The Computational Mathematics Group conducts research and development of algorithms and software for solving linear and nonlinear systems, which are often obtained from approximations of partial differential equations and arise in numerous areas of science and engineering, including fluid dynamics, solid mechanics, combustion, elasticity, electromagnetics, large scale data mining, and cybersecurity. Our customers are primarily scientists and engineers working in these fields. The algorithms we investigate include scalable algorithms, such as multigrid and multilevel methods, adaptive mesh refinement, overlapping grid techniques, projection methods, higher-order upwind schemes, embedded boundary methods, interface tracking methods, and turbulence models. We also seek scalable spectral methods for extremely large graph matrices and rank-revealing decompositions of matrices in data mining applications. Our research includes the development of object-oriented code frameworks for the implementation of these algorithms on a wide range of serial and parallel architectures.

In summary, our goals are to develop both innovative grid based techniques for the computational modeling of physical problems and code infrastructures that facilitate the software implementation of such algorithms.

Ulrike Meier Yang: Iterative linear solvers, algebraic multigrid, parallel computing, mathematical algorithms for multicore architectures, scientific software, performance analysis

Andrew Barker: numerical methods for partial differential equations, domain decomposition, parallel computing

Jakub Cerveny: higher-order finite elements, adaptive mesh refinement algorithms, parallel computing

Kyle Chand: numerical methods for partial differential equations, mesh generation

Veselin Dobrev: finite elements, discontinuous Galerkin methods, multigrid

Rob Falgout: multilevel methods, parallel computing

Hormozd Gahvari: parallel performance modeling and analysis of numerical algorithms

Van Emden Henson: multigrid and algebraic multigrid, eigenvalues and eigenvectors, large-scale graphs, multilinear (tensor) algebra, Krylov methods

Christine Klymko: network analysis, numerical linear algebra, graph algorithms, data mining, scientific computing, numerical analysis, matrix analysis

Tzanio Kolev: scalable preconditioners, finite elements, electromagnetic problems

Anders Petersson: numerical methods for wave propagation and fluid mechanics, summation by parts discretizations, embedded boundary methods, parallel computing, large scale seismic wave simulations, high quality scientific software

Deepak Rajan: computational optimization and integer programming; optimization problems in scientific distributed computing.

Geoffrey Sanders: algebraic multigrid, eigenspectra, multilinear (tensor) algebra, large-scale graphs

Claudio Santiago: combinatorial optimization, integer programming, conic programming, convex relaxations, and nonlinear programming.

Jacob Schroder: numerical analysis, partial differential equations (PDEs), iterative solvers for large sparse linear systems

Bjorn Sjogreen: numerical methods for partial differential equations, high order finite difference and finite volume methods with applications to fluid mechanics , large scale parallel computing

Vladimir Tomov: development and analysis of finite element and finite volume methods for PDEs, radiation hydrodynamics

Panayot Vassilevski: numerical linear algebra, finite elements

Umberto Villa: numerical methods for partial differential equations, high performance computing, finite element analysis, fluid-dynamics, multigrid and upscaling techniques

Lu Wang: multigrid method for partial differential equations (PDEs), parallel solvers for coupled PDE systems, like fluid dynamics, reservoir simulation, and fluid-structure interaction