Parametric PDEs: Sparse or Low-Rank Approximations?

TL;DR

This paper compares adaptive approximation strategies for parametric PDEs, including sparse polynomial, low-rank, and tensor methods, analyzing their efficiency and complexity for high-dimensional problems.

Contribution

It introduces a unified framework for adaptive algorithms and provides near-optimal complexity bounds for each approximation type, including new operator compression results.

Findings

Low-rank expansions can be more efficient for certain problems.

Sparse polynomial methods are preferable for others depending on problem structure.

Complexity estimates vary significantly based on the parametric problem class.

Abstract

We consider adaptive approximations of the parameter-to-solution map for elliptic operator equations depending on a large or infinite number of parameters, comparing approximation strategies of different degrees of nonlinearity: sparse polynomial expansions, general low-rank approximations separating spatial and parametric variables, and hierarchical tensor decompositions separating all variables. We describe corresponding adaptive algorithms based on a common generic template and show their near-optimality with respect to natural approximability assumptions for each type of approximation. A central ingredient in the resulting bounds for the total computational complexity are new operator compression results for the case of infinitely many parameters. We conclude with a comparison of the complexity estimates based on the actual approximability properties of classes of parametric model problems, which shows that the computational costs of optimized low-rank expansions can be significantly lower or higher than those of sparse polynomial expansions, depending on the particular type of parametric problem.