两类PDE约束优化问题的数值解法

Many problems in science and technology are formulated as optimization constrained with partial differential equation. Such research topic is challenging and full of vitality, since it involves theories of optimization in function spaces, discretization methods of PDEs and algorithms for the discretized problem. It is attracting more and more attentions in recent years. In this project, we will concerntrate on numerical solutions for two kinds of optimization with PDE constraints, specifically on three issues: 1. For sparse optimal control problem with control constraints and L1 or Lp regularization, when the usual finite element method is employed, the resulting discretized problem does not possess a decouple form as the finite dimensional sparse optimization usually does. we will consider discretization method which enables a decouple form and preserve accuracy of usual order, and analyse the discretization error. We will design algorithms for the corresponding discretized problem based on its structure and analyse the convergence of the algorithm. 2. For optimal control problem with state constraints, we will approximate the state function by piecewise polynomial of degree two and discretize the problem into a semi-definite program. We will design algorithms for such semi-definite program and analysed its convergence analogously. 3. For nonsmooth nonlinear systems arising from the two kinds of optimization with PDEs constraints mentioned above, we will consider efficient Jacobian-free semismooth Newton-Krylov method as well as preconditioning techniques. We will also develop corresponding software and try to use it to solve practical problems.

科学与工程中的很多问题可以表述为带PDE约束的优化问题。这类问题涉及函数空间中的优化理论、离散化方法和离散问题的数值方法，是一个既有挑战性又有生命力的课题。本项目将研究两类特殊的PDE约束优化问题的数值方法，拟解决问题如下：1.对带控制约束和L1或Lp正则化项的稀疏最优控制问题，在采用通常的有限元法离散时，得到的问题不具有有限维稀疏优化问题的可分结构。本项目将设计既能保持可分结构又保持离散精度的离散方法，以及相应的快速算法，并分析离散误差和算法的收敛性；2.对带状态约束的最优控制问题，采用分片二次多项式对状态函数进行逼近，把离散化问题转化成半定规划问题，设计高效算法，并分析离散误差和算法收敛性；3.对这两类问题求解中涉及的非光滑非线性方程组，将研究高效率的Jacobian-Free半光滑Newton-Krylov方法及预条件技术。我们还将考虑所设计的算法的软件实现及其应用。

带偏微分方程(PDE)约束的优化问题在现代工业、医学、经济学等领域中都有非常重要的应用。值得注意的是，PDE约束最优控制问题是无穷维的优化问题，其求解涉及函数空间的离散方法、最优性理论、优化算法等许多面，所以其无论在理论分析方面还是在数值解法方面都是具有挑战性的。此外，实际应用中的PDE约束最优控制问题随着科学和工程的发展变得越来越复杂，而正是由于问题的复杂性，其精确的最优解在一般情况下是难以求解的。那么发展快速、高效、鲁棒的数值解法来求解此类问题就显得十分重要。本项目主要研究了两类PDE约束最优控制问题：以流体流动的控制等实际问题为背景的带状态约束的PDE约束最优控制问题；以压电磁盘上制动器的位置设计等实际问题为背景的带控制约束的PDE约束稀疏最优控制问题。针对带L1控制成本的稀疏椭圆最优控制问题、带L2控制成本的椭圆最优控制问题、带箱型状态约束的L2控制成本的椭圆最优控制问题以及带积分型状态梯度约束的L2控制成本的椭圆最优控制问题等问题，利用有效的一些一阶算法，如交替方向乘子法(ADMM)、加速块坐标下降法(ABCD)，并综合利用多重网格法、预条件技术等方法和理论，构造有效的数值解法。在理论上，给出误差分析，分析所设计算法的收敛性以及有效性，并通过数值实验进行验证。