Collaborative Research: Multigrid Methods for PDE Constrained Optimization

该建议的目的是开发，分析和实现一类优化算法，集成多级迭代求解器和所谓的“一次全部”优化方法。多级技术提供了有效的偏微分方程（PDE）求解算法的复杂性。基于一次性方法的优化方法，如序列二次规划（SQP）方法和原始-对偶牛顿邻域点方法，将PDE作为约束纳入优化程序，并有望节省大量的计算工作相比，将PDE解视为控制/设计变量的隐函数的方法。本研究整合多层技术与优化演算法，以从原本无限维的优化问题中撷取足够的结构资讯，而这是单一网格无法达到的。除了一般的偏微分方程约束优化算法的发展，这项建议也将有助于两个具有挑战性的现实生活中的应用：电流变装置的形状优化和大气气溶胶建模中的不同阶段的识别的解决方案的方法的发展。这两种应用都是由具有非线性的偏微分方程的复杂系统控制的，例如，偏微分方程的本构方程或复杂的耦合条件。此外，这两个优化问题涉及额外的等式和不等式的约束，由于设计规格或问题的化学。这些问题的解决方案是一个重要的任务，在越来越多的现实生活中的应用，如形状优化的技术设备和大气和地球物理过程中的物理量的识别。尽管最近的进展，这些优化问题的可靠的数值解仍然是一项具有挑战性的任务。 挑战出现，例如，从底层偏微分方程的复杂性，从优化问题的大规模和从底层应用程序的结构，偏微分方程的数值解和数值优化的相互作用。除了一般的算法开发，这项研究还解决了两个重要的和具有挑战性的现实生活中的PDE约束优化应用：电流变装置，如减震器的形状优化，并识别不同阶段的大气气溶胶建模，在环境研究中的一个重要组成部分。

The aim of this proposal is to develop, analyze and implement a class of optimization algorithms that integrate multilevel iterative solvers and so-called `all-at-once' optimization methods. Multilevel techniques provide efficient partial differential equation (PDE) solvers with regard to algorithmic complexity. Optimization methods based on the all-at-once approach, such as sequential quadratic programming (SQP) methods and primal-dual Newton interior-point methods, incorporate the PDEs as constraints into the optimization routine and hold the promise to save a considerable amount of computational work compared to methods that view the PDE solution as an implicit function of the control/design variables. This research integrates multilevel techniques and optimization algorithms to extract an adequate amount of structural information from the originally infinite dimensional optimization problem which can not be achieved when only relying on a single grid. In addition to general PDE constrained optimization algorithm development, this proposal will also contribute to the development of solution methods for two challenging real-life applications: the shape optimization of electrorheological devices and the identification of different phases in atmospheric aerosol modeling. Both applications are governed by complex systems of PDEs with nonlinearities due to, e.g., the constitutive equations or the intricate coupling conditions for the PDEs. Moreover, both optimization problems involve additional equality and inequality constraints due to design specifications or problem chemistry.This research provides new algorithmic tools for optimization problems with constraints given by systems of partial differential equations (PDEs). The solution of such problems is an important task in an increasing number real-life applications such as the shape optimization of technological devices and the identification of physical quantities in atmospheric and geophysical processes. Despite recent progress, the reliable numerical solution of these optimization problems still represents a challenging task. Challenges arise, e.g., from the complexity of the underlying PDEs, from the large scale of the optimization problems and from the interactions of the structure of the underlying application, the numerical solution of PDEs and the numerical optimization. In addition to general algorithm development, this research also tackles two important and challenging real-life PDE constrained optimization applications: the shape optimization of electrorheological devices, such as shock absorbers, and the identification of different phases in atmospheric aerosol modeling, a crucial component in environmental research.