# 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.