A stochastic collocation approach for parabolic PDEs with random domain deformations

TL;DR

This paper introduces a stochastic collocation method using sparse grids to efficiently compute statistical moments of solutions to parabolic PDEs with random domain deformations, providing convergence analysis and numerical validation.

Contribution

It develops a novel approach combining domain remapping and stochastic collocation for parabolic PDEs with random geometries, including convergence rate derivations.

Findings

The method accurately computes statistical moments of the QoI.

Convergence rates are established and validated numerically.

The approach effectively handles complex random domain deformations.

Abstract

This work considers the problem of numerically approximating statistical moments of a Quantity of Interest (QoI) that depends on the solution of a linear parabolic partial differential equation. The geometry is assumed to be random and is parameterized by $N$ random variables. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit an extension on a well defined region embedded in the complex hyperplane. A Stochastic collocation method with an isotropic Smolyak sparse grid is used to compute the statistical moments of the QoI. In addition, convergence rates for the stochastic moments are derived and compared to numerical experiments.