Optimal PDE control using COMSOL MULTIPHYSICS

by Ira Neitzel, Uwe Prüfert, and Thomas Slawig

Introduction

In the following you will find the COMSOL MULTIPHYSICS® scripts from the articels Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part one: problems without inequality constraints and Strategies for time-dependent PDE control using an integrated modeling and simulation environment. Part two: constrained problems by Ira Neitzel, Uwe Prüfert, and Thomas Slawig.

In the papers we presented the code for solving the examples in a very compressed form. To provide the reader an opportunity to reproduce our results, we offer the reader here a zip-archive for downloading. Feel free to test it.


Part One:  problems without inequality constraints

Abstract

We show how time-dependent optimal control for partial differential equations can be realized in a modern high-level modeling and simulation package. We summarize the general formulation for distributed and boundary control for initial-boundary value problems for parabolic PDEs and derive the optimality system including the adjoint equation. The main difficulty therein is that the latter has to be integrated backwards in time. This implies that complicated implementation effort is necessary to couple state and adjoint equations to compute an optimal solution. Furthermore a large amount of computational effort or storage is required to provide the needed information (i.e the trajectories) of the state and adjoint variables. We show how this can be realized in the modeling and simulation package Comsol Multiphysics, taking advantage of built-in discretization, solver and post-processing technologies and thus minimizing the implementation effort. We present two strategies: The treatment of the coupled optimality system in the space-time cylinder, and the iterative approach by sequentially solving state and adjoint system and updating the controls. Numerical examples show the elegance of the implementation and the effciency of the two strategies.

You can download the preprint here.


Part two: constrainded problems

Abstract

In this second part we extend our approach to (state) constrained problems. Pure state constraints in a function space setting lead to non-regular Lagrange multipliers (if they exist), i.e. the Lagrange multipliers are in general Borel measures. This will be overcome by different regularization techniques. To implement inequality constraints, active set methods and interior point methods (or barrier methods) are widely in use.
We show how these techniques can be realized in the modeling and simulation package Comsol Multiphysics. In contrast to the first part, only the one-shot-approach based on space-time elements is considered. We implemented a projection method based on active sets as well as a barrier method and compare these methods by a specialized PDE optimization program, and a program that optimizes the discrete version of the given problem.

You can download the preprint here.



The authors, Nov. 1th, 2007. Last update: August, 28th, 2009.

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