Goal Oriented Mesh Adaptivity for Constrained Optimal Control and Optimization Problems

University Of Houston, Houston TX

Investigators

Abstract

The investigator proposes to develop concepts of goal oriented mesh adaptivity for the numerical solution of constrained optimal control and structural optimization problems for partial differential equations. In case of pure simulation, i.e., without optimization, mesh adaptivity on the basis of efficient and reliable a posteriori error estimators for finite element discretized partial differential equations is a well-established tool. Residual type a posteriori error estimators rely on the appropriate evaluation of the residual with respect to an approximate solution of the problem and lead to cheaply computable, local error terms by means of element and face or edge residuals. The error is typically estimated in norms associated with the underlying function space. The dual approach involves the adjoint problem and allows one to derive sharp upper bounds for the error with respect to various error functionals ranging from global norms to local, even pointwise, quantities. The idea is to consider the target functional as the right hand side in the adjoint of the given differential equation which provides a multiplicative relation between the error in the original and the adjoint equation. The investigator will systematically study goal oriented mesh adaptivity for control and state constrained optimal control problems and structural optimization problems such as shape and topology optimization with equality and inequality constraints on the state and design variables. In particular, he will consider different target quantities, including the objective functional and constraint satisfaction/violation, and investigate their impact on mesh adaptation and accuracy of the approximate solution. On this basis, he will develop, analyze and implement goal oriented a posteriori error estimators. This will be complemented by extensive numerical studies to document the efficiency and reliability of the developed tools for selected optimal control and optimization problems. The optimal control and structural optimization of systems described by partial differential equations has a deep impact on the cost effective development of technologically relevant devices and systems. Adaptive mesh refinement and coarsening on the basis of efficient and reliable goal oriented a posteriori error estimators is a significant algorithmic tool for numerical design studies which contribute to improve the functionality of the devices and systems without resorting to the cost intensive production of prototypes. The project will introduce graduate students to both state-of-the-art optimization and numerical simulation methods. The material will be used in graduate and undergraduate courses.