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的条件下寻找使目标函数最小化的设计参数。主要挑战在于计算目标函数对设计参数的梯度，因为应用链式法则需要计算设计参数对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避免了这一雅可比矩阵。计算这些偏导数需要求解一个具有伴随微分算子的第二偏微分方程，从而产生时间回溯模拟。基于粒子的蒙特卡洛求解器常用于计算高维偏微分方程的解。然而，这类求解器存在在计算结果中引入噪声的缺陷，因而需要随机优化方法。为保证在此场景下的收敛性，约束方程和伴随蒙特卡洛模拟应模拟相同的粒子轨迹。对于大规模模拟，因存储约束方程完整路径供伴随方程重用的方式受限于内存而不可行。本文提出了一种可逆扩展的置换同余伪随机数生成器（PCG）族。进而利用该生成器对热方程的时间回溯路径进行重新计算，从而避免了这些内存问题。