Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

Abstract

We establish summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions for countably-parametric solutions of linear elliptic and parabolic divergenceform partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments and induction on the differentiation order of solution derivatives with respect to the parameters. The present holomorphybased argument allows a unified, "differentiation-free" sparsity analysis of Wiener-Hermite polynomial chaos expansions in various scales of function spaces. The analysis also implies corresponding results for posterior densities in Bayesian inverse problems subject to Gaussian priors on uncertain inputs from function spaces. Our results furthermore yield dimension-independent convergence rates of various constructive high-dimensional deterministic numerical approximation schemes such as single-level and multi-level versions of anisotropic sparse-grid Hermite-Smolyak interpolation and quadrature in both forward and inverse computational uncertainty quantification.

Suboptimality of Gauss–Hermite Quadrature and Optimality of the Trapezoidal Rule for Functions with Finite Smoothness

“…Integration problems in high dimensions arise, for example in computing statistics of solutions of partial differential equations parametrised by random variables. In particular, integration with respect to the Gaussian measure has been attracting increasing attention; see for example [25,11,30,28,17,18]. The measure being Gaussian, algorithms that use Gauss-Hermite quadrature as a building block have gained popularity [11,17,18].…”

“…In particular, integration with respect to the Gaussian measure has been attracting increasing attention; see for example [25,11,30,28,17,18]. The measure being Gaussian, algorithms that use Gauss-Hermite quadrature as a building block have gained popularity [11,17,18]. The key to proving error estimates in this context is the regularity of the quantity of interest with respect to the parameter, and for elliptic and parabolic problems such smoothness, even an analytic regularity, has been shown [1,37,18].…”

Sub-optimality of Gauss--Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness

“…Integration problems in high-dimension arise, for example in computing statistics of solutions of partial differential equations parametrised by random variables. In particular, integration with respect to the Gaussian measure has been attracting increasing attention; see for example [21,8,26,24,13,14]. The measure being Gaussian, algorithms that use Gauss-Hermite quadrature as a building block have gained popularity [8,13,14].…”

“…In particular, integration with respect to the Gaussian measure has been attracting increasing attention; see for example [21,8,26,24,13,14]. The measure being Gaussian, algorithms that use Gauss-Hermite quadrature as a building block have gained popularity [8,13,14].…”

“…The key to proving error estimates in this context is the regularity of the quantity of interest with respect to the parameter, and for elliptic and parabolic problems such smoothness, even an analytic regularity, has been shown [1,33,14]. In contrast, the solutions of parametric hyperbolic systems suffer from limited regularity under mild assumptions [31,32].…”