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

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dc.contributor.authorBiccari, Umberto
dc.contributor.authorSong, Yongcun
dc.contributor.authorYuan, Xiaoming
dc.contributor.authorZuazua, Enrique
dc.date.accessioned2024-12-12T10:40:22Z
dc.date.available2024-12-12T10:40:22Z
dc.date.issued2023-09
dc.date.updated2024-12-12T10:40:22Z
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.en
dc.description.sponsorshipThis 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 Kong.en
dc.identifier.citationBiccari, U., Song, Y., Yuan, X., & Zuazua, E. (2023). A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation. Inverse Problems, 39(9). https://doi.org/10.1088/1361-6420/ACE548
dc.identifier.doi10.1088/1361-6420/ACE548
dc.identifier.eissn1361-6420
dc.identifier.issn0266-5611
dc.identifier.urihttp://hdl.handle.net/20.500.14454/2159
dc.language.isoeng
dc.publisherInstitute of Physics
dc.rights© 2023 The Author(s)
dc.subject.otherDiffusion–advection equations
dc.subject.otherInitial source identification
dc.subject.otherInverse problem
dc.subject.otherNon-smooth optimization
dc.subject.otherOptimal control
dc.subject.otherPrimal-dual algorithm
dc.subject.otherSparse control
dc.titleA two-stage numerical approach for the sparse initial source identification of a diffusion–advection equationen
dc.typejournal article
dcterms.accessRightsopen access
oaire.citation.issue9
oaire.citation.titleInverse Problems
oaire.citation.volume39
oaire.licenseConditionhttps://creativecommons.org/licenses/by/4.0/
oaire.versionVoR
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