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.

翻译：运动式方程式的应用,如最佳设计和逆向问题,往往涉及通过基于梯度的优化算法寻找未知参数。根据联合状态法,我们得出两种不同的框架,以近似非线性波尔茨曼方程式制约的客观功能梯度。虽然前方问题可以通过DSMC方法解决,但很难有效解决“优化-当时的分解”方法获得的高维连续方程式。这一挑战促使我们提出一种采用“分解-正对称-优化”方法的DSMC辅助方法,用于博尔茨曼的制约优化。我们还分析了这两个框架的特性及其关联性。提出了几个数字例子,以表明其准确性和效率。