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

Abstract: The collection of solutions of discrete parameter-dependent partial differential equations often takes the form of a low-rank matrix. 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.

This is joint work with Tengfei Su.

computational engineering, finance, and sciencenumerical analysis

Audience: researchers in the topic

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