论文标题
非结构化时空有限元方法，用于抛物线方程的最佳稀疏控制
Unstructured Space-Time Finite Element Methods for Optimal Sparse Control of Parabolic Equations
论文作者
论文摘要
我们在完全非结构化的简单网格上考虑了一种时空有限元方法，以最佳地稀疏对半线性抛物线方程的控制。目的是组合标准的二次跟踪型功能，包括Tikhonov正则化项和$ l^1 $ norm的控件，该功能是其时空的稀疏性。我们使用相关离散的最佳稀疏控制问题的一阶必要最优系统，使用时空Petrov-Galerkin有限元离散化。离散化是基于一种变性公式，该公式在空间和时间上同时采用分段线性有限元。最后，通过半齿牛顿方法解决了由耦合前后状态和伴随状态方程组成的离散的非线性最优系统。
We consider a space-time finite element method on fully unstructured simplicial meshes for optimal sparse control of semilinear parabolic equations. The objective is a combination of a standard quadratic tracking-type functional including a Tikhonov regularization term and of the $L^1$-norm of the control that accounts for its spatio-temporal sparsity. We use a space-time Petrov-Galerkin finite element discretization for the first-order necessary optimality system of the associated discrete optimal sparse control problem. The discretization is based on a variational formulation that employs piecewise linear finite elements simultaneously in space and time. Finally, the discrete nonlinear optimality system that consists of coupled forward-backward state and adjoint state equations is solved by a semismooth Newton method.
