Simultaneous spatial-parametric collocation approximation for parametric PDEs with log-normal random inputs

TL;DR

This paper proves improved convergence rates for a multi-level collocation method solving parametric elliptic PDEs with log-normal inputs, matching best n-term approximation rates up to logarithmic factors.

Contribution

It introduces a fully discrete multi-level collocation approach with significantly improved convergence rates for parametric PDEs with log-normal inputs.

Findings

Convergence rates are significantly improved over previous methods.

Rates match best n-term approximation rates up to logarithmic factors.

The approach applies multi-level linear sampling recovery theory in Bochner spaces.

Abstract

We establish convergence rates for a fully discrete, multi-level, linear collocation method solving parametric elliptic PDEs on bounded polygonal domains with log-normal inputs. The method uses a finite set of function evaluations in the spatial-parametric domain. Compared with the best-known fully discrete collocation rates, these rates are significantly improved and, up to logarithmic factors, match the rates of best n-term approximations. The results follow from applying general multi-level linear sampling recovery theory in abstract Bochner spaces -- via extended least-squares -- to infinite-dimensional holomorphic functions. The abstract multi-level recovery in Bochner spaces guarantees yield the improved rates when specialized to the parametric PDE setting.