Collaborative Research: Multigrid Methods for PDE Constrained Optimization

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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.