Applied Mathematics and Scientific Computing
Our research focuses on mathematical problems that arise in real-world applications. These require often the use of computing ressources for simulation, computation, data analysis, and visualization. Key problems are the numerical approximation of differential equations, and the solution of large systems of equations. Both the mathematical modeling of real-world applications and the design of modern numerical approaches require a solid theoretical background in functional analysis, differential equations, calculus of variations, graph-theory, etc.
Faculty
Yury Grabovsky's interests are in the area of Calculus of Variations, PDE and applications to Continuum Mechanics. He has worked on mathematical theory of composite materials, non-linear elasticity and phase transitions in solids. His work on exact relations for effective tensors of polycrystalline composites has combined methods of modern algebra and PDE. His recent work (joint with a former graduate student) includes a proof of Ball's conjecture in Calculus of Variations.
Marlliny Monsalve's research interests include numerical linear algebra, scientific computing, matrix theory, perturbation analysis, and their application to science and engineering. Currently she is working on the development of numerical methods for the solution of nonlinear matrix problems.
Igor Rivin works in the area of computational geometry and combinatorics.
Benjamin Seibold's primary research fields are computational partial differential equations and numerical analysis. He works on meshfree and particle methods for fluid flow simulations, level set methods, traffic modeling, radiative transfer, and molecular dynamics simulations.
Daniel B. Szyld has worked on many aspects of numerical linear algebra and matrix computations, including eigenvalue problems, sparse matrix techniques, Schwarz preconditioning and domain decomposition, and Krylov subspace methods. He is currently the Chair of the SIAM Activity Group on Linear Algebra, and editor of several leading journals on applied and numerical linear algebra. He is also an editor-in-chief of the Electronic Transactions on Numerical Analysis.
Louis Theran's research interests are in discrete and computational geometry. His main focus is problems in combinatorial rigidity. He also works in matroids, algorithms, and random graphs.
Fei Xue's research interests include numerical linear algebra, matrix computations and scientific computing in general. Specifically, he is working on design and analysis of efficient inexact eigenvalue algorithms for large sparse matrices. He is also interested in Krylov subspace methods for the solution of linear systems and preconditioning techniques for saddle point problems.
Wei-Shih Yang's research interests are in the fields of probability theory, mathematical physics and mathematical finance. He has worked on phase transitions and critical phenomena in statistical mechanics, Gaussian and non-Gaussian random fields with applications to quantum field theory, percolation, Ising model, and self-avoiding random walks. His more recent work includes the following topics: quantum random walks with applications to quantum computing, ruin probability of risk processes, and probabilistic models for terrorism.
Research Profile
The Applied Mathematics and Scientific Computing group is active in a variety of research areas, such as:
- meshfree particle methods for fluid flow simulations
- gradient-augmented level set approaches
- modeling of traffic flow
- moment closures in radiative transfer
- iterative solution of large linear systems
- numerical solution of eigenvalue problems
- solution of matrix equations
- modern Krylov subspace methods
- theory of composite materials
- non-linear elasticity and phase transitions
Seminars
Graduate Program and Courses
In the recent years, several graduate students have completed a Ph.D. or masters degree in the area of Applied Mathematics and Scientific Computing.
Information of about the graduate program in Mathematics can be found on the
Graduate Program website.
Students who are interested in specializing in Applied Mathematics and Scientific Computing can achieve a M.A. in Mathematics with Applied Concentration, as well as a Ph.D in Mathematics, with an advisor in the applied areas. In both cases, students are advised to take (many of) the courses listed below.
More detailed syllabi can be found on the
Course listing by the Graduate School of the College of Science and Technology.
The courses are not be taught every semester. Please check the website of the Department of Mathematics for the course schedule.
Central Courses
5043. Introduction to Numerical Analysis provides the basis in numerical analysis and fundamental numerical methods.
8013/8014. Numerical Linear Algebra I / II cover modern concepts and methods to solve linear systems and eigenvalue problems.
8023/8024. Numerical Differential Equations I / II present modern methods for the numerical solution of partial differential equations, their analysis, and their practical application.
9200/9210. Topics in Numerical Analysis I / II
focus on specialized topics relating to the faculties' research areas.
Theoretical Basis
In addition, we recommend courses that provide fundamental theoretical background.
8141/8142. Partial Differential Equations I / II provide a theoretical understanding of many of the equations considered in 8023/8024.
8200/8210. Topics in Applied Mathematics I / II focus on specialized topics relating to the faculties' research areas.
9005. Combinatorial Mathematics relates to many key problems in Scientific Computing, such as mesh generation, load balancing, and multigrid.
9041. Functional Analysis is a theoretical basis for many numerical approximation approaches, such as the finite element method.
9043. Calculus of Variations provides powerful tools for the theoretical study of dynamics, structural mechanics, and material properties.