Collaborative Research: Multigrid Methods for PDE Constrained Optimization

本提案的目的是开发、分析和实现一类优化算法，该算法集成了多层迭代求解器和所谓的“一次性”优化方法。多层技术提供了有效的偏微分方程（PDE）求解算法复杂性。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内点方法，将偏微分方程作为约束纳入优化程序，与将偏微分方程解视为控制/设计变量的隐式函数的方法相比，有望节省大量的计算工作。本研究将多层技术与优化算法相结合，从原来的无限维优化问题中提取出足够的结构信息，这是仅依靠单个网格无法实现的。除了一般的PDE约束优化算法开发之外，本提案还将有助于开发两个具有挑战性的现实应用的解决方法：电流变装置的形状优化和大气气溶胶建模中不同阶段的识别。这两种应用都是由具有非线性的微分方程的复杂系统控制的，例如，微分方程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，这两个优化问题都涉及额外的等式和不等式约束。本研究为求解具有偏微分方程组约束的优化问题提供了新的算法工具。在越来越多的实际应用中，解决这些问题是一项重要的任务，例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。尽管最近取得了一些进展，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。挑战出现了，例如，从底层偏微分方程的复杂性，从大规模的优化问题，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。除了通用算法开发之外，本研究还解决了两个重要且具有挑战性的现实PDE约束优化应用：电流变装置（如减震器）的形状优化，以及大气气溶胶建模中不同阶段的识别，这是环境研究的关键组成部分。

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.