Reversible random number generation for adjoint Monte Carlo simulation of the heat equation

TL;DR

This paper introduces a reversible pseudorandom number generator to efficiently recompute particle trajectories in adjoint Monte Carlo simulations of the heat equation, reducing memory requirements in PDE-constrained optimization.

Contribution

It presents a reversible extension to permuted congruential pseudorandom generators enabling path recomputation without extensive memory use in Monte Carlo PDE solvers.

Findings

Reversible generator allows path recomputation in backward simulations

Reduces memory usage in high-dimensional Monte Carlo methods

Enables noise-free gradient computation in PDE-constrained optimization

Abstract

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.