Numerical Analysis and Scientific Computing Seminar

Low-rank multigrid methods for stochastic partial differential equations

Speaker: Howard Elman, University of Maryland

Location: Warren Weaver Hall 1302

Date: April 3, 2020, 10 a.m.

Synopsis:

The collection of solutions of discrete parameter-dependent partial
differential equations often takes the form of a low-rank object.
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, eigenvalues and eigenvectors can be represented
as generalized polynomial chaos expansions having low-rank structure, and
we introduce a variant of inverse subspace 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.