Abstract
We consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides whether the region contains eigenvalue(s) or not. It is particularly suitable to test whether zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.
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References
An J, Shen J. A Fourier-spectral-element method for transmission eigenvalue problems. J Sci Comput, 2013, 57: 670–688
Austin A P, Kravanja P, Trefethen L N. Numerical algorithms based on analytic function values at roots of unity. SIAM J Numer Anal, 2014, 52: 1795–1821
Beyn W J. An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl, 2012, 436: 3839–3863
Cakoni F, Colton D, Monk P, et al. The inverse electromagnetic scattering problem for anisotropic media. Inverse Problems, 2010, 26: 074004
Cakoni F, Monk P, Sun J. Error analysis of the finite element approximation of transmission eigenvalues. Comput Methods Appl Math, 2014, 14: 419–427
Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed. New York: Springer-Verlag, 2013
Colton D, Monk P, Sun J. Analytical and computational methods for transmission eigenvalues. Inverse Problems, 2010, 26: 045011
Cossonnière A. Valeurs propres de transmission et leur utilisation dans l’identification d’inclusions à partir de mesures électromagnétiques. PhD Thesis. Toulouse: Université de Toulouse, 2011
Cossonnière A, Haddar H. Surface integral formulation of the interior transmission problem. J Integral Equations Appl, 2013, 25: 341–376
Gintides D, Pallikarakis N. A computational method for the inverse transmission eigenvalue problem. Inverse Problems, 2013, 29: 104010
Goedecker S. Linear scaling electronic structure methods. Rev Modern Phys, 1999, 71: 1085–1123
Hsiao G, Liu F, Sun J, et al. A coupled BEM and FEM for the interior transmission problem in acoustics. J Comp Appl Math, 2011, 235: 5213–5221
Hsiao G C, Xu L. A system of boundary integral equations for the transmission problem in acoustics. Appl Num Math, 2011, 61: 1017–1029
Huang R, Struthers A, Sun J, et al. Recursive integral method for transmission eigenvalues. ArXiv:1503.04741, 2015
Huang T, Huang W, Lin W. A robust numerical algorithm for computing maxwell’s transmission eigenvalue problems. SIAM J Sci Comput, 2015, 37: A2403–A2423
Ji X, Sun J. A multi-level method for transmission eigenvalues of anisotropic media. J Comput Phys, 2013, 255: 422–435
Ji X, Sun J, Turner T. A mixed finite element method for Helmholtz Transmission eigenvalues. ACM Trans Math Software, 2012, 38: Algorithm 922
Ji X, Sun J, Xie H. A multigrid method for Helmholtz transmission eigenvalue problems. J Sci Comput, 2014, 60: 276–294
Kato T. Perturbation Theory of Linear Operators. New York: Springer-Verlag, 1966
Kleefeld A. A numerical method to compute interior transmission eigenvalues. Inverse Problems, 2013, 29: 104012
Krämer L, Di Napoli E, Galgon M, et al. Dissecting the FEAST algorithm for generalized eigenproblems. J Comput Appl Math, 2013, 244: 1–9
Li T, Huang W, Lin W W, et al. On spectral analysis and a novel algorithm for transmission eigenvalue problems. J Sci Comput, 2015, 64: 83–108
Olver F, Lozier D, Boisvert R, et al. NIST Handbook of Mathematical Functions. Cambridge: Cambridge University Press, 2010
Osborn J. Spectral approximation for compact operators. Math Comp, 1975, 29: 712–725
Polizzi E. Density-matrix-based algorithms for solving eigenvalue problems. Phys Rev B, 2009, 79: 115112
Sakurai T, Sugiura H. A projection method for generalized eigenvalue problems using numerical integration. J Comput Appl Math, 2003, 159: 119–128
Sauter S, Schwab C. Boundary Element Methods. Berlin: Springer, 2011
Sun J. Iterative methods for transmission eigenvalues. SIAM J Numer Anal, 2011, 49: 1860–1874
Sun J, Xu L. Computation of the Maxwell’s transmission eigenvalues and its application in inverse medium problems. Inverse Problems, 2013, 29: 104013
Tang P, Polizzi E. FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection. SIAM J Matrix Anal Appl, 2014, 35: 354–390
Yang Y, Han J, Bi H. Non-conforming finite element methods for transmission eigenvalue problem. ArXiv:1601.01068, 2016
Yin G. A contour-integral based method for counting the eigenvalues inside a region in the complex plane. ArXiv:1503.05035, 2015
Yin G, Chan R, Yeung M. A FEAST algorithm with oblique projection for generalized eigenvalue problems. ArXiv:1404.1768, 2014
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Zeng, F., Sun, J. & Xu, L. A spectral projection method for transmission eigenvalues. Sci. China Math. 59, 1613–1622 (2016). https://doi.org/10.1007/s11425-016-0289-8
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DOI: https://doi.org/10.1007/s11425-016-0289-8