RB-Based PDE-Constrained Non-Smooth Optimization

Bernreuther, Marco

RB-Based PDE-Constrained Non-Smooth Optimization

Dateien

Bernreuther_2-t4k1djyj77yn3.pdfGröße: 1.86 MBDownloads: 385

Datum

2019

Open Access-Veröffentlichung

Open Access Green

Sammlungen

Mathematik und Statistik

Publikationstyp

Masterarbeit/Diplomarbeit

Publikationsstatus

Published

Zusammenfassung

We investigate the Reduced Basis (RB) method for a semilinear, non-smooth, parameter dependent, elliptic PDE with a max-type response term, the so called max-PDE. Thereby, we first show the existence and uniqueness of so- lutions to the max-PDE with the help of monotone operator theory, as well as the Lipschitz continuity and compactness of the solution operator. Based on that we introduce a RB-greedy method, analyze its convergence behav- ior, introduce error estimates and an a posteriori error estimator. To solve the nonlinear equation on Finite Element (FE) and RB-level, a semismooth Newton method is used. The necessary theory on subdifferentials and semis- moothness is introduced and subsequently locally quadratic convergence of the semismooth Newton method for the max-PDE is shown. Additionally, the Discrete Empirical Interpolation method (DEIM) is used to approximate the nonlinearity in the RB-system. As application optimization problems con- strained by the parameter dependent max-PDE are analyzed. The theoretical results are verified through numerical examples for both the max-PDE itself and the optimization problems. It becomes apparent that RB can be a suitable approach to solve the max-PDE more efficient.

Fachgebiet (DDC)

510 Mathematik

Schlagwörter

non-smooth, semismooth, optimization, reduced basis, RB, max-PDE, DEIM

Zitieren

ISO 690

BERNREUTHER, Marco, 2019. RB-Based PDE-Constrained Non-Smooth Optimization [Master thesis]. Konstanz: Universität Konstanz

BibTex

@mastersthesis{Bernreuther2019RBBas-47244,
  year={2019},
  title={RB-Based PDE-Constrained Non-Smooth Optimization},
  address={Konstanz},
  school={Universität Konstanz},
  author={Bernreuther, Marco}
}

RDF

<rdf:RDF
    xmlns:dcterms="http://purl.org/dc/terms/"
    xmlns:dc="http://purl.org/dc/elements/1.1/"
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
    xmlns:bibo="http://purl.org/ontology/bibo/"
    xmlns:dspace="http://digital-repositories.org/ontologies/dspace/0.1.0#"
    xmlns:foaf="http://xmlns.com/foaf/0.1/"
    xmlns:void="http://rdfs.org/ns/void#"
    xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > 
  <rdf:Description rdf:about="https://kops.uni-konstanz.de/server/rdf/resource/123456789/47244">
    <dcterms:title>RB-Based PDE-Constrained Non-Smooth Optimization</dcterms:title>
    <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/>
    <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dc:rights>terms-of-use</dc:rights>
    <dcterms:abstract xml:lang="eng">We investigate the Reduced Basis (RB) method for a semilinear, non-smooth, parameter dependent, elliptic PDE with a max-type response term, the so called max-PDE. Thereby, we first show the existence and uniqueness of so- lutions to the max-PDE with the help of monotone operator theory, as well as the Lipschitz continuity and compactness of the solution operator. Based on that we introduce a RB-greedy method, analyze its convergence behav- ior, introduce error estimates and an a posteriori error estimator. To solve the nonlinear equation on Finite Element (FE) and RB-level, a semismooth Newton method is used. The necessary theory on subdifferentials and semis- moothness is introduced and subsequently locally quadratic convergence of the semismooth Newton method for the max-PDE is shown. Additionally, the Discrete Empirical Interpolation method (DEIM) is used to approximate the nonlinearity in the RB-system. As application optimization problems con- strained by the parameter dependent max-PDE are analyzed. The theoretical results are verified through numerical examples for both the max-PDE itself and the optimization problems. It becomes apparent that RB can be a suitable approach to solve the max-PDE more efficient.</dcterms:abstract>
    <bibo:uri rdf:resource="https://kops.uni-konstanz.de/handle/123456789/47244"/>
    <dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/>
    <dc:creator>Bernreuther, Marco</dc:creator>
    <dc:language>eng</dc:language>
    <dc:contributor>Bernreuther, Marco</dc:contributor>
    <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2019-10-17T07:12:39Z</dc:date>
    <dcterms:hasPart rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/47244/3/Bernreuther_2-t4k1djyj77yn3.pdf"/>
    <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2019-10-17T07:12:39Z</dcterms:available>
    <foaf:homepage rdf:resource="http://localhost:8080/"/>
    <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dcterms:issued>2019</dcterms:issued>
    <dspace:hasBitstream rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/47244/3/Bernreuther_2-t4k1djyj77yn3.pdf"/>
  </rdf:Description>
</rdf:RDF>

Hochschulschriftenvermerk

Konstanz, Universität Konstanz, Masterarbeit/Diplomarbeit, 2019

Universitätsbibliographie

Ja