Efficient Solution of Advection Dominated PDE Constrained Optimization Problems

这一提议的目的是开发、分析和实施优化算法，该算法集成了多级迭代求解器、自适应网格加密方法和所谓的“一次全部”方法，用于解决由平流控制的偏微分方程组(PDE)控制的优化问题。控制偏微分方程组中强平流的存在给这些优化问题的数值解带来了许多挑战，这不仅是单个平流占主导地位的偏微分方程组的数值模拟所遇到的挑战，也是解决弱平流或无平流的偏微分方程组约束优化问题的许多困难。在优化背景下产生额外挑战的一个原因是存在所谓的伴随方程，它也是平流主导的偏微分方程组，但平流是由控制方程中的平流的负值给出的。当离散化格式和迭代求解器从应用程序扩展到单平流控制偏微分方程组，再到求解偏微分方程组约束优化问题时，这可能会导致离散化格式和迭代求解器行为上的显著差异，甚至可能是意想不到的。这项研究将分析这些差异的来源，它们对计算解的质量和求解算法效率的影响。另一个目标是设计多层迭代求解器、自适应网格加密方法和所谓的KKT求解器的修改，以使它们对平流存在时具有鲁棒性。许多重要的现实应用，如技术设备的形状优化、系统的最优控制和环境过程中的参数辨识，都导致了由平流控制偏微分方程组控制的优化问题。这项研究解决了数学和算法问题，这些问题对于可靠和有效地解决这些问题至关重要。这将有助于从理论上更好地理解这些优化问题的解的性质，并为它们的可靠和有效解提供新的计算工具。此外，它还将在计算科学的一个重要领域为学生提供培训机会。

The aim of this proposal is to develop, analyze and implement optimizationalgorithms that integrate multilevel iterative solvers, adaptive mesh refinement methods, and so-called `all-at-once' methods for the solution of optimization problems governed by advection dominated partial differential equations (PDEs). The presence of strong advection in the governing PDE creates many challenges forthe numerical solution of these optimization problems, beyond the challenges alreadyencountered in the numerical simulation of single advection dominated PDEs and beyondthe many difficulties in solving PDE constrained optimization problems with weak or no advection. One reason for the additional challenges arising in the optimization context is the presence of the so-called adjoint equation, which is also an advection dominated PDE, but with advection given by the negative of the advection in the governing equation. This can cause significant and perhaps unexpected differences in the behavior of discretization schemes and iterative solvers when they are extended from the application to single advection dominated PDEs to the solution of PDE constrained optimization problems. This research will analyze the sources of these differences, their impact on the quality of computed solutionand on the efficiency of solution algorithms. Another goal is to devise modifications of multilevel iterative solvers, adaptive mesh refinement methods, and so-called KKT solvers tomake them robust against the presence of advection.Many important real-life applications such as the shape optimization of technological devices, the optimal control of systems, and the identification of parameters in environmentalprocesses lead to optimization problems governed by systems of advection dominatedpartial differential equations. This research addresses mathematical and algorithmicissues that are crucial for the reliable and efficient solution of these problems. It will leadto a better theoretical understanding of the solution properties of these optimization problems as well as to new computational tools for their reliable and efficient solution. Furthermore, it will provide training opportunities for students in an important area of computational science.