用于热方程伴随蒙特卡洛模拟的可逆随机数生成
Reversible random number generation for adjoint Monte Carlo simulation of the heat equation
AI总结：
针对PDE约束优化中伴随蒙特卡洛模拟需复用粒子轨迹但内存不足的问题，提出PCG伪随机数生成器的可逆扩展，通过重算热方程反向路径解决内存限制问题。
摘要：
在偏微分方程（PDE）约束优化问题中，研究目标是找到满足偏微分方程约束、使某目标函数最小化的设计参数。该领域的一大挑战是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PDE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规避该雅可比矩阵的计算，求解这些偏导数需要得到带约束的伴随微分算子对应的第二个PDE的解，最终需要开展时间反向的模拟。基于粒子的蒙特卡洛求解器常被用于求解高维PDE，但这类求解器的缺点是会给计算结果引入噪声，因此需要使用随机优化方法。为保证该场景下的收敛性，约束蒙特卡洛模拟和伴随蒙特卡洛模拟需要模拟相同的粒子轨迹。对于大规模模拟，由于内存限制，存储约束方程的完整路径以供伴随方程复用并不可行。本文对置换同余伪随机数生成器（PCG）系列提出了可逆扩展，随后使用这类生成器重新计算热方程的时间反向路径，从而规避上述内存问题。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
