Adjoint Monte Carlo Method

TL;DR

The article surveys adjoint Monte Carlo methods for optimization problems governed by kinetic PDEs, addressing the challenge of high-dimensional phase space and stochastic objective evaluations. It develops a unified framework that blends Monte Carlo gradient estimators (score function, pathwise/reparameterization, and coupling) with PDE-constrained optimization via the adjoint-state method, and demonstrates two kinetic case studies: the Radiative Transport Equation and the Boltzmann equation. Key contributions include deriving adjoint MC formulations under both OTD and DTO paradigms, introducing adjoint particle methods and adjoint DSMC for efficient, correlated forward-adjoint computations, and providing numerical evidence of variance-controlled gradient estimation and scalability in high-dimensional settings. The results highlight the practical impact of adjoint MC methods for design, control, and topology optimization in radiative transfer, rarefied gas dynamics, and related kinetic systems. Overall, the work offers a robust, extensible framework for integrating MC gradient estimators with PDE-constrained optimization to tackle complex, high-dimensional kinetic problems.

Abstract

This survey explores the development of adjoint Monte Carlo methods for solving optimization problems governed by kinetic equations, a common challenge in areas such as plasma control and device design. These optimization problems are particularly demanding due to the high dimensionality of the phase space and the randomness in evaluating the objective functional, a consequence of using a forward Monte Carlo solver. To overcome these difficulties, a range of ``adjoint Monte Carlo methods'' have been devised. These methods skillfully combine Monte Carlo gradient estimators with PDE-constrained optimization, introducing innovative solutions tailored for kinetic applications. In this review, we begin by examining three primary strategies for Monte Carlo gradient estimation: the score function approach, the reparameterization trick, and the coupling method. We also delve into the adjoint-state method, an essential element in PDE-constrained optimization. Focusing on applications in the radiative transfer equation and the nonlinear Boltzmann equation, we provide a comprehensive guide on how to integrate Monte Carlo gradient techniques within both the optimize-then-discretize and the discretize-then-optimize frameworks from PDE-constrained optimization. This approach leads to the formulation of effective adjoint Monte Carlo methods, enabling efficient gradient estimation in complex, high-dimensional optimization problems.

Figures (7)

• Figure 1: The diagram illustrates the main structure of this survey paper. The adjoint Monte Carlo (MC) method integrates effective techniques from Monte Carlo gradient estimation and problem-solving frameworks in PDE-constrained optimization to tackle challenging optimization problems based on various kinetic PDEs.
• Figure 2: A diagram showing the dependence of the approximated objective function $\widehat{J}$ on the samples $\{X_i\} \sim \mu_\theta$, which are determined by the parameter $\theta$ through the push-forward map $T_\theta(\boldsymbol{\cdot})$path by path in the "Pathwise Derivative Method" (see Section \ref{['subsubsec:pw_grad']}).
• Figure 3: The OTD approach (solid line) and the DTO approach (dash line) to compute the gradient with respect to the unknown parameter for a general PDE-constrained optimization problem \ref{['eq:PDE-OPT']}.
• Figure 4: An illustration of the RTE forward Monte Carlo trajectory (black) and its adjoint Monte Carlo trajectory (blue) for particle indices $n$ and $n+1$, as examples. The forward and adjoint trajectories coincide since they share the same randomness in the adjoint Monte Carlo method. Moreover, the adjoint variable $\gamma(t,x,v)$ carries a fixed value $\psi_n$ (resp. $\psi_{n+1}$) along the time trajectory; see \ref{['eq:rte_adj_MC']}.
• Figure 5: Illustration of the approximated gradient for an RTE-constrained optimization problem based on various numerical methods (left) and the standard deviation for the adjoint Monte Carlo methods based on the OTD and DTO approaches (right).

Theorems & Definitions (1)

• Remark 1