Sampling recovery in Bochner spaces and applications to parametric PDEs

TL;DR

This paper establishes convergence rates for sampling recovery in Bochner spaces, applying the results to improve approximation methods for parametric PDEs with uncertain inputs in computational uncertainty quantification.

Contribution

It introduces a unified framework for sampling recovery in Bochner spaces and enhances convergence rates for approximating solutions to parametric PDEs with complex inputs.

Findings

Improved convergence rates for log-normal input models.

Enhanced approximation accuracy for affine input models.

Unified treatment of different parametric PDE input types.

Abstract

We prove convergence rates of linear sampling recovery of functions in abstract Bochner spaces satisfying weighted summability of their generalized polynomial chaos expansion coefficients. The underlying algorithm is a function-valued extension of the least squares method widely used and thoroughly studied in scalar-valued function recovery. We apply our theory to two core problems in Computational Uncertainty Quantification. First, we address non-intrusive approximations of solutions to parametric elliptic or parabolic PDEs with log-normal or affine inputs using a finite set of particular solvers. Second, we consider approximating infinite-dimensional holomorphic functions that arise as solutions to more general parametric PDEs with Gaussian random field inputs. Our framework allows a unified treatment of the log-normal and affine input models and yields substantial improvements in the state of the art of these problems. More specifically, we obtain convergence rates that improve known results by a polynomial factor in the log-normal case and by a logarithmic factor in the affine case.