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

by   Emil Løvbak, et al.

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.


page 1

page 2

page 3

page 4


Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

We study the application of a tailored quasi-Monte Carlo (QMC) method to...

Solving Inverse PDE Problems using Grid-Free Monte Carlo Estimators

Modeling physical phenomena like heat transport and diffusion is crucial...

Monte Carlo Gradient in Optimization Constrained by Radiative Transport Equation

Can Monte Carlo (MC) solvers be directly used in gradient-based methods ...

Particle simulation methods for the Landau-Fokker-Planck equation with uncertain data

The design of particle simulation methods for collisional plasma physics...

Approximate Bayesian Model Inversion for PDEs with Heterogeneous and State-Dependent Coefficients

We present two approximate Bayesian inference methods for parameter esti...

Deterministic particle flows for constraining SDEs

Devising optimal interventions for diffusive systems often requires the ...

Statistical deconvolution of the free Fokker-Planck equation at fixed time

We are interested in reconstructing the initial condition of a non-linea...

Please sign up or login with your details

Forgot password? Click here to reset