Multilevel Monte Carlo Finite Element Method for A Stochastic Optimal Control Problem

TL;DR

This paper develops a multi-level Monte Carlo finite element approach for stochastic optimal control problems with log-normal coefficients, optimizing sample sizes and demonstrating improved efficiency over traditional methods.

Contribution

It introduces a novel multi-level Monte Carlo algorithm tailored for stochastic control problems with log-normal coefficients, including formulas for optimal sampling and practical implementation tricks.

Findings

The method reduces computational cost compared to traditional Monte Carlo.

Numerical results validate the effectiveness of the approach for elliptic SPDEs.

The approach successfully handles multi-level log-normal coefficients.

Abstract

In this paper, we consider the implementation of multi-level Monte Carlo method to a stochastic optimal control problem with log-normal coefficients and its surrogate model problem. From the perspective of two optimization problems, i.e., minimizing the accuracy using a fixed computational cost and minimizing the total computational cost to attain a given accuracy, we derive formulas to determine the optimal sample sizes for each level of multi-level Monte Carlo method. Furthermore, we put forward the multi-level Monte Carlo algorithm for our stochastic optimal control problem and some tricks to deal with the multi-level log-normal coefficients. Finally, we present the numerical results of both the elliptic SPDEs and our control problem to validate the effectiveness over the traditional Monte Carlo method.