Reduced order model for parametric Boltzmann equation and its application to inverse problems

Shanyin Tong; Jingwei Hu; Fengyan Li; Zhaiming Shen; Yunan Yang; Jiajia Yu

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Reduced order model for parametric Boltzmann equation and its application to inverse problems

Shanyin Tong, Jingwei Hu, Fengyan Li, Zhaiming Shen, Yunan Yang, Jiajia Yu

arXiv:2609.02578v1Sep 2, 2026math.NAmath.OC

TL;DR

The paper tackles the computational cost of steady-state parametric Boltzmann simulations and the repeated forward solves required by inverse problems. It develops a greedy reduced-order model with efficient collision evaluation and applies it to collision-parameter reconstruction from macroscopic temperature data. The ROM accelerates forward solves by more than three orders of magnitude and makes representative inverse solves take less than 10 seconds while maintaining approximately 10^-4 accuracy and comparable reconstruction accuracy to the full-order model.

Problem
High-dimensional steady-state Boltzmann simulations and repeated full-order solves make parametric studies and inverse problems computationally expensive.
Method
A greedy reduced-basis ROM uses residual minimization with mass conservation, quadratic collision structure, separable kernel approximation, and bilevel or KKT-based single-level inverse formulations.
Results
More than three orders of magnitude acceleration is achieved as a surrogate solver with approximately 10^-4 accuracy, while representative inverse solves take less than 10 seconds.
Takeaways & Limitations
The ROM makes the thermally driven collision-parameter inverse problem computationally tractable while preserving reconstruction accuracy comparable to the full-order model.
Takeaways & Limitations
The numerical demonstrations use one-dimensional physical space and two-dimensional velocity space, with extensions to higher-dimensional settings, richer kernels, and noisy data left for future work.

Abstract

from arXiv · show

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.

1 Introduction

The paper addresses the high cost of steady-state parametric Boltzmann simulations and inverse problems by developing a reduced-order surrogate and applying it to collision-parameter inference.

Steady-state Boltzmann solves are costly because they require high-dimensional phase-space resolution, nonlinear collision-operator approximation, and potentially hundreds to hundreds of thousands of iterations.
Parameter studies and inverse problems further increase the burden by requiring repeated full-order forward simulations for varying collision-kernel parameters.
The proposed MOR strategy builds a reduced surrogate for steady-state parametric solutions and uses it to infer collision-kernel parameters from observational data.
The ROM combines greedy reduced-basis construction, residual minimization, quadratic collision-operator structure, and a separable collision-kernel approximation.
The framework targets a relatively underdeveloped area because prior MOR work focused mainly on linear steady-state kinetic models, while nonlinear steady-state Boltzmann problems received less attention.
Numerical experiments cover collision-dominated and transport-dominated regimes, and the inverse problem is formulated through bilevel or single-level optimization.

2 The Boltzmann equation and its inverse problem

This section formulates the steady-state Boltzmann model and an inverse problem that calibrates a parameterized collision kernel from macroscopic observations such as temperature.

The Boltzmann equation describes a phase-space distribution whose transport and nonlinear collision terms govern particle-system dynamics.
The collision kernel determines collision frequency through relative speed and scattering angle, while collisions conserve momentum and energy.
The steady-state model uses an inflow boundary condition prescribing the distribution for velocities entering the domain.
Because the exact collision kernel is generally unavailable, the model uses a parametric kernel whose parameters are calibrated against experimentally observed transport properties.
The inverse problem seeks collision-kernel parameters that minimize mismatch between predicted and observed macroscopic quantities while enforcing the Boltzmann equation and boundary condition.
The study calibrates a physically motivated kernel family from low-order macroscopic data rather than recovering an unrestricted kernel from velocity-resolved boundary measurements.

3 Numerical scheme for the Boltzmann equation

The full-order model combines first-order spatial discretization, iterative source updates, and a fast Fourier spectral collision solver for a one-dimensional physical and two-dimensional velocity setting.

The FOM uses a first-order finite-difference iterative scheme coupled with a fast Fourier spectral method for collision evaluation.
The numerical scheme studies a solution homogeneous in one spatial dimension, leaving one physical coordinate and two velocity coordinates.
The collision model uses an angularly independent kernel parameterized by µ=(a,b), spanning hard and soft potentials and transport- to collision-dominated regimes.
The discrete formulation represents phase-space functions on a flattened grid-function space and evaluates transport through a linear discrete operator.
A single fast Fourier spectral collision evaluation costs O(MN_v^3 log N_v) and remains a computational bottleneck despite outperforming the direct method's O(N_v^4) complexity.
Mass conservation is enforced during iteration by rescaling each update, and iterations stop when the discrete L2 residual falls below a prescribed threshold.
The source-iteration scheme is known to converge slowly in the fluid regime, but the study does not employ proposed preconditioning or high-dimensional low-rank alternatives.

4 Reduced order model for the Boltzmann equation

The ROM builds a greedy reduced basis offline and solves many parameter instances online using residual minimization, mass conservation, and precomputed collision and transport quantities.

Reduced-basis construction: The offline stage constructs nested reduced-basis spaces from solution snapshots selected greedily over a training parameter set, while the online stage solves in the terminal space.
Collision evaluation: The ROM exploits the collision operator's quadratic structure and a separable empirical-interpolation approximation of the parameterized kernel.
Collision evaluation: EIM selects parameter points greedily according to interpolation error, with the interpolation error measured in the infinity norm in this work.
Collision evaluation: Precomputed terms L(u_i), Q(u_k,u_l;˜µ_m), and β_m(a) enable efficient evaluation of reduced residuals for many coefficient vectors and parameters.
Reduced-basis construction: The next training parameter is chosen where the current reduced solution has the largest residual, and its full-order solution is added to the basis.
Offline efficiency: Compared with POD, the greedy strategy requires the minimum number of full-order solves during the offline stage, which matters because those solves dominate cost near the fluid regime.
Online reduced solve: The reduced solution is obtained by residual minimization over the reduced space while preserving total mass.

5 Solving inverse problem using RB space and ROM

The paper uses a reduced basis and ROM to make thermally driven inverse reconstruction of collision-kernel parameters more tractable. It evaluates temperature-data misfit through reduced solutions, using either bilevel or single-level optimization.

Inverse problem formulation: The inverse problem reconstructs collision-kernel parameters from temperature observations in a thermally driven flow.Different parameter choices produce different interior temperature profiles, but some profiles differ only subtly, making reconstruction difficult.
Offline stage: The offline ROM builds a reduced basis from full-order solutions using greedy sampling, orthogonalization, and precomputed transport and collision terms.The construction uses EIM interpolation for the collision kernel and stops when the spectral ratio reaches the prescribed tolerance.
Online stage: The online ROM computes reduced coefficients by minimizing the residual in the reduced space and returns the corresponding reduced solution for each parameter value.The residual uses precomputed terms and the EIM kernel approximation, reducing the cost of repeated forward evaluations.
Computational motivation: Temperature profiles can be nearly indistinguishable for different parameters, while ROM-based inverse solvers address the repeated high-dimensional forward solves required by optimization.The method combines greedy reduced representations with the quadratic collision structure and separable kernel approximation.
Bilevel optimization: In the bilevel formulation, the ROM solves the lower-level constrained residual problem while the upper level minimizes temperature-data misfit.The inner optimization is treated as a black-box evaluator, allowing gradient-free pattern search or gradient-based BFGS optimization.
Single-level optimization: The single-level KKT reformulation uses the low-dimensional reduced basis in the optimality conditions, but it is a relaxation rather than an equivalent reformulation.This reduced representation also supports extensions involving spatially varying quantities without a significant increase in computational complexity.

6 Numerical experiments

The ROM achieves substantial forward-solve acceleration in both collision- and transport-dominated regimes, while inverse reconstructions depend on reduced-basis dimension and optimization formulation. Accuracy typically improves before a plateau, with single-level optimization offering the strongest computational savings.

Collision-dominated forward problem: The collision-dominated ROM uses an RB space of dimension N = 13, with singular values dropping about three orders of magnitude in the first three modes.The experiment evaluates training and testing performance over M = [−1, 1]×[20, 50].
Computational costs: Offline Q-term precomputation grows linearly with RB dimension and can become comparable to or larger than FOM cost when n > 10.EIM eliminates these collision-term evaluations from the online stage and also reduces offline work relative to computing them without EIM.
Collision-dominated forward problem: The online ROM requires less than one second at n = 10, compared with an average FOM time of about 2345 seconds.EIM kernel approximation and Q-term precomputation reduce the online task to a single n-dimensional polynomial optimization problem.
Transport-dominated forward problem: The transport-dominated ROM achieves a speedup of at least 200 over the FOM, whose average test-set solve time is approximately 264 seconds.The transport-dominated parameter manifold has slower singular-value decay and requires a relatively larger reduced-basis space.
Collision-dominated inverse problem: Inverse-problem errors generally decrease before the ROM testing-error plateau, with collision-dominated cases reaching approximately 1% at n = 10 and 0.1% at n = 12 for positive a∗.For negative a∗ cases, errors can rise from or beyond n = 10, making n = 9 or 10 a balance of accuracy and robustness.
Inverse-solver comparisons: At n = 10, the gradient-free bilevel solver takes about 100 seconds, the gradient-based bilevel solver about half as long, and the single-level formulation less than 10 seconds.In the transport-dominated case a∗ = 0.5, b∗ = 2, the best reconstruction occurs at n = 9 with relative errors of approximately 0.1%.

7 Conclusion and discussion

The paper develops a ROM for steady-state parametric Boltzmann equations and applies it to thermally driven inverse problems. Numerical evidence shows substantial acceleration with approximately 10^-4 accuracy, while reduced-space optimization may improve reconstruction quality.

The ROM solves steady-state parametric Boltzmann equations using greedy low-dimensional spaces, collision-operator precomputation, EIM kernel approximation, and enforced mass conservation.
The ROM supports thermally driven inverse problems through direct bilevel optimization or a single-level nonlinear constrained formulation.
More than three orders of magnitude acceleration is achieved as a surrogate solver while maintaining approximately 10^-4 accuracy relative to the full-order model.
Numerical evidence indicates that reduced-space optimization may provide a more favorable landscape and improve reconstruction quality.
The study is a first step, with future work needed for higher-dimensional settings, more general kernels, richer observations, and stable reconstruction from noisy data.