Adjoint DSMC for nonlinear Boltzmann equation constrained optimization
Adjoint DSMC for nonlinear Boltzmann equation constrained optimization
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DOI：
10.1016/j.jcp.2021.110404
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发表时间：
2020-09
期刊：
影响因子：
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通讯作者：
R. Caflisch;Denis A. Silantyev;Yunan Yang
中科院分区：
文献类型：
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作者：
R. Caflisch;Denis A. Silantyev;Yunan Yang
优化设计和反问题等动力学方程的应用通常涉及通过基于梯度的优化算法查找未知参数。基于伴随状态方法，我们推导了两种不同的框架来近似受非线性玻尔兹曼方程约束的目标函数的梯度。虽然DSMC方法可以解决前向问题，但通过“优化然后离散”方法很难有效地求解高维连续伴随方程。这一挑战促使我们提出一种遵循玻尔兹曼约束优化的“离散然后优化”方法的伴随 DSMC 方法。我们还分析了两个框架的属性及其联系。给出了几个数值例子来证明其准确性和效率。
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.