论文标题
PDE受限的优化模型和伪谱方法多尺寸粒子动力学
PDE-Constrained Optimization Models and Pseudospectral Methods for Multiscale Particle Dynamics
论文作者
论文摘要
我们得出了新的算法，以通过描述多尺寸粒子动力学的部分微分方程来限制优化问题，包括代表粒子之间相互作用的非本地积分术语。特别是，我们研究了控制作为对流“流”向量或部分微分方程的源项的问题，并且约束配备了Dirichlet或No-Flux类型的边界条件。在得出了此类问题的连续一阶最佳条件之后，我们通过与统计力学的计算方法建立链接，在时空和时间变量中得出伪谱方法，并利用现有固定点方法的变体来解决所得系统。数值实验表明我们的方法在一系列问题设置，边界条件以及正则化和模型参数方面的有效性。
We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we investigate problems where the control acts as an advection 'flow' vector or a source term of the partial differential equation, and the constraint is equipped with boundary conditions of Dirichlet or no-flux type. After deriving continuous first-order optimality conditions for such problems, we solve the resulting systems by developing a link with computational methods for statistical mechanics, deriving pseudospectral methods in both space and time variables, and utilizing variants of existing fixed point methods. Numerical experiments indicate the effectiveness of our approach for a range of problem set-ups, boundary conditions, as well as regularization and model parameters.
