Stochastic Gradient Descent for Semilinear Elliptic Equations with Uncertainties

TL;DR

This paper presents a stochastic gradient descent framework for efficiently solving semilinear elliptic PDEs with random coefficients, addressing issues of ill-conditioning and variance, and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a novel two-step approach combining calculus of variations and stochastic gradient descent for PDEs with uncertainties, including convergence analysis and variance reduction techniques.

Findings

The proposed method converges under specified criteria.

Numerical experiments show high accuracy and efficiency.

The framework effectively handles ill-conditioning and large variance.

Abstract

Randomness is ubiquitous in modern engineering. The uncertainty is often modeled as random coefficients in the differential equations that describe the underlying physics. In this work, we describe a two-step framework for numerically solving semilinear elliptic partial differential equations with random coefficients: 1) reformulate the problem as a functional minimization problem based on the direct method of calculus of variation; 2) solve the minimization problem using the stochastic gradient descent method. We provide the convergence criterion for the resulted stochastic gradient descent algorithm and discuss some useful technique to overcome the issues of ill-conditioning and large variance. The accuracy and efficiency of the algorithm are demonstrated by numerical experiments.