{"title":"具有高阶跳跃条件的薄表面的光谱目标特征","authors":"F. Cakoni, Heejin Lee, P. Monk, Yangwen Zhang","doi":"10.3934/ipi.2022020","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in <inline-formula><tex-math id=\"M1\">\\begin{document}$ {\\mathbb R}^m $\\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M2\">\\begin{document}$ m = 2, 3 $\\end{document}</tex-math></inline-formula> from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open <inline-formula><tex-math id=\"M3\">\\begin{document}$ m-1 $\\end{document}</tex-math></inline-formula> dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [<xref ref-type=\"bibr\" rid=\"b20\">20</xref>]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":" ","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A spectral target signature for thin surfaces with higher order jump conditions\",\"authors\":\"F. Cakoni, Heejin Lee, P. Monk, Yangwen Zhang\",\"doi\":\"10.3934/ipi.2022020\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p style='text-indent:20px;'>In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in <inline-formula><tex-math id=\\\"M1\\\">\\\\begin{document}$ {\\\\mathbb R}^m $\\\\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\\\"M2\\\">\\\\begin{document}$ m = 2, 3 $\\\\end{document}</tex-math></inline-formula> from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open <inline-formula><tex-math id=\\\"M3\\\">\\\\begin{document}$ m-1 $\\\\end{document}</tex-math></inline-formula> dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [<xref ref-type=\\\"bibr\\\" rid=\\\"b20\\\">20</xref>]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach</p>\",\"PeriodicalId\":50274,\"journal\":{\"name\":\"Inverse Problems and Imaging\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2022-01-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Inverse Problems and Imaging\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3934/ipi.2022020\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Inverse Problems and Imaging","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/ipi.2022020","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
A spectral target signature for thin surfaces with higher order jump conditions
In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in \begin{document}$ {\mathbb R}^m $\end{document}, \begin{document}$ m = 2, 3 $\end{document} from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open \begin{document}$ m-1 $\end{document} dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [20]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach
期刊介绍:
Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing.
This journal is committed to recording important new results in its field and will maintain the highest standards of innovation and quality. To be published in this journal, a paper must be correct, novel, nontrivial and of interest to a substantial number of researchers and readers.