A robust finite element solver for a multiharmonic parabolic optimal control problem
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| Title: | A robust finite element solver for a multiharmonic parabolic optimal control problem |
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| Authors: | Kollmann, M.1 markus.kollmann@dk-compmath.jku.at, Kolmbauer, M.2 kolmbauer@numa.uni-linz.ac.at, Langer, U.2 ulanger@numa.uni-linz.ac.at, Wolfmayr, M.2 monika.wolfmayr@numa.uni-linz.ac.at, Zulehner, W.2 zulehner@numa.uni-linz.ac.at |
| Source: | Computers & Mathematics with Applications. Feb2013, Vol. 65 Issue 3, p469-486. 18p. |
| Subjects: | Finite element method data processing, Optimal control theory, Coupled mode theory (Wave-motion), Parabolic differential equations, Coefficients (Statistics), Fourier series |
| Abstract: | Abstract: This paper presents the analysis of a distributed parabolic optimal control problem in a multiharmonic setting. In particular, the desired state is assumed to be multiharmonic. After eliminating the control from the optimality system, we arrive at the reduced optimality system for the state and the co-state that is nothing but a coupled system of a forward and a backward parabolic partial differential equation. Due to the linearity, the state and the co-state are multiharmonic as well. We discretize the Fourier coefficients by the finite element method. This leads to a large system of algebraic equations, which fortunately decouples into smaller systems each of them defining the cosine and sine Fourier coefficients for the state and co-state with respect to a single frequency. For these smaller systems, we construct preconditioners resulting in a fast converging minimal residual solver with a parameter-independent convergence rate. All these systems can be solved totally in parallel. [Copyright &y& Elsevier] |
| Copyright of Computers & Mathematics with Applications is the property of Pergamon Press - An Imprint of Elsevier Science and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 85008839 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: A robust finite element solver for a multiharmonic parabolic optimal control problem – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Kollmann%2C+M%2E%22">Kollmann, M.</searchLink><relatesTo>1</relatesTo><i> markus.kollmann@dk-compmath.jku.at</i><br /><searchLink fieldCode="AR" term="%22Kolmbauer%2C+M%2E%22">Kolmbauer, M.</searchLink><relatesTo>2</relatesTo><i> kolmbauer@numa.uni-linz.ac.at</i><br /><searchLink fieldCode="AR" term="%22Langer%2C+U%2E%22">Langer, U.</searchLink><relatesTo>2</relatesTo><i> ulanger@numa.uni-linz.ac.at</i><br /><searchLink fieldCode="AR" term="%22Wolfmayr%2C+M%2E%22">Wolfmayr, M.</searchLink><relatesTo>2</relatesTo><i> monika.wolfmayr@numa.uni-linz.ac.at</i><br /><searchLink fieldCode="AR" term="%22Zulehner%2C+W%2E%22">Zulehner, W.</searchLink><relatesTo>2</relatesTo><i> zulehner@numa.uni-linz.ac.at</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Computers+%26+Mathematics+with+Applications%22">Computers & Mathematics with Applications</searchLink>. Feb2013, Vol. 65 Issue 3, p469-486. 18p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Finite+element+method+data+processing%22">Finite element method data processing</searchLink><br /><searchLink fieldCode="DE" term="%22Optimal+control+theory%22">Optimal control theory</searchLink><br /><searchLink fieldCode="DE" term="%22Coupled+mode+theory+%28Wave-motion%29%22">Coupled mode theory (Wave-motion)</searchLink><br /><searchLink fieldCode="DE" term="%22Parabolic+differential+equations%22">Parabolic differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Coefficients+%28Statistics%29%22">Coefficients (Statistics)</searchLink><br /><searchLink fieldCode="DE" term="%22Fourier+series%22">Fourier series</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Abstract: This paper presents the analysis of a distributed parabolic optimal control problem in a multiharmonic setting. In particular, the desired state is assumed to be multiharmonic. After eliminating the control from the optimality system, we arrive at the reduced optimality system for the state and the co-state that is nothing but a coupled system of a forward and a backward parabolic partial differential equation. Due to the linearity, the state and the co-state are multiharmonic as well. We discretize the Fourier coefficients by the finite element method. This leads to a large system of algebraic equations, which fortunately decouples into smaller systems each of them defining the cosine and sine Fourier coefficients for the state and co-state with respect to a single frequency. For these smaller systems, we construct preconditioners resulting in a fast converging minimal residual solver with a parameter-independent convergence rate. All these systems can be solved totally in parallel. [Copyright &y& Elsevier] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Computers & Mathematics with Applications is the property of Pergamon Press - An Imprint of Elsevier Science and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.camwa.2012.06.012 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 18 StartPage: 469 Subjects: – SubjectFull: Finite element method data processing Type: general – SubjectFull: Optimal control theory Type: general – SubjectFull: Coupled mode theory (Wave-motion) Type: general – SubjectFull: Parabolic differential equations Type: general – SubjectFull: Coefficients (Statistics) Type: general – SubjectFull: Fourier series Type: general Titles: – TitleFull: A robust finite element solver for a multiharmonic parabolic optimal control problem Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Kollmann, M. – PersonEntity: Name: NameFull: Kolmbauer, M. – PersonEntity: Name: NameFull: Langer, U. – PersonEntity: Name: NameFull: Wolfmayr, M. – PersonEntity: Name: NameFull: Zulehner, W. IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 02 Text: Feb2013 Type: published Y: 2013 Identifiers: – Type: issn-print Value: 08981221 Numbering: – Type: volume Value: 65 – Type: issue Value: 3 Titles: – TitleFull: Computers & Mathematics with Applications Type: main |
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