal multicomponent, multiphase field systems.
The subject of this thesis is the derivation and analysis of numerical approximations of multicomponent, multiphase field systems. Recent approximations of solutions to such models are mostly based on explicit time stepping schemes and require the computation of many time steps. Implicit methods exhibit inherent numerical challenges, in particular due to the nonsmoothness of the underlying energy functionals.
Our focus lies on the derivation of numerical approximations within the thermodynamically consistent context with high efficiency and robustness. We aim to exploit the special mathematical structure of the model and the underlying thermodynamics without introducing additional regularizations.
We introduce the thermodynamic and multiphase setting in chapter 2 and continue by motivating and presenting a thermodynamically consistent multicomponent, multiphase field model in chapter 3. Based on Rothe’s method, we obtain a semidiscretization allowing for adaptive meshes in chapter 4 and the implicit problems are analyzed. In chapter 5, a full discretization with adaptive finite elements based on hierarchical a posteriori error estimation is set up. We transition to a purely algebraic formulation and present the iterative approximation of solutions with a nonsmooth Schur–Newton multigrid approach in chapter 6. Finally, in chapter 7, we perform numerical experiments to underline the thermodynamical consistency and numerical efficiency of our method.
Kies, Tobias: Gradient methods for membranemediated particle interactions
Discretecontinuous hybrid models are a popular means for describing elastic membranemediated particle interactions in and on lipid bilayers. Here, the continuous part is usually given by an approximation of the lipid membrane by an infinitely thin and sufficiently smooth hypersurface, whose elastic energy is determined by a CanhamHelfrich type functional. The discrete component results from modeling nonmembrane particles as rigid discrete entities, which, depending on their configuration, induce local constraints on the membrane along the membraneparticle interfaces. In this context, the interaction potential describes the optimal elastic energy of such hybrid systems with a fixed particle configuration. Correspondingly, the energy minimization principle yields that stationary particle configurations are given by the local minima of the interaction potential. The main goal of this work is the proof of differentiability of the interaction potential for a selected class of models. This is accomplished using a variational approach that is already established in the literature in order to develop and apply robust numerical optimization methods for computing stationary particle configurations. Correspondingly, an additional focus is the derivation of a numerically accessible representation of the gradient, including its discretization and relevant numerical analysis. The proof of differentiability is brought forward by an application of the implicit function theorem. The basis for this is socalled boundary preserving domain transformations, which are induced by suitable families of vector fields and which locally admit the reformulation of the minimization problem that is implicitly defined by the interaction potential with respect to a fixed particle configuration. This subsequently enables the representation of the gradient as a volume integral using matrix analysis methods. The discretization of the partial differential equations for describing optimal membrane shapes is done via finite element methods. For particle methods with socalled curve restrictions a fictitious domain stabilized Nitsche method is developed, and for models with point value restrictions a conforming Galerking discretization is made possible by local QR transformations of the nodal finite element basis. For both cases suitable a priori error estimates are proven, and in addition also error estimates for the volume representation of the gradient are shown within that context. These developed methods open up the domain of efficient simulation of macro structures by isotropic and anisotropic particles, which is illustrated with the aid of various example applications and by means of perturbed gradient methods.
Djurdjevac, Ana: Random partial differential equations on evolving hypersurfaces
Partial differential equations with random coefficients (random PDEs) is a very developed and popular field. The variety of applications, especially in biology, motivate us to consider the random PDEs on curved moving domains. We introduce and analyse the advectiondiffusion equations with random coefficients on moving hypersurfaces. We consider both cases, uniform and lognormal distributions of coefficients. Furthermore, we will introduce and analyse a surface finite element discretisation of the equation. We show unique solvability of the resulting semidiscrete problem and prove optimal error bounds for the semidiscrete solution and Monte Carlo samplings of its expectation. Our theoretical findings are illustrated by numerical experiments. In the end we present an outlook for the case when the velocity of a hypersurface is an uniformly bounded random field and the domain is flat.
Youett, Evgenia: Adaptive multilevel Monte Carlo methods for random elliptic problems
In this thesis we introduce a novel framework for uncertainty quantification in problems with random coefficients. The developed framework utilizes the ideas of multilevel Monte Carlo (MLMC) methods and allows for exploiting the advantages of adaptive finite element techniques. In contrast to the standard MLMC method, where levels are characterized by a hierarchy of uniform meshes, we associate the MLMC levels with a chosen sequence of tolerances. Each deterministic problem corresponding to a MC sample on a given level is then approximated up to the corresponding accuracy. This can be done, for example, using pathwise a posteriori error estimation and adaptive mesh refinement techniques. We further introduce an adaptive MLMC finite element method for random linear elliptic problems based on a residualbased a posteriori error estimation technique. We provide a careful analysis of the novel method based on a generalization of existing results, for deterministic residualbased error estimation, to the random setting. We complement our theoretical results by numerical simulations illustrating the advantages of our approach compared to the standard MLMC finite element method when applied to problems with random singularities.
Youett, Jonathan William: Dynamic large deformation contact problems and applications in virtual medicine
Dynamic large deformation contact problems arise in many industrial applications like auto mobile engineering or biomechanics but only few methods exists for their numerical solution, all having their advantages and disadvantages. In this thesis the numerical solution of large deformation contact problems is tackled from an optimisation point of view and an application of this approach within a femoroacetabular impingement analysis is described. In this thesis we use a nonsmooth Hamilton principle and Fréchet subdifferential calculus to derive a weak formulation of the problem. The resulting subdifferential inclusion is discretised in time by constructing a contactstabilised midpoint rule. For the spatial discretisation the stateof theart dual mortar method is applied which results in nonconvex constrained minimisation problems that have to be solved solved during each time step. For the solution of these problems an inexact filter trustregion method is derived which allows to use inexact linearisations of the nonpenetration constraints. This method in combination with fast monotone multigrid method is then shown to be globally convergent.
Hardering, Hanne: Intrinsic Discretization Error Bounds for Geodesic Finite Elements
This work is concerned with the proof of optimal error bounds for the discretization of $H^1$elliptic minimization problem with solutions taking values in a Riemannian manifold. The discretization is done using Geodesic Finite Elements, a method of arbitrary order that is invariant under isometries. The discretization error is considered both intrinsically in a specially introduced Sobolevdistance as well as extrinsically. Optimal estimates of $H^1$ and $L^2$type are shown, that have been observed experimentally in previous works of other authors. Using the Rothe method consisting of an implicit Euler method for the time discretization and Geodesic Finite Elements for the spatial discretization, error estimates for $L^2$gradient flows of $H^1$elliptic energies are derived as well. The core of the work is formed by the discretization error estimates for minimization problems in instrinsic $H^1$ and $L^2$distances. To derive these, inverse estimates and interpolation errors for Geodesic Finite Elements and their discrete variations are shown. Using a nonlinear Cea's Lemma, this leads to the $H^1$error estimate for minimizers of $H^1$elliptic energies. A generalization of the AubinNitscheLemma shows optimal $L^2$error estimates for (essentially) semilinear energies, as long as the dimension of the domain of the minimizer is limited to $d<4$ for technical reasons. All results are illustrated using harmonic maps into a smooth Riemannian manifold satisfying certain curvature bounds as an example.
Pipping, Elias: Dynamic problems of rateandstate friction in viscoelasticity
In this work, the model of rateandstate friction, which can be viewed as central to the numerical simulation of earthquakes, is considered from a mathematical point of view. First, a framework is presented through which a general class of such friction laws can be understood and analysed. A prototypical viscoelastic problem of earthquake rupture is then formulated, both in strong and in variational form. Analysis of this problem is difficult, since the incorporation of rateandstate friction leads to a coupling of variables. In a timediscrete setting, nonetheless, results on existence, uniqueness, and continuous parameter dependence of solutions can be obtained. The principal idea is to reformulate the variable interdependence as a fixed point problem and to prove convergence for a corresponding iteration. With that in mind, next, a numerical algorithm is proposed that resolves the coupling through a fixed point iteration. Since it puts a stateoftheart solver and adaptive time stepping to use, it is not only stable but also fast. Its applicability to problems of interest is demonstrated in the penultimate chapter, which focuses on simulations of megathrust earthquakes that form at the base of a subduction zone. The main assumptions made throughout this work are summarised and discussed in the last chapter.
Sack, Uli: Numerical Simulation of Phase Separation in Binary and Multicomponent Systems
The ban of lead in electronics solder by EU directives results in the technological challenge to develop leadfree alternatives with comparable life span and processing properties. Numerical simulations of the microstructure evolution may contribute to identify promising candidates and thus focus the immense experimental effort. Aim of this work is on the one hand to develop a numerical framework for the efficient and robust simulation of the microstructure evolution in binary alloys combining adaptive finite element methods with fast solvers for the CahnHilliard model. On the other hand we will extend the existing fast solvers for the discrete scalar CahnHilliard equation to the vectorvalued case. After some preliminary remarks on phase diagrams, phase separation, and phasefield models in Chapter 1 we will firstly discuss anisotropic AllenCahn equations in Chapter 2. AlleCahnlike problems arise as subproblems in the Nonsmooth SchurNewton (NSNMG) method for Cahn Hilliard equations in Chapters 3 and 4. Here we prove existence and uniqueness of solutions to the anisotropic AllenCahn equation with logarithmic potential using the theory of maximal monotone operators. For the numerical solution we introduce an adaptive spatial mesh refinement cycle for evolution problems and several variants of implicit Euler time discretization. We prove stability for the latter and numerical experiments conclude the chapter. Chapter 3 combines existing and newly developed numerical tools to a simulation software for microstructure evolution in binary alloys. Key ingredients are the adaptive mesh refinement cycle of Chapter 2, the NSNMG solver, a quantification algorithm for measuring "coarseness" of microstructures and a quotient space multigrid method for indefinite problems. An application of this software to simulate the microstructure evolution in a eutectic AgCu alloy shows only marginal impact of elastic stresses on coarsening in the setting considered; while the use of a smooth interpolant of the logarithmic potential affects the coarsening dynamics considerably. In the final chapter we consider the multicomponent CahnHilliard equation and derive a unified formulation for the discrete problems which allows a direct application of the NSNMG method. Existence and uniqueness of discrete solution are proved and numerical examples illustrate the robustness of the scheme with respect to temperature, mesh size, and number of components.
Gräser, Carsten: Convex minimization and phase field models
Phase field models are a widely used approach to describe physical processes that are characterized by thin interfacial regions between large almost homogeneous domains. Important application areas of phase field models are transition processes of the state of matter and the separation of alloys. A fundamental property of these models is, that the transition and separation of phases is driven by a doublewell potential with distinct minima for the different phases. Already the pioneering work of Cahn and Hilliard used a temperature dependent logarithmic potential that is differentiable with singular derivatives. If the temperature tends to zero it degenerates to the nondifferentiable obstacle potential. The goal of this thesis is to develop methods for the efficient numerical solution of such equations that are also robust for nonsmooth potentials and anisotropic surface energies. These methods are derived for the CahnHilliard equation that are prototypic for a multitude of such models. The main result of the thesis is the development of a fast iterative solver for nonlinear saddle point problems like the ones that arise from finite element discretization of CahnHilliard equations. The solver relies on a reformulation of the problem as dual minimization problem whose energy functional is differentiable. The gradient of this functional turns out to be the nonlinear Schur complement of the saddle point problem. Generalized linearizations for the Schur complement are derived and used for a nonsmooth Newton method. Global convergence for this 'Schur Nonsmooth Newton' method and inexact versions is proved using the fact the equivalence to a descent method for the dual minimization problem. Each step of this method requires the solution of a nonlinear convex minimization problem. To tackle this problem the 'Truncated Nonsmooth Newton Multigrid' (TNNMG) method is developed. In contrast to other nonlinear multigrid methods the TNNMG method is significantly easier to implement and can also be applied to anisotropic problems while its convergence speed is in general comparable or sometimes even faster. Numerical examples show that the derived methods exhibit mesh independent convergence. Furthermore they turn out to be robust with respect to the temperature including the limiting case zero. The reason for this robustness is, that all methods do not rely on smoothness but on the inherent convex structure of the problems.
Forster, Ralf: On the stochastic Richards equation
A frequent problem during numerical computations consists in the uncertainty of certain model parameters due to measuring errors or their high variability. In the last years, one could observe an increasing interest in the quantification of these uncertainties and their effects to the solution of numerical simulations; a powerful tool which has been proven to be an efficient approach in this context is the socalled polynomial chaos method which is based on a spectral decomposition of the covariance function of the uncertain parameters and a representation of the solution in a polynomial basis. The aim of this thesis is the application of this method to the Richards equation modeling groundwater flow in saturated and unsaturated porous media. The main difficulty consists in the saturation and the hydraulic conductivity appearing in the time derivative and in the spatial derivatives, since both depend nonlinearly on the pressure. Considering uncertain parameters like random initial and boundary conditions and, in particular, a random permeability leads to a stochastic variational inequality of second kind with obstacle conditions and a nonlinear convex functional as superposition operator. Considering variational inequalities in the context of uncertain parameters and the polynomial chaos method is new, and we start by deriving a weak formulation of the problem and approximating the parameters by a KarhunenLoève expansion. The existence of a unique solution u in a tensor space can be proven for the timediscrete problem by reformulation as a convex minimization problem. We proceed by discretizing with finite elements and polynomial ansatz functions and by approximating the convex functional with Gaussian quadrature. The convergence of the solution of the discretized problem to the solution u is proved in a special case for a stochastic obstacle problem. Moreover, we perform numerical experiments to determine the discretization error. In the second part of this thesis, we develop an efficient numerical method to solve the discretized minimization problems. It is based on a global converging Block Gauß Seidel method and exploits a transformation which decouples the stochastic coefficients and connects the stochastic Galerkin with the stochastic collocation approach. This also allows us to establish a multigrid solver to accelerate the convergence. We conclude this thesis by demonstrating the power of our approach on a realistic example with lognormal permeability and exponential covariance.
Sander, Oliver: Multidimensional coupling in a human knee model
The thesis presents a new model for the numerical simulation of the mechanics of the human knee. In this model bones are described using linear elasticity. Ligaments instead are modelled as onedimensional Cosserat rods. The simulations give insight into the mechanical behavior of human joints. This can be helpful for a number of applications. For example, it is possible to estimate the longterm effect of certain surgical interventions. Also, the design of prosthetic devices can be improved. The main mathematical focus is on the correct formulation of the coupling conditions between one and three dimensional objects. Starting from the case of two threedimensional objects, for which coupling conditions can be derived rigorously, conditions for the multidimensional case are formulated. A solution algorithm for this coupled problem is presented, and the existence of solutions is shown under certain symmetry assumptions. For the subproblems, large contact problems and minimization problems on Riemannian manifolds have to be solved. For both problems, robust and efficient numerical methods are introduced. Numerical experiments show the applicability for realworld problems.
Berninger, Heiko: Domain decomposition methods for elliptic problems with jumping nonlinearities and application to the Richards equation
The thesis presents a new method for the solution of saturatedunsaturated groundwater flow problems in heterogeneous porous media. Concretely, highly nonlinear degenerate elliptic problems arising from a certain time discretization of the Richards equation are the basis of this work. The problems are considered as homogeneous in subdomains where a single soil prevails and, therefore, the parameter functions do not depend on space. These nonlinearities, however, may jump across the interfaces between the subdomains and, thus, account for the heterogeneous setting of different soil types in different subdomains. As a consequence, nonoverlapping domain decomposition problems in which subproblems are coupled via nonlinear transmission conditions are obtained. In this work these problems are solved without any linearization. By Kirchhoff transformation the homogeneous subproblems are transformed into convex minimization problems. Here, additional constraints like Signorinitype boundary conditions, which occur on seapage faces around lakes, can be taken into account. Finite elements are chosen for the space discretization, and convex analysis is applied as the solution theory. Finally, monotone multigrid methods provide efficient solvers which are robust with respect to degenerating soil parameters. In order to deal with the coupling of the homogeneous subproblems, nonlinear DirichletNeumann and Robin methods are used. Here, the thesis provides new convergence results for these iterations applied to nonlinear elliptic problems in 1D as well as well posedness results, which generalize existing linear theory. On the other hand, detailed numerical experiments demonstrate that the methods can also be applied successfully to problems in 2D. Finally, based on the artificial viscosity method, an upwind discretization with finite elements is developed in order to account for gravity. Hence, stability of the numerical solutions is obtained. In a closing numerical example the Richards equation is solved in 2D with four different soils and coupled to a surface water reservoir. The result demonstrates the applicability of the developed solution technique to a heterogeneous problem with realistic hydrological data.
Gebauer, Susanna: Hierarchical Domain Decomposition Methods for saturated Groundwaterflow in fractured porous Media
The focus of this thesis ist the numerical computation of flow in special geometries dominated by jumps in the flow coefficients and large differences in the scales of the main flow pathes and the surrounding materials. These characteristics result in difficulties in the numerical computation of the modelling equations. By means of groundwaterflow in fractured porous media we present a hierarchical domain decomposition method for the numerical computation of flow. Under certain assumptions this new method converges independently of the fracture width, the refinement depth and the jump in the flow coefficient. The theoretical results are confirmed by practical computations of a model problem and a fracture network. Thus for a new class of completely overlapping domain decomposition methods multigrid efficiency is shown for a class of problems, for which so far no comparably theoretically validated method existed.
Krause, Rolf H.: Monotone Multigrid Methods for Signorini's Problem with Friction
In this work, we consider the numerical simulation of contact problems. Since the numerical realization of contact problems is of high importance in many application areas, there is a strong demand for fast and reliable simulation method. We introduce and analyze a new nonlinear multigrid method for solving contact problems with and without friction. As it turns out, by means of our new method nonlinear contact problems can be solved with a computational amount comparable that of linear problems. In particular, in our numerical experiments we observe our method to be of optimal complexity. Moreover, since we do not use any regularization techniques, the computed discrete boundary stresses as well as the computed displacements turn out to be highly accurate. The new method is based on the succesive minimization of the associated energy functional in direction of properly choosen functions. We show the global convergence of our method and give several numerical examples in two and three space dimensions, illustrating the robustness and the performance of the method. In addition to the theoretical analysis, the method has been implemented in an object oriented way. We explain the concepts of our implementation and show the flexibility of our approach by deriving a nonlinear algebraic multigrid method. To include frictional effects, we use a discrete fixed point iteration. As a faster alternative, also a GaussSeidel like iteration scheme is proposed. Both methods are compared in numerical examples. The resulting nonlinear algorithm turns out to be fast and reliable. Finally, we consider the case of contact between elastic bodies. Here, the information transfer at the interface is realized by means of non conforming domain decomposition methods (mortar methods). This gives rise to a nonlinear Dirichlet Neumann Algorithm.
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Opportunities for PhD research are available in a wide range of topics in Mathematics. For more information, please see Numerical Analysis research page or the Numerical Analysis group website and find out more about specific projects.
Up to two funded PhD projects are available in the Department of Mathematics at the University of Manchester on Connections between Numerical Analysis of Differential Equations and Machine Learning. Read more. Supervisor: Dr C Powell. Year round applications PhD Research Project Funded PhD Project (UK Students Only)
Research includes mathematical analysis, partial differential equations, numerical analysis, applied probability, dynamical systems, multiscale modeling, high performance scientific computation, and numerical optimization with applications in optics and photonics, material science, machine learning, data science, imaging science, biology, and climate modeling, to name a few.
Numerical analysis is the study of algorithms for the problems of continuous mathematics. Find out about this expertise at The University of Manchester. ... Research seminars. Research seminars on topics associated with numerical analysis and scientific computing take place regularly in the following series: ... PhD research opportunities. We ...
Numerical Analysis at Department of Mathematics, Texas A&M University ... As well as doing research into theoretical numerical analysis, the group works closely with the Institute for Scientific Computation and the Institute for Applied Mathematics and Computational Science on the development of large scale scientific simulations. The graduate ...
Computational Science & Numerical Analysis. Computational science is a key area related to physical mathematics. The problems of interest in physical mathematics often require computations for their resolution. Conversely, the development of efficient computational algorithms often requires an understanding of the basic properties of the ...
PhD training in applied mathematics at Courant focuses on a broad and deep mathematical background, techniques of applied mathematics, computational methods, and specific application areas. Descriptions of several appliedmath graduate courses are given below. Numerical analysis is the foundation of applied mathematics, and all PhD students in ...
The Department is one of the lead partners in the MAGIC project and research students can attend any of the postgraduate courses offered by the MAGIC consortium. Opportunities for PhD research are available in a wide range of topics in Mathematics . For more information, please see Numerical Analysis research page or the Numerical Analysis ...
The Department is one of the lead partners in the MAGIC project and research students can attend any of the postgraduate courses offered by the MAGIC consortium. Opportunities for PhD research are available in a wide range of topics in Numerical Analysis. For some of the available areas of possible PhD research please visit the Numerical ...
Major research areas in this department include computational fluid dynamics (CFD), interface and front tracking methods, iterative methods in numerical linear algebra, and algorithms for parallel computers.Current research topics in CFD include: front tracking methods for fluid flow problems with free surfaces or immersed interfaces in the ...
The Numerical Analysis Group of the University of WisconsinMilwaukee Department of Mathematical Sciences offers many opportunities for students to enter the important and exciting field of research in numerical analysis and scientific computing. The cornerstone of the program is the PhD in Mathematics with specialty in Numerical Analysis ...
This Numerical Analysis programme is offered at The University of Manchester. Probability, financial mathematics and actuarial science. Statistics, inverse problems, uncertainty quantification and data science. Your GPA, or Grade Point Average, is a number that indicates how well or how high you scored in your courses on average.
Research topics include. Numerical methods for partial differential equations; ... numerical analysis, applied probability, machine learning, image processing and computer vision. ... More faculty research areas. School of Mathematics. 127 Vincent Hall 206 Church St. SE Minneapolis, MN 55455
For entry in the academic year beginning September 2023, the tuition fees are as follows: PhD (fulltime) UK students (per annum): Band A £4,712; Band B £6,500; Band C £9,500; Band D £13,500. International, including EU, students (per annum): Band A £26,500; Band B £28,500; Band C £33,500; Band D £40,500. PhD (parttime)
PhD in Computer Arithmetic and Numerical Analysis. University of Leeds Faculty of Engineering and Physical Sciences. Research Group: School of Computing. In 1985, a floatingpoint standard has been introduced, which, among multiple things, defined a set of required and recommended arithmetic operations and mathematical functions.
The first exam is based on the Numerical Analysis I (MATH602) and Mathematical Modeling I, II (MATH622, 722). The second exam is based on the student's concentration foundation courses and additional material deemed appropriate by the committee and consists of a short research project. Dissertation Research Advisor and Committee
Numerical Analysis & Scientific Computing. Scientific computing and numerical simulation are playing an ever more important role in science and technology. Hardly any new developments, e.g., in engineering or the geosciences, take place without careful mathematical modelling, analysis and optimisation. More and more complicated systems are ...
PhD Theses 2017. Author. Title. Cong Wu. Stability and Control of Caputo Fractional Order Systems. Monjur Morshed. Efficient Finitedifference Methods for Sensitivity Analysis of Stiff Stochastic Discrete Models of Biochemical Systems. Alexander James Maxwell Howse. Nonlinear Preconditioning Methods for Optimization and ParallelInTime Methods ...
ToomCook multiplication — generalization of Karatsuba multiplication. SchönhageStrassen algorithm — based on Fourier transform, asymptotically very fast. Fürer's algorithm — asymptotically slightly faster than SchönhageStrassen. Division algorithm — for computing quotient and/or remainder of two numbers.
Students enroll in courses based on research topics that they choose in consultation with faculty. Typically, students take regular or reading courses with their advisers in the three topic areas of their choice, completing the regular exams and course work for these courses. ... Asymptotics, analysis, numerical analysis, and signal processing ...
List of Research Topics and Ideas of Numerical Analysis for MS and Ph.D. Thesis. A comprehensive numerical analysis of heat and mass transfer phenomenons during cavitation sonoprocess. Numerical analysis of mechanical behaviour of lattice and porous structures.
L.P. Gibson  Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model. 1997. A.K. Griffith  Data assimilation for numerical weather prediction using control theory.. J. Bryans  Denotational semantic models for realtime LOTOS. 1996. I. MacDonald  Analysis and computation of steady open channel flow.
PhD Theses. Search terms. 15 Theses. Podlesny, Joscha: Multiscale modelling and simulation of deformation accumulation in fault networks. ... The subject of this thesis is the derivation and analysis of numerical approximations of multicomponent, multiphase field systems. Recent approximations of solutions to such models are mostly based on ...