论文标题
非线性Boltzmann方程的伴随DSMC约束优化
Adjoint DSMC for nonlinear Boltzmann equation constrained optimization
论文作者
论文摘要
动力学方程（例如最佳设计和反问题）的应用通常涉及通过基于梯度的优化算法找到未知参数。基于伴随状态方法，我们得出了两个不同的框架，用于近似于非线性玻尔兹曼方程约束的客观功能的梯度。虽然可以通过DSMC方法来解决正向问题，但很难有效地解决通过“优化 - 然后降低”方法获得的高维连续伴随方程。这项挑战激发了我们在“离散化 - 优化”方法的玻尔兹曼受限优化方法之后提出一种伴随的DSMC方法。我们还分析了两个框架及其连接的属性。提出了几个数值示例，以证明它们的准确性和效率。
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.
