In PDE-constrained optimization, one aims to find design parameters that minimize some objective, subject to the satisfaction of a partial differential equation. A major challenges is computing gradients of the objective to the design parameters, as applying the chain rule requires computing the Jacobian of the design parameters to the PDE's state. The adjoint method avoids this Jacobian by computing partial derivatives of a Lagrangian. Evaluating these derivatives requires the solution of a second PDE with the adjoint differential operator to the constraint, resulting in a backwards-in-time simulation. Particle-based Monte Carlo solvers are often used to compute the solution to high-dimensional PDEs. However, such solvers have the drawback of introducing noise to the computed results, thus requiring stochastic optimization methods. To guarantee convergence in this setting, both the constraint and adjoint Monte Carlo simulations should simulate the same particle trajectories. For large simulations, storing full paths from the constraint equation for re-use in the adjoint equation becomes infeasible due to memory limitations. In this paper, we provide a reversible extension to the family of permuted congruential pseudorandom number generators (PCG). We then use such a generator to recompute these time-reversed paths for the heat equation, avoiding these memory issues.

翻译：在偏微分方程约束优化中，目标是在满足偏微分方程约束的前提下，寻找使目标函数最小化的设计参数。主要挑战之一在于计算目标函数关于设计参数的梯度，因为应用链式法则需要计算设计参数关于偏微分方程状态的雅可比矩阵。伴随方法通过计算拉格朗日函数的偏导数来避免这一雅可比矩阵。计算这些偏导数需要求解第二个带有约束伴随微分算子的偏微分方程，从而产生一个时间逆向的模拟过程。基于粒子的蒙特卡罗求解器常用于计算高维偏微分方程的解。然而，这类求解器的缺点是为计算结果引入噪声，因此需要采用随机优化方法。为在此类设定下保证收敛性，约束方程与伴随方程的蒙特卡罗模拟应模拟相同的粒子轨迹。对于大规模模拟，存储约束方程中的完整路径以供伴随方程重用，会因内存限制而变得不可行。本文为排列同余伪随机数生成器（PCG）族提供了可逆扩展，并利用此类生成器重新计算热方程的时间逆向路径，从而避免了这些内存问题。