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Article: A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation

TitleA two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation
Authors
Keywordsdiffusion–advection equations
initial source identification
inverse problem
non-smooth optimization
optimal control
primal-dual algorithm
sparse control
Issue Date1-Sep-2023
PublisherIOP Publishing
Citation
Inverse Problems, 2023, v. 39, n. 9 How to Cite?
AbstractWe consider the problem of identifying a sparse initial source condition to achieve a given state distribution of a diffusion–advection partial differential equation after a given final time. The initial condition is assumed to be a finite combination of Dirac measures. The locations and intensities of this initial condition are required to be identified. This problem is known to be exponentially ill-posed because of the strong diffusive and smoothing effects. We propose a two-stage numerical approach to treat this problem. At the first stage, to obtain a sparse initial condition with the desire of achieving the given state subject to a certain tolerance, we propose an optimal control problem involving sparsity-promoting and ill-posedness-avoiding terms in the cost functional, and introduce a generalized primal-dual algorithm for this optimal control problem. At the second stage, the initial condition obtained from the optimal control problem is further enhanced by identifying its locations and intensities in its representation of the combination of Dirac measures. This two-stage numerical approach is shown to be easily implementable and its efficiency in short time horizons is promisingly validated by the results of numerical experiments. Some discussions on long time horizons are also included.
Persistent Identifierhttp://hdl.handle.net/10722/348357
ISSN
2023 Impact Factor: 2.0
2023 SCImago Journal Rankings: 1.185

 

DC FieldValueLanguage
dc.contributor.authorBiccari, Umberto-
dc.contributor.authorSong, Yongcun-
dc.contributor.authorYuan, Xiaoming-
dc.contributor.authorZuazua, Enrique-
dc.date.accessioned2024-10-09T00:30:59Z-
dc.date.available2024-10-09T00:30:59Z-
dc.date.issued2023-09-01-
dc.identifier.citationInverse Problems, 2023, v. 39, n. 9-
dc.identifier.issn0266-5611-
dc.identifier.urihttp://hdl.handle.net/10722/348357-
dc.description.abstractWe consider the problem of identifying a sparse initial source condition to achieve a given state distribution of a diffusion–advection partial differential equation after a given final time. The initial condition is assumed to be a finite combination of Dirac measures. The locations and intensities of this initial condition are required to be identified. This problem is known to be exponentially ill-posed because of the strong diffusive and smoothing effects. We propose a two-stage numerical approach to treat this problem. At the first stage, to obtain a sparse initial condition with the desire of achieving the given state subject to a certain tolerance, we propose an optimal control problem involving sparsity-promoting and ill-posedness-avoiding terms in the cost functional, and introduce a generalized primal-dual algorithm for this optimal control problem. At the second stage, the initial condition obtained from the optimal control problem is further enhanced by identifying its locations and intensities in its representation of the combination of Dirac measures. This two-stage numerical approach is shown to be easily implementable and its efficiency in short time horizons is promisingly validated by the results of numerical experiments. Some discussions on long time horizons are also included.-
dc.languageeng-
dc.publisherIOP Publishing-
dc.relation.ispartofInverse Problems-
dc.rightsThis work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.-
dc.subjectdiffusion–advection equations-
dc.subjectinitial source identification-
dc.subjectinverse problem-
dc.subjectnon-smooth optimization-
dc.subjectoptimal control-
dc.subjectprimal-dual algorithm-
dc.subjectsparse control-
dc.titleA two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation-
dc.typeArticle-
dc.identifier.doi10.1088/1361-6420/ace548-
dc.identifier.scopuseid_2-s2.0-85167509557-
dc.identifier.volume39-
dc.identifier.issue9-
dc.identifier.eissn1361-6420-
dc.identifier.issnl0266-5611-

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