参数化玻尔兹曼方程的降阶模型及其在逆问题中的应用

中文摘要 AI 辅助

玻尔兹曼方程在众多科学与工程应用中对介观行为的建模发挥着重要作用。然而，由于该模型的高维性以及非线性非局部碰撞算子的存在，其数值求解的计算成本极高，尤其是对于需要迭代求解器的稳态问题。当涉及逆问题时，由此产生的优化问题需要反复进行正向求解，这使得计算成本变得难以承受。在本研究中，我们针对参数化玻尔兹曼方程提出了一种降阶模型（ROM），以应对这一计算挑战。该ROM通过基于残差的贪婪策略，为参数诱导的解流形构建了一个低维近似空间，随后在该降维空间中结合质量守恒约束，通过残差最小化得到降阶解。ROM的整体效率通过利用碰撞算子的二次结构以及碰撞核的预计算可分离近似来实现。我们将所得ROM进一步应用于热驱动逆问题，以从观测到的宏观温度数据中重构碰撞参数，具体实现方式有两种：一是直接用ROM替代偏微分方程约束，形成双层优化格式；二是通过Karush-Kuhn-Tucker（KKT）条件将任务重构为单层优化问题。我们在碰撞主导和输运主导两种情形下进行了数值实验，以验证所提ROM的效率、精度及其在逆问题中的有效性。特别地，所得到的逆问题在计算上更易处理，与基于全阶模型的逆问题相比，实现了数个数量级的加速，同时保持了相当的精度。

英文摘要

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.

发表机构

• University of Pennsylvania（宾夕法尼亚大学）
• University of Washington（华盛顿大学）
• Rensselaer Polytechnic Institute（伦斯勒理工学院）
• Georgia Institute of Technology（佐治亚理工学院）
• Cornell University（康奈尔大学）
• Duke University（杜克大学）

机构由 AI 辅助整理，请以论文原文为准。