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.

翻译：本综述探讨了伴随蒙特卡洛方法在求解受动力学方程约束的优化问题方面的发展，这类问题在等离子体控制和器件设计等领域中普遍存在。由于相空间的高维性以及使用正向蒙特卡洛求解器导致目标泛函评估的随机性，这些优化问题尤其具有挑战性。为克服这些困难，一系列“伴随蒙特卡洛方法”被设计出来。这些方法巧妙地将蒙特卡洛梯度估计器与偏微分方程约束优化相结合，并引入了针对动力学应用定制的创新解决方案。在本综述中，我们首先考察了蒙特卡洛梯度估计的三种主要策略：得分函数法、重参数化技巧以及耦合方法。我们还深入探讨了伴随状态法，这是偏微分方程约束优化中的一个核心要素。聚焦于辐射传输方程和非线性玻尔兹曼方程中的应用，我们提供了一个全面的指南，阐述如何将蒙特卡洛梯度技术整合到偏微分方程约束优化中的“先优化后离散化”和“先离散化后优化”两种框架内。这种方法促成了高效伴随蒙特卡洛方法的构建，从而能够在复杂的高维优化问题中实现高效的梯度估计。