Spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem

We first develop an efficient spectral-Galerkin method and a rigorous error analysis for the generalized eigenvalue problems associated to a transmission eigenvalue problem. Then, we present an iterative scheme, based on computation of the first transmission eigenvalue, to estimate the index of refraction of an inhomogeneous medium. We present ample numerical results to demonstrate the effectiveness and accuracy of our approach.

[1]  H. Haddar,et al.  On the existence of transmission eigenvalues in an inhomogeneous medium , 2009 .

[2]  Fioralba Cakoni,et al.  Qualitative Methods in Inverse Scattering Theory: An Introduction , 2005 .

[3]  Jie Shen,et al.  Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials , 1994, SIAM J. Sci. Comput..

[4]  Houssem Haddar,et al.  The interior transmission problem for anisotropic Maxwell's equations and its applications to the inverse problem , 2004 .

[5]  Peter Monk,et al.  On the use of transmission eigenvalues to estimate the index of refraction from far field data , 2007 .

[6]  Jie Shen,et al.  Spectral and High-Order Methods with Applications , 2006 .

[7]  D. Colton,et al.  Transmission eigenvalues and the nondestructive testing of dielectrics , 2008 .

[8]  Jiguang Sun,et al.  Estimation of transmission eigenvalues and the index of refraction from Cauchy data , 2010 .

[9]  P. Oswald,et al.  Hierarchical conforming finite element methods for the biharmonic equation , 1992 .

[10]  Xia Ji,et al.  Algorithm 922: A Mixed Finite Element Method for Helmholtz Transmission Eigenvalues , 2012, TOMS.

[11]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[12]  Jie Shen,et al.  Efficient Spectral-Galerkin Methods III: Polar and Cylindrical Geometries , 1997, SIAM J. Sci. Comput..

[13]  Peter Hähner,et al.  On the uniqueness of the shape of a penetrable, anisotropic obstacle , 2000 .

[14]  Fioralba Cakoni,et al.  On the determination of Dirichlet or transmission eigenvalues from far field data , 2010 .

[15]  D. Colton,et al.  Analytical and computational methods for transmission eigenvalues , 2010 .

[16]  D. Colton,et al.  The inverse electromagnetic scattering problem for anisotropic media , 2010 .

[17]  Fioralba Cakoni,et al.  The Existence of an Infinite Discrete Set of Transmission Eigenvalues , 2010, SIAM J. Math. Anal..

[18]  Jiguang Sun,et al.  Error Analysis for the Finite Element Approximation of Transmission Eigenvalues , 2014, Comput. Methods Appl. Math..

[19]  Jie Shen,et al.  A Spectral-Element Method for Transmission Eigenvalue Problems , 2013, J. Sci. Comput..

[20]  Gene H. Golub,et al.  Matrix computations , 1983 .

[21]  Bryan P. Rynne,et al.  The interior transmission problem and inverse scattering from inhomogeneous media , 1991 .

[22]  Efficient Spectral Methods for Transmission Eigenvalues and Estimation of the Index of Refraction , 2014 .

[23]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[24]  A. Kirsch.  On the existence of transmission eigenvalues , 2009 .

[25]  D. Colton,et al.  The interior transmission problem , 2007 .

[26]  Fioralba Cakoni,et al.  Transmission Eigenvalues , 2021, Applied Mathematical Sciences.

[27]  D. Colton,et al.  The computation of lower bounds for the norm of the index of refraction in an anisotropic media from far field data , 2009 .

[28]  Jie Shen,et al.  Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials , 2006, J. Sci. Comput..

[29]  Jiguang Sun.  Iterative Methods for Transmission Eigenvalues , 2011, SIAM J. Numer. Anal..