Efficient Solution of Advection Dominated PDE Constrained Optimization Problems

本提案的目的是开发、分析和实现优化算法，该算法集成了多层迭代求解器、自适应网格细化方法和所谓的“一次性”方法，用于解决由平流主导的偏微分方程（PDEs）控制的优化问题。在控制偏微分方程中，强平流的存在给这些优化问题的数值求解带来了许多挑战，这不仅仅是在单一平流主导偏微分方程的数值模拟中遇到的挑战，也不仅仅是在求解弱平流或无平流的偏微分方程约束优化问题时遇到的许多困难。在优化环境中出现额外挑战的一个原因是所谓的伴随方程的存在，这也是一个平流主导的PDE，但是平流是由控制方程中的平流的负值给出的。当离散化方案和迭代求解器的应用从单一平流主导的偏微分方程扩展到偏微分方程约束优化问题的求解时，这可能会导致它们的行为发生重大的甚至可能是意想不到的差异。本研究将分析这些差异的来源，以及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的修改，自适应网格细化方法，以及所谓的KKT求解器，使它们对平流的存在具有鲁棒性。许多重要的现实应用，如工艺设备的形状优化，系统的最优控制，以及环境过程中参数的识别，都会导致由平流主导的偏微分方程系统控制的优化问题。本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。这将导致对这些优化问题的解的性质有更好的理论认识，并为其可靠和有效的解决提供新的计算工具。此外，它将为学生在计算科学的一个重要领域提供培训机会。

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.