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罗方法在求解受动力学方程控制的优化问题中的发展，这类问题常见于等离子体控制与器件设计等领域。由于相空间的高维性以及使用正向蒙特卡罗求解器评估目标泛函时引入的随机性，此类优化问题尤为棘手。为克服这些困难，学界提出了多种"伴随蒙特卡罗方法"。这些方法巧妙地将蒙特卡罗梯度估计器与偏微分方程约束优化相结合，为动力学应用领域量身定制了创新性解方案。本文首先考察蒙特卡罗梯度估计的三种主要策略：得分函数法、重参数化技巧及耦合方法，并深入阐述偏微分方程约束优化中的核心要素——伴随状态方法。聚焦于辐射传输方程与非线性玻尔兹曼方程的应用场景，我们系统阐述了如何将蒙特卡罗梯度技术融入"先优化再离散"与"先离散再优化"两种偏微分方程约束优化框架，从而构建高效伴随蒙特卡罗方法，实现在复杂高维优化问题中的有效梯度估计。