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

We 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...

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Detalles Bibliográficos
Autores: Biccari, Umberto, Song, Yongcun, Yuan, Xiaoming, Zuazua Iriondo, Enrique
Tipo de recurso: artículo
Fecha de publicación:2023
País:España
Institución:Universidad Autónoma de Madrid
Repositorio:Biblos-e Archivo. Repositorio Institucional de la UAM
Idioma:inglés
OAI Identifier:oai:repositorio.uam.es:10486/713245
Acceso en línea:http://hdl.handle.net/10486/713245
https://dx.doi.org/10.1088/1361-6420/ace548
Access Level:acceso abierto
Palabra clave:Diffusion–advection equations
initial source identification
inverse problem
non-smooth optimization
optimal control
primal-dual algorithm
sparse control
Matemáticas
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spelling A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equationBiccari, UmbertoSong, YongcunYuan, XiaomingZuazua Iriondo, EnriqueDiffusion–advection equationsinitial source identificationinverse problemnon-smooth optimizationoptimal controlprimal-dual algorithmsparse controlMatemáticasWe 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 includedThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No: 694126-DyCon). The work of U B and E Z is partially supported by the Grant PID2020-112617GB-C22 KILEARN of MINECO (Spain) and the Elkartek Grant KK-2020/00091 CONVADP of the Basque Government. E Z has been funded by the Alexander von Humboldt-Professorship program, the ModConFlex Marie Curie Action, HORIZON-MSCA-2021-DN-01, the COST Action MAT-DYN-NET, the Tran- sregio 154 Project “Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks” of the DFG, grants PID2020-112617GB-C22 and TED2021-131390B-I00 of MINECO (Spain), and by the Madrid Goverment – UAM Agreement for the Excellence of the University Research Staff in the context of the V PRICIT (Regional Programme of Research and Technological Innovation). The work of: X Y is supported by Seed Fund for Basic Research (Project Number: 202011159106) from The University of Hong KongInstitute of PhysicsDepartamento de MatemáticasFacultad de Ciencias20232023-09-01research articlehttp://purl.org/coar/resource_type/c_2df8fbb1VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10486/713245https://dx.doi.org/10.1088/1361-6420/ace548reponame:Biblos-e Archivo. Repositorio Institucional de la UAMinstname:Universidad Autónoma de MadridInglésengEuropean Commission http://dx.doi.org/10.13039/501100000780 Horizon 2020 Framework Programme 694126-DyConopen accesshttp://purl.org/coar/access_right/c_abf2Attribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:repositorio.uam.es:10486/7132452026-06-23T12:46:27Z
dc.title.none.fl_str_mv A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation
title A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation
spellingShingle A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation
Biccari, Umberto
Diffusion–advection equations
initial source identification
inverse problem
non-smooth optimization
optimal control
primal-dual algorithm
sparse control
Matemáticas
title_short A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation
title_full A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation
title_fullStr A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation
title_full_unstemmed A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation
title_sort A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation
dc.creator.none.fl_str_mv Biccari, Umberto
Song, Yongcun
Yuan, Xiaoming
Zuazua Iriondo, Enrique
author Biccari, Umberto
author_facet Biccari, Umberto
Song, Yongcun
Yuan, Xiaoming
Zuazua Iriondo, Enrique
author_role author
author2 Song, Yongcun
Yuan, Xiaoming
Zuazua Iriondo, Enrique
author2_role author
author
author
dc.contributor.none.fl_str_mv Departamento de Matemáticas
Facultad de Ciencias
dc.subject.none.fl_str_mv Diffusion–advection equations
initial source identification
inverse problem
non-smooth optimization
optimal control
primal-dual algorithm
sparse control
Matemáticas
topic Diffusion–advection equations
initial source identification
inverse problem
non-smooth optimization
optimal control
primal-dual algorithm
sparse control
Matemáticas
description We 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
publishDate 2023
dc.date.none.fl_str_mv 2023
2023-09-01
dc.type.none.fl_str_mv research article
http://purl.org/coar/resource_type/c_2df8fbb1
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10486/713245
https://dx.doi.org/10.1088/1361-6420/ace548
url http://hdl.handle.net/10486/713245
https://dx.doi.org/10.1088/1361-6420/ace548
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv European Commission http://dx.doi.org/10.13039/501100000780 Horizon 2020 Framework Programme 694126-DyCon
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Institute of Physics
publisher.none.fl_str_mv Institute of Physics
dc.source.none.fl_str_mv reponame:Biblos-e Archivo. Repositorio Institucional de la UAM
instname:Universidad Autónoma de Madrid
instname_str Universidad Autónoma de Madrid
reponame_str Biblos-e Archivo. Repositorio Institucional de la UAM
collection Biblos-e Archivo. Repositorio Institucional de la UAM
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,198674

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