SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2744-2774, December 2023.
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 discretize-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.

中文翻译：

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辐射传输方程约束优化中的蒙特卡洛梯度

《SIAM 数值分析杂志》，第 61 卷，第 6 期，第 2744-2774 页，2023 年 12 月。
摘要。蒙特卡罗 (MC) 求解器可以直接用于基于梯度的方法来解决偏微分方程约束的优化问题吗？在这些问题中，损失函数的梯度通常表示为两个 PDE 解的乘积，一个用于前向方程，另一个用于伴随。当使用 MC 求解器时，数值解是狄拉克测度。因此，人们立即面临着解释两种度量相乘的困难。这表明 MC 求解器自然与偏微分方程约束下基于梯度的优化不兼容。在本文中，我们研究了两种不同的策略来克服这一困难。一是采用先离散后优化的技术，对代数系统进行全面优化，避免狄拉克测度。第二种策略属于“优化然后离散”框架。我们提出了一种相关模拟，其中我们不单独使用 MC 求解器来解决前向和伴随问题，而是在伴随求解器中回收前向模拟中的样本。这将伴随解构建为测试函数，因此允许严格的收敛分析。研究是通过辐射传递方程的透镜来呈现的，无论是在光学成像的逆设置中还是在最优控制框架中。我们详细介绍了算法开发、收敛性分析和复杂性成本。还提供了数字证据来证明这些主张。