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:在受PDE限制的优化中,我们的目标是找到设计参数,在满足部分差异方程式的前提下,最大限度地降低某些目标。一个重大挑战是将目标的梯度计算成设计参数,因为适用链规则需要将设计参数的Jacobian计算成PDE的状态。联合方法通过计算一个拉格兰江的局部衍生物,避免了这个Jacobian。评估这些衍生物需要第二个PDE的解决方案,与连接差运算器的连接差运算到制约,从而导致逆向模拟。基于粒子的蒙特卡洛解析器常常被用来计算高维度的PDE的解决方案。然而,这些解算器有向计算结果输入噪音的退缩,因此需要随机优化方法。为了保证这一环境的趋同,制约和连接蒙特卡洛的模拟应该模拟同样的粒子导体。对于大型模拟而言,由于记忆限制,无法储存用于重新使用的制约方程式的完整路径。在本文中,我们为计算结果结果的结果结果结果结果结果结果,我们提供了一种可逆的扩展面延伸方法,因此需要精确地将这种温度变的模型模拟模型,这些模拟模拟模型的模型用于这些模拟的模型的模型。