Adaptive Multilevel Interior-Point Methods in PDE Constrained Optimization

  • We are concerned with structural optimization problems where the state variables are supposed to satisfy a PDE or a system of PDEs and the design variables are subject to inequality constraints. Within a primal-dual setting, we suggest an all-at-once approach based on interior-point methods. Coupling the inequality constraints by logarithmic barrier functions involving a barrier parameter and the PDE by Lagrange multipliers, the KKT conditions for the resulting saddle point problem represent a parameter dependent nonlinear system. The efficient numerical solution relies on multilevel path-following predictor-corrector techniques with an adaptive choice of the continuation parameter where the discretization is taken care of by finite elements with respect to nested hierarchies of simplicial triangulations of the computational domain. In particular, the predictor is a nested iteration type tangent continuation, whereas the corrector is a multilevel inexact Newton method featuringWe are concerned with structural optimization problems where the state variables are supposed to satisfy a PDE or a system of PDEs and the design variables are subject to inequality constraints. Within a primal-dual setting, we suggest an all-at-once approach based on interior-point methods. Coupling the inequality constraints by logarithmic barrier functions involving a barrier parameter and the PDE by Lagrange multipliers, the KKT conditions for the resulting saddle point problem represent a parameter dependent nonlinear system. The efficient numerical solution relies on multilevel path-following predictor-corrector techniques with an adaptive choice of the continuation parameter where the discretization is taken care of by finite elements with respect to nested hierarchies of simplicial triangulations of the computational domain. In particular, the predictor is a nested iteration type tangent continuation, whereas the corrector is a multilevel inexact Newton method featuring transforming null space iterations. As an application in life sciences, we consider the optimal shape design of capillary barriers in microfluidic biochips.show moreshow less

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Metadaten
Author:Harbir AntilGND, Ronald H. W. HoppeGND, Christopher LinsenmannGND
URN:urn:nbn:de:bvb:384-opus4-5203
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/659
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2008-19)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Contributing Corporation:University of Houston
Release Date:2008/04/30
Tag:shape optmization; interior-point methods; adaptive path-following; multilevel methods
GND-Keyword:Gestaltoptimierung; Innere-Punkte-Methode; Partielle Differentialgleichung; Biochip; Mikrofluidik
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik