Adjoint DSMC for nonlinear Boltzmann equation constrained optimization

TL;DR

This paper develops and compares two frameworks for gradient approximation in nonlinear Boltzmann equation constrained optimization, introducing an adjoint DSMC method for improved efficiency and accuracy in inverse problems.

Contribution

It proposes an adjoint DSMC method based on the discretize-then-optimize approach, addressing challenges in efficiently solving high-dimensional adjoint equations.

Findings

The adjoint DSMC method achieves accurate gradient approximations.

Numerical examples demonstrate the method's efficiency and effectiveness.

The frameworks are analyzed for their properties and connections.

Abstract

Applications for kinetic equations such as optimal design and inverse problems often involve finding unknown parameters through gradient-based optimization algorithms. Based on the adjoint-state method, we derive two different frameworks for approximating the gradient of an objective functional constrained by the nonlinear Boltzmann equation. While the forward problem can be solved by the DSMC method, it is difficult to efficiently solve the high-dimensional continuous adjoint equation obtained by the "optimize-then-discretize" approach. This challenge motivates us to propose an adjoint DSMC method following the "discretize-then-optimize" approach for Boltzmann-constrained optimization. We also analyze the properties of the two frameworks and their connections. Several numerical examples are presented to demonstrate their accuracy and efficiency.