# Adjoint DSMC for nonlinear Boltzmann equation constrained optimization

arXiv: https://arxiv.org/abs/2009.01363 · HTML: https://mlsanity.com/paper/2009.01363/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Sep 2020.

## Built on by (most cited first, 3 of 3)

- [Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow](https://mlsanity.com/paper/2410.00229/) (2024)
- [Data-Driven Discovery of PDEs via the Adjoint Method](https://mlsanity.com/paper/2401.17177/) (2024)
- [Divergence-kernel method for linear responses of densities and generative models](https://mlsanity.com/paper/2509.03992/) (2025)

## Abstract (arXiv)

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.