Publication
Huygens’ principle and iterative methods in inverse obstacle scattering
| datacite.subject.fos | Ciências Naturais::Matemáticas | |
| datacite.subject.sdg | 03:Saúde de Qualidade | |
| datacite.subject.sdg | 07:Energias Renováveis e Acessíveis | |
| datacite.subject.sdg | 11:Cidades e Comunidades Sustentáveis | |
| dc.contributor.author | Ivanyshyn, Olha | |
| dc.contributor.author | Kress, Rainer | |
| dc.contributor.author | Serranho, Pedro | |
| dc.date.accessioned | 2025-11-25T18:34:07Z | |
| dc.date.available | 2025-11-25T18:34:07Z | |
| dc.date.issued | 2009-07-22 | |
| dc.description.abstract | The inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens' principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i. e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens' principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations. | eng |
| dc.description.sponsorship | The research of O.I. was partially supported by the German Research Foundation DFG through the collaborative research center SFB 755. The research of P.S. was partially supported by the Calouste Gulbenkian Foundation. | |
| dc.identifier.citation | Ivanyshyn, O., Kress, R. & Serranho, P. Huygens’ principle and iterative methods in inverse obstacle scattering. Adv Comput Math 33, 413–429 (2010). https://doi.org/10.1007/s10444-009-9135-6. | |
| dc.identifier.doi | 10.1007/s10444-009-9135-6 | |
| dc.identifier.eissn | 1572-9044 | |
| dc.identifier.issn | 1019-7168 | |
| dc.identifier.uri | http://hdl.handle.net/10400.8/14726 | |
| dc.language.iso | eng | |
| dc.peerreviewed | yes | |
| dc.publisher | Springer Nature | |
| dc.relation.hasversion | https://link.springer.com/article/10.1007/s10444-009-9135-6 | |
| dc.relation.ispartof | Advances in Computational Mathematics | |
| dc.rights.uri | http://creativecommons.org/licenses/by-nc/4.0/ | |
| dc.subject | Inverse scattering | |
| dc.subject | Nonlinear integral equations | |
| dc.subject | Sound-soft obstacle | |
| dc.title | Huygens’ principle and iterative methods in inverse obstacle scattering | eng |
| dc.type | journal article | |
| dspace.entity.type | Publication | |
| oaire.citation.endPage | 429 | |
| oaire.citation.startPage | 413 | |
| oaire.citation.title | Advances in Computational Mathematics | |
| oaire.citation.volume | 33 | |
| oaire.version | http://purl.org/coar/version/c_970fb48d4fbd8a85 | |
| person.familyName | Serranho | |
| person.givenName | Pedro | |
| person.identifier.orcid | 0000-0003-2176-3923 | |
| relation.isAuthorOfPublication | dac740d0-c72f-4bf6-95b7-e229bb0471df | |
| relation.isAuthorOfPublication.latestForDiscovery | dac740d0-c72f-4bf6-95b7-e229bb0471df |
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- The inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens' principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i. e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens' principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations.
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