Efficient Solution of Advection Dominated PDE Constrained Optimization Problems

该提案的目的是开发，分析和实施optimizationalgorithms，集成多级迭代求解器，自适应网格细化方法，以及所谓的“一次性”方法，用于解决由对流主导的偏微分方程（PDE）的优化问题。强平流的存在下，在执政的PDE创造了许多挑战，为这些优化问题的数值解，超越了挑战已经遇到的数值模拟单平流为主的PDE和超越了许多困难，在求解PDE约束优化问题，弱或没有平流。在优化上下文中出现的额外挑战的一个原因是存在所谓的伴随方程，其也是平流主导的PDE，但是平流由控制方程中的平流的负值给出。这可能会导致显着的，也许是意想不到的差异，离散化方案和迭代求解器的行为时，他们从应用程序扩展到单对流主导偏微分方程的偏微分方程约束优化问题的解决方案。 本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的效率的影响。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。许多重要的实际应用，如技术设备的形状优化，系统的最优控制，环境过程中参数的识别导致了由对流主导的偏微分方程系统所支配的优化问题。这项研究解决了这些问题的可靠和有效的解决方案是至关重要的数学和算法问题。这将导致更好的理论理解这些优化问题的解决方案的性质，以及新的计算工具，其可靠和有效的解决方案。此外，它将为学生提供计算科学重要领域的培训机会。

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.