Gaussian mixture Taylor approximations of risk measures constrained by PDEs with Gaussian random field inputs

TL;DR

This paper introduces a Gaussian mixture Taylor approximation method for efficiently computing risk measures governed by PDEs with Gaussian random field inputs, significantly reducing computational cost while maintaining high accuracy.

Contribution

It develops a novel Gaussian mixture-based approach combined with Taylor approximations to improve risk measure estimation for PDEs with high-dimensional Gaussian inputs.

Findings

Achieves less than 1% relative error in CVaR estimation.

Requires only about 10 PDE solves, comparable to Monte Carlo with 10,000 samples.

Demonstrates effectiveness on advection-diffusion-reaction and Helmholtz equations.

Abstract

This work considers the computation of risk measures for quantities of interest governed by PDEs with Gaussian random field parameters using Taylor approximations. While efficient, Taylor approximations are local to the point of expansion, and hence may degrade in accuracy when the variances of the input parameters are large. To address this challenge, we approximate the underlying Gaussian measure by a mixture of Gaussians with reduced variance in a dominant direction of parameter space. Taylor approximations are constructed at the means of each Gaussian mixture component, which are then combined to approximate the risk measures. The formulation is presented in the setting of infinite-dimensional Gaussian random parameters for risk measures including the mean, variance, and conditional value-at-risk. We also provide detailed analysis of the approximations errors arising from two sources: the Gaussian mixture approximation and the Taylor approximations. Numerical experiments are conducted for a semilinear advection-diffusion-reaction equation with a random diffusion coefficient field and for the Helmholtz equation with a random wave speed field. For these examples, the proposed approximation strategy can achieve less than $1\%$ relative error in estimating CVaR with only $\mathcal{O}(10)$ state PDE solves, which is comparable to a standard Monte Carlo estimate with $\mathcal{O}(10^4)$ samples, thus achieving significant reduction in computational cost. The proposed method can therefore serve as a way to rapidly and accurately estimate risk measures under limited computational budgets.