MONTE CARLO GRADIENT IN OPTIMIZATION CONSTRAINED BY RADIATIVE TRANSPORT EQUATION

Qin Li, Li Wang, Yunan Yang

Research output: Contribution to journalArticlepeer-review

Abstract

Can Monte Carlo (MC) solvers be directly used in gradient-based methods for PDEconstrained 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 discretizethen- 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.

Original languageEnglish (US)
Pages (from-to)2744-2774
Number of pages31
JournalSIAM Journal on Numerical Analysis
Volume61
Issue number6
DOIs
StatePublished - 2023

Bibliographical note

Publisher Copyright:
© 2023 Society for Industrial and Applied Mathematics Publications. All rights reserved.

Keywords

  • Monte Carlo gradient
  • adjoint-state method
  • particle method
  • radiative transport equation

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