# Computational Mathematics Group

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.

## Group Lead

Ulrike Meier Yang: Iterative linear solvers, algebraic multigrid, parallel computing, scientific software, performance analysis

## Research Staff

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

Veselin Dobrev: finite elements, discontinuous Galerkin methods, multigrid

Yohann Dudouit: finite elements, discontinuous Galerkin methods, local time stepping, parallel computing

Alyson Fox: numerical linear algebra, graph Laplacian linear systems algebraic multigrid solvers, error analysis of ZFP compression

Rob Falgout: multilevel methods, parallel computing

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: high-order finite elements, compressible shock hydrodynamics, scalable linear solvers, parallel time integration, computational electromagnetics

Tim La Fond: network analysis, dynamic graph algorithms, data mining, anomaly detection

Chak Shing Lee: numerical methods for partial differential equations, scalable linear solvers, multiscale methods

Ruipeng Li: sparse matrix computations, parallel computing, iterative methods for solving linear systems, preconditioning techniques, eigenvalue problems

Sarah Osborn: numerical linear algebra, numerical methods for partial differential equations, uncertainty quantification

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

Colin Ponce: numerical linear algebra, algebraic multigrid, numerical methods for powergrid analysis

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

Jacob Schroder: parallel computing, numerical linear algebra, numerical methods for partial differential equations, iterative solvers for large sparse linear systems, algebraic multigrid, parallel-in-time methods

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