Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

TL;DR

This paper introduces a sparse approximation framework for multilinear problems, particularly in uncertainty quantification, using a generalized Smolyak's algorithm to reduce the curse of dimensionality in high-dimensional kernel-based applications.

Contribution

It develops a generalized Smolyak's algorithm for sparse multilinear approximation and demonstrates its effectiveness in uncertainty quantification and kernel-based PDE applications.

Findings

Convergence rates mitigate curse of dimensionality.

Framework applies to response surface approximation.

Numerical experiments validate theoretical results.

Abstract

We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak's algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments.