Hybrid collocation perturbation for PDEs with random domains

TL;DR

This paper introduces a hybrid collocation-perturbation method for efficiently approximating the statistics of PDE solutions on random domains, reducing computational complexity especially in high-dimensional stochastic problems.

Contribution

It proposes a novel hybrid approach combining sparse grid collocation and perturbation techniques to handle large and small domain variations separately, improving efficiency.

Findings

Convergence rates for the variance of the QoI are derived.

The method significantly reduces the stochastic problem's dimensionality.

Computational cost increases at most quadratically, linearly if variations are independent.

Abstract

In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by $N$ random variables. The random domain is split into large and small variations contributions. The large variations are approximated by applying a sparse grid stochastic collocation method. The small variations are approximated with a stochastic collocation-perturbation method. Convergence rates for the variance of the QoI are derived and compared to those obtained in numerical experiments. Our approach significantly reduces the dimensionality of the stochastic problem. The computational cost of this method increases at most quadratically with respect to the number of dimensions of the small variations. Moreover, for the case that the small and large variations are independent the cost increases linearly.