Doctoral Dissertation Defense: Jyoti Saraswat
Advisor: Dr. Andrei Draganescu
Location
The Commons : 329
Date & Time
September 30, 2014, 1:30 pm – 2:30 pm
Description
Title: Multigrid solution of distributed optimal control problems constrained by
semilinear elliptic PDEs
Abstract:
Partial differential equations (PDEs) constrained optimization is a research area in which the scientific and engineering communities have developed a growing interest over the last decade. The recent rise in interest was fostered by the tremendous increase in computing power over the last twenty years. This can be attributed both to the tremendous advances in high-performance computing technologies and to its wide range of applicability. However, just growth in computing power is insufficient for tackling large-scale PDE-constrained optimization problems and there is always an ever increasing need for efficient algorithms. The objective of this dissertation is to develop, analyze, and implement multigrid preconditioners for the preconditioning of the linear systems arising in the Newton-Krylov solution process for optimal control problems constrained by semilinear elliptic equations. We analyze the preconditioners both theoretically and numerically. In this work we show that the multigrid preconditioners are of optimal order (p = 2) for piecewise linear finite element approximations. We also study control-constrained linear-elliptic constrained optimization problems. This problem is nontrivial from the optimization point of view as the KKT system is now a complementarity system. We employ semismooth Newton methods to solve this problem efficiently. The multigrid preconditioners discussed here are extensions of preconditioners developed previously for the unconstrained case. We present some new results and techniques that yield optimal order multigrid preconditioners, at least for the case when the control is discretized using piecewise-constant finite elements.
semilinear elliptic PDEs
Abstract:
Partial differential equations (PDEs) constrained optimization is a research area in which the scientific and engineering communities have developed a growing interest over the last decade. The recent rise in interest was fostered by the tremendous increase in computing power over the last twenty years. This can be attributed both to the tremendous advances in high-performance computing technologies and to its wide range of applicability. However, just growth in computing power is insufficient for tackling large-scale PDE-constrained optimization problems and there is always an ever increasing need for efficient algorithms. The objective of this dissertation is to develop, analyze, and implement multigrid preconditioners for the preconditioning of the linear systems arising in the Newton-Krylov solution process for optimal control problems constrained by semilinear elliptic equations. We analyze the preconditioners both theoretically and numerically. In this work we show that the multigrid preconditioners are of optimal order (p = 2) for piecewise linear finite element approximations. We also study control-constrained linear-elliptic constrained optimization problems. This problem is nontrivial from the optimization point of view as the KKT system is now a complementarity system. We employ semismooth Newton methods to solve this problem efficiently. The multigrid preconditioners discussed here are extensions of preconditioners developed previously for the unconstrained case. We present some new results and techniques that yield optimal order multigrid preconditioners, at least for the case when the control is discretized using piecewise-constant finite elements.
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