Reduced order model for parametric Boltzmann equation and its application to inverse problems
Abstract
The Boltzmann equation plays an important role in modeling mesoscopic behavior in a wide range of scientific and engineering applications. However, its numerical solution is computationally expensive due to the high dimensionality of the model and the nonlinear nonlocal collision operator, especially for steady-state problems that require iterative solvers. This cost becomes prohibitive for inverse problems, where the induced optimization problem requires repeated forward solves. In this work, we propose a reduced-order model (ROM) for the parametric Boltzmann equation to address this computational challenge. The ROM constructs a low-dimensional approximation space for the parameter-induced solution manifold through a residual-based greedy strategy, and the reduced solution is then obtained via residual minimization over the reduced space, subject to mass conservation. The overall efficiency of the ROM is achieved by exploiting the quadratic structure of the collision operator and a precomputed separable approximation of the collision kernel. The resulting ROM is further applied to a thermally-driven inverse problem for reconstructing collision parameters from the observed macroscopic temperature data. This is accomplished either by directly replacing the PDE constraint with the ROM, leading to a bilevel optimization formulation, or by reformulating the task as a single-level optimization problem through the Karush--Kuhn--Tucker (KKT) conditions. Numerical experiments in both collision-dominated and transport-dominated cases are performed to demonstrate the efficiency and accuracy of the proposed ROM and its effectiveness in inverse problems. In particular, the resulting inverse problem is computationally much more tractable, achieving speedups of several orders of magnitude over that based on the full-order model while maintaining comparable accuracy.
Related
- A data-based conservative reduced collision model for the Boltzmann equation
- On the existence problem for a double mean field equation and its application to prescribed curvature problems
- Fluid dynamic limit of Boltzmann equation for granular hard-spheres in a nearly elastic regime
- Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem
- Hydrodynamic Limit of a Viscous BGK Model of the Boltzmann Equation
BibTeX
@misc{tong2026reduced,
title = {{Reduced order model for parametric Boltzmann equation and its application to inverse problems}},
author = {Shanyin Tong and Jingwei Hu and Fengyan Li and Zhaiming Shen and Yunan Yang and Jiajia Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02578},
eprint = {2609.02578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02578},
}
RIS
TY - UNPB
TI - Reduced order model for parametric Boltzmann equation and its application to inverse problems
AU - Shanyin Tong
AU - Jingwei Hu
AU - Fengyan Li
AU - Zhaiming Shen
AU - Yunan Yang
AU - Jiajia Yu
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02578
UR - https://arxiv.org/abs/2609.02578
AB - The Boltzmann equation plays an important role in modeling mesoscopic behavior in a wide range of scientific and engineering applications. However, its numerical solution is computationally expensive due to the high dimensionality of the model and the nonlinear nonlocal collision operator, especially for steady-state problems that require iterative solvers. This cost becomes prohibitive for inverse problems, where the induced optimization problem requires repeated forward solves. In this work, we propose a reduced-order model (ROM) for the parametric Boltzmann equation to address this computational challenge. The ROM constructs a low-dimensional approximation space for the parameter-induced solution manifold through a residual-based greedy strategy, and the reduced solution is then obtained via residual minimization over the reduced space, subject to mass conservation. The overall efficiency of the ROM is achieved by exploiting the quadratic structure of the collision operator and a precomputed separable approximation of the collision kernel. The resulting ROM is further applied to a thermally-driven inverse problem for reconstructing collision parameters from the observed macroscopic temperature data. This is accomplished either by directly replacing the PDE constraint with the ROM, leading to a bilevel optimization formulation, or by reformulating the task as a single-level optimization problem through the Karush--Kuhn--Tucker (KKT) conditions. Numerical experiments in both collision-dominated and transport-dominated cases are performed to demonstrate the efficiency and accuracy of the proposed ROM and its effectiveness in inverse problems. In particular, the resulting inverse problem is computationally much more tractable, achieving speedups of several orders of magnitude over that based on the full-order model while maintaining comparable accuracy.
ER -