Unbiased Sampling of Multidimensional Partial Differential Equations with Random Coefficients

TL;DR

This paper introduces an unbiased estimator for PDE solutions with random coefficients, ensuring finite variance and cost, and demonstrating applicability across disciplines through connection with stochastic differential equations.

Contribution

It presents a novel unbiased estimator for PDE solutions with random coefficients, with finite variance and cost, insensitive to problem dimension.

Findings

Estimator has finite variance and expected computational cost.

Estimator is dimension-insensitive and applicable across disciplines.

Error analysis connects PDEs with stochastic differential equations.

Abstract

We construct an unbiased estimator for function value evaluated at the solution of a partial differential equation with random coefficients. We show that the variance and expected computational cost of our estimator are finite and our estimator is unsensitive to the problem dimension so that it can be applied in problems across various disciplines. For the error analysis, we connect the parabolic partial differential equations with the expectations of stochastic differential equations and perform rough path estimation.