Monte Carlo Gradient in Optimization Constrained by Radiative Transport Equation

TL;DR

This paper explores how Monte Carlo solvers can be integrated into gradient-based PDE-constrained optimization, proposing strategies to handle measure multiplication issues and demonstrating their effectiveness through radiative transfer applications.

Contribution

It introduces two strategies for incorporating Monte Carlo methods into PDE optimization, including a correlated simulation approach with convergence analysis.

Findings

Correlated Monte Carlo simulation improves gradient estimation accuracy.

The proposed methods are effective for radiative transfer inverse problems.

Convergence and complexity analyses support the algorithms' validity.

Abstract

Can Monte Carlo (MC) solvers be directly used in gradient-based methods for PDE-constrained optimization problems? In these problems, a gradient of the loss function is typically presented as a product of two PDE solutions, one for the forward equation and the other for the adjoint. When MC solvers are used, the numerical solutions are Dirac measures. As such, one immediately faces the difficulty in explaining the multiplication of two measures. This suggests that MC solvers are naturally incompatible with gradient-based optimization under PDE constraints. In this paper, we study two different strategies to overcome the difficulty. One is to adopt the Discrete-Then-Optimize technique and conduct the full optimization on the algebraic system, avoiding the Dirac measures. The second strategy stays within the Optimize-Then-Discretize framework. We propose a correlated simulation where, instead of using MC solvers separately for both forward and adjoint problems, we recycle the samples in the forward simulation in the adjoint solver. This frames the adjoint solution as a test function, and hence allows a rigorous convergence analysis. The investigation is presented through the lens of the radiative transfer equation, either in the inverse setting from optical imaging or in the optimal control framework. We detail the algorithm development, convergence analysis, and complexity cost. Numerical evidence is also presented to demonstrate the claims.