Multigrid Methods for Computing Low-Rank Solutions to Parameter-Dependent Partial Differential Equations


Speaker:       Howard Elman, University of Maryland

Time:            21:00-22:00  Apr.27.2022

Host:            Yang Wang

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The collection of solutions of discrete parameter-dependent partial differential equations often takes the form of a low-rank matrix. Together with Tengfei Su, we show that in this scenario, iterative algorithms for computing these solutions can take advantage of low-rank structure to reduce both computational effort and memory requirements. Implementation of such solvers requires that explicit rank-compression computations be done to truncate the ranks of intermediate quantities that must be computed. We prove that when truncation strategies are used as part of a multigrid solver, the resulting algorithms retain textbook (grid-independent) convergence rates, and we demonstrate how the truncation criteria affect convergence behavior. In addition, we show that these techniques can be used to construct efficient solution algorithms for computing the eigenvalues of parameter-dependent operators. In this setting, parameterized eigenvectors can be grouped into matrices of low-rank structure, and we introduce a variant of inverse subspace iteration for computing them.  We demonstrate the utility of this approach on two benchmark problems, a stochastic diffusion problem with some poorly separated eigenvalues, and an operator derived from a discrete Stokes problem whose minimal eigenvalue is related to the inf-sup stability constant.



Howard Elman is a Professor in the Department of Computer Science and Institute of Advanced Computer Studies at the University of Maryland, College Park and affiliate Professor in the Department of Mathematics, where he is currently the Director of the Applied Mathematics Program.  He received his doctorate in Computer Science from Yale University in 1982.  He has had visiting positions at Stanford University, the University of Manchester Institute of Science and Technology and the University of Oxford. He is a SIAM Fellow, currently serves as Vice President for Publications at SIAM, and has served as Editor-in-Chief of SIAM Journal of Scientific Computing. His research concerns numerical solution of partial differential equations, sparse matrix methods and uncertainty quantification.