Collaborative Research: Multigrid Methods for PDE Constrained Optimization

该方案的目的是开发、分析和实现一类集成了多级迭代求解器和所谓的“一次全部”优化方法的优化算法。在算法复杂性方面，多层技术提供了高效的偏微分方程(PDE)解算器。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-对偶牛顿内点方法，将偏微分方程组作为约束纳入优化程序，与将偏微分方程组的解视为控制/设计变量的隐函数的方法相比，有望节省大量的计算工作量。本研究将多层次技术和优化算法相结合，从原来的无限维优化问题中提取出足够的结构信息，而这些信息仅依靠单一网格是无法实现的。除了开发一般的偏微分方程组约束优化算法外，这一建议还将有助于开发两种具有挑战性的实际应用的求解方法：电流变器件的形状优化和大气气溶胶建模中不同阶段的识别。这两个应用都是由具有非线性的复杂偏微分方程组系统控制的，例如，由于偏微分方程组的本构方程或复杂的耦合条件。此外，由于设计规范或问题化学的原因，这两个优化问题都涉及到额外的等式和不等式约束，该研究为偏微分方程组约束优化问题提供了新的算法工具。这些问题的解决在越来越多的实际应用中是一项重要的任务，如技术设备的形状优化以及大气和地球物理过程中物理量的识别。尽管最近取得了进展，但这些优化问题的可靠数值解仍然是一项具有挑战性的任务。挑战例如来自底层偏微分方程的复杂性、来自大规模的优化问题以及来自底层应用的结构、偏微分方程组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且具有挑战性的实际PDE约束优化应用：电流变器件(如减振器)的形状优化，以及大气气溶胶模拟中不同阶段的识别，这是环境研究中的关键组成部分。

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.