{"title":"反弹性空腔散射的因式分解法","authors":"Shuxin Li, Junliang Lv, Yi Wang","doi":"arxiv-2409.09434","DOIUrl":null,"url":null,"abstract":"This paper is concerned with the inverse elastic scattering problem to\ndetermine the shape and location of an elastic cavity. By establishing a\none-to-one correspondence between the Herglotz wave function and its kernel, we\nintroduce the far-field operator which is crucial in the factorization method.\nWe present a theoretical factorization of the far-field operator and rigorously\nprove the properties of its associated operators involved in the factorization.\nUnlike the Dirichlet problem where the boundary integral operator of the\nsingle-layer potential involved in the factorization of the far-field operator\nis weakly singular, the boundary integral operator of the conormal derivative\nof the double-layer potential involved in the factorization of the far-field\noperator with Neumann boundary conditions is hypersingular, which forces us to\nprove that this operator is isomorphic using Fredholm's theorem. Meanwhile, we\npresent theoretical analyses of the factorization method for various\nillumination and measurement cases, including compression-wave illumination and\ncompression-wave measurement, shear-wave illumination and shear-wave\nmeasurement, and full-wave illumination and full-wave measurement. In addition,\nwe also consider the limited aperture problem and provide a rigorous\ntheoretical analysis of the factorization method in this case. Numerous\nnumerical experiments are carried out to demonstrate the effectiveness of the\nproposed method, and to analyze the influence of various factors, such as\npolarization direction, frequency, wavenumber, and multi-scale scatterers on\nthe reconstructed results.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"204 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Factorization method for inverse elastic cavity scattering\",\"authors\":\"Shuxin Li, Junliang Lv, Yi Wang\",\"doi\":\"arxiv-2409.09434\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper is concerned with the inverse elastic scattering problem to\\ndetermine the shape and location of an elastic cavity. By establishing a\\none-to-one correspondence between the Herglotz wave function and its kernel, we\\nintroduce the far-field operator which is crucial in the factorization method.\\nWe present a theoretical factorization of the far-field operator and rigorously\\nprove the properties of its associated operators involved in the factorization.\\nUnlike the Dirichlet problem where the boundary integral operator of the\\nsingle-layer potential involved in the factorization of the far-field operator\\nis weakly singular, the boundary integral operator of the conormal derivative\\nof the double-layer potential involved in the factorization of the far-field\\noperator with Neumann boundary conditions is hypersingular, which forces us to\\nprove that this operator is isomorphic using Fredholm's theorem. Meanwhile, we\\npresent theoretical analyses of the factorization method for various\\nillumination and measurement cases, including compression-wave illumination and\\ncompression-wave measurement, shear-wave illumination and shear-wave\\nmeasurement, and full-wave illumination and full-wave measurement. In addition,\\nwe also consider the limited aperture problem and provide a rigorous\\ntheoretical analysis of the factorization method in this case. Numerous\\nnumerical experiments are carried out to demonstrate the effectiveness of the\\nproposed method, and to analyze the influence of various factors, such as\\npolarization direction, frequency, wavenumber, and multi-scale scatterers on\\nthe reconstructed results.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"204 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.09434\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09434","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Factorization method for inverse elastic cavity scattering
This paper is concerned with the inverse elastic scattering problem to
determine the shape and location of an elastic cavity. By establishing a
one-to-one correspondence between the Herglotz wave function and its kernel, we
introduce the far-field operator which is crucial in the factorization method.
We present a theoretical factorization of the far-field operator and rigorously
prove the properties of its associated operators involved in the factorization.
Unlike the Dirichlet problem where the boundary integral operator of the
single-layer potential involved in the factorization of the far-field operator
is weakly singular, the boundary integral operator of the conormal derivative
of the double-layer potential involved in the factorization of the far-field
operator with Neumann boundary conditions is hypersingular, which forces us to
prove that this operator is isomorphic using Fredholm's theorem. Meanwhile, we
present theoretical analyses of the factorization method for various
illumination and measurement cases, including compression-wave illumination and
compression-wave measurement, shear-wave illumination and shear-wave
measurement, and full-wave illumination and full-wave measurement. In addition,
we also consider the limited aperture problem and provide a rigorous
theoretical analysis of the factorization method in this case. Numerous
numerical experiments are carried out to demonstrate the effectiveness of the
proposed method, and to analyze the influence of various factors, such as
polarization direction, frequency, wavenumber, and multi-scale scatterers on
the reconstructed results.