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On Spectral Analysis and a Novel Algorithm for Transmission Eigenvalue Problems

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Abstract

The transmission eigenvalue problem, besides its critical role in inverse scattering problems, deserves special interest of its own due to the fact that the corresponding differential operator is neither elliptic nor self-adjoint. In this paper, we provide a spectral analysis and propose a novel iterative algorithm for the computation of a few positive real eigenvalues and the corresponding eigenfunctions of the transmission eigenvalue problem. Based on approximation using continuous finite elements, we first derive an associated symmetric quadratic eigenvalue problem (QEP) for the transmission eigenvalue problem to eliminate the nonphysical zero eigenvalues while preserve all nonzero ones. In addition, the derived QEP enables us to consider more refined discretization to overcome the limitation on the number of degrees of freedom. We then transform the QEP to a parameterized symmetric definite generalized eigenvalue problem (GEP) and develop a secant-type iteration for solving the resulting GEPs. Moreover, we carry out spectral analysis for various existence intervals of desired positive real eigenvalues, since a few lowest positive real transmission eigenvalues are of practical interest in the estimation and the reconstruction of the index of refraction. Numerical experiments show that the proposed method can find those desired smallest positive real transmission eigenvalues accurately, efficiently, and robustly.

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Notes

  1. Here \(\mathcal {O}\) denotes the “big O”.

  2. Since we are only interested in finding the positive real eigenvalues of the QEP (10), we restrict our discussion to the case \(\tau \ge 0\).

References

  1. Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H.: Templates for the Solution of algebraic eigenvalue problems: a practical guide. SIAM, Philadelphia, PA (2000)

    Book  Google Scholar 

  2. Cakoni, F., Çayören, M., Colton, D.: Transmission eigenvalues and the nondestructive testing of dielectrics. Inverse Probl. 24(6), 065016 (2008)

    Article  Google Scholar 

  3. Cakoni, F., Colton, D., Haddar, H.: On the determination of Dirichlet or transmission eigenvalues from far field data. C. R. Math. 348(7–8), 379–383 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cakoni, F., Colton, D., Monk, P.: On the use of transmission eigenvalues to estimate the index of refraction from far field data. Inverse Probl. 23(2), 507–522 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cakoni, F., Colton, D., Monk, P., Sun, J.: The inverse electromagnetic scattering problem for anisotropic media. Inverse Probl. 26(7), 074004 (2010)

    Article  MathSciNet  Google Scholar 

  6. Cakoni, F., Gintides, D., Haddar, H.: The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Anal. 42(1), 237–255 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cakoni, F., Haddar, H.: On the existence of transmission eigenvalues in an inhomogeneous medium. Appl. Anal. 88(4), 475–493 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cakoni, F., Haddar, H.: Transmission eigenvalues in inverse scattering theory. In: Uhlmann, G. (ed.) Inverse problems and applications: inside Out II, MSRI Publications, vol. 60, pp. 527–578. Cambridge University Press (2012)

  9. Colton, D., Kress, R.: Inverse Acoustic and electromagnetic scattering theory, applied mathematical sciences, vol. 93, 3rd edn. Springer, New York (2013)

  10. Colton, D., Monk, P., Sun, J.: Analytical and computational methods for transmission eigenvalues. Inverse Probl. 26(4), 045011 (2010)

    Article  MathSciNet  Google Scholar 

  11. Colton, D., Päivärinta, L., Sylvester, J.: The interior transmission problem. Inverse Probl. Imaging 1(1), 13–28 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Golub, G.H., Van Loan, C.F.: Matrix computations, 4th edn. The Johns Hopkins University Press, Baltimore, MD (2012)

    Google Scholar 

  13. Guo, J.S., Lin, W.W., Wang, C.S.: Numerical solutions for large sparse quadratic eigenvalue problems. Linear Algebra Appl. 225, 57–89 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ji, X., Sun, J., Turner, T.: Algorithm 922: a mixed finite element method for Helmholtz transmission eigenvalues. ACM Trans. Math. Softw. 38(4), 29:1–29:8 (2012)

    Article  MathSciNet  Google Scholar 

  15. Ji, X., Sun, J., Xie, H.: A multigrid method for Helmholtz transmission eigenvalue problems. J. Sci. Comput. 60(2), 276–294 (2014)

  16. Kirsch, A.: On the existence of transmission eigenvalues. Inverse Probl. Imaging 3(2), 155–172 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Moler, C.B., Stewart, G.W.: An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10(2), 241–256 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  18. Monk, P., Sun, J.: Finite element methods for Maxwell’s transmission eigenvalues. SIAM J. Sci. Comput. 34(3), B247–B264 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Päivärinta, L., Sylvester, J.: Transmission eigenvalues. SIAM J. Math. Anal. 40(2), 738–753 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Parlett, B.N.: The symmetric eigenvalue problem. SIAM, Philadelphia, PA (1998)

    Book  MATH  Google Scholar 

  21. Persson, P.O., Strang, G.: A simple mesh generator in MATLAB. SIAM Rev. 46(2), 329–345 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Sun, J.: Estimation of transmission eigenvalues and the index of refraction from Cauchy data. Inverse Probl. 27(1), 015,009 (2011)

    Article  Google Scholar 

  23. Sun, J.: Iterative methods for transmission eigenvalues. SIAM J. Numer. Anal. 49(5), 1860–1874 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43(2), 235–286 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. van der Vorst, H.A.: A generalized Lanczos scheme. Math. Comp. 39(160), 559–561 (1982)

    MATH  MathSciNet  Google Scholar 

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Acknowledgments

The first author is supported by NSFC (No. 11471074) and the Fundamental Research Funds for the Central Universities. The fourth author is supported by NSFC (No. 91330109). The second and third authors would like to acknowledge the Grant support from the Ministry of Science and Technology (W.-Q. Huang: MOST 103-2811-M-009-029; W.-W. Lin: MOST 103-2628-M-009-005) and the National Center for Theoretical Sciences in Taiwan, and they would also like to thank the ST Yau Center at the National Chiao Tung University for the support.

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Tiexiang Li & Jijun Liu

  2. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan

    Wei-Qiang Huang & Wen-Wei Lin

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  1. Tiexiang Li
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  2. Wei-Qiang Huang
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  3. Wen-Wei Lin
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  4. Jijun Liu
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Correspondence to Wei-Qiang Huang.

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Li, T., Huang, WQ., Lin, WW. et al. On Spectral Analysis and a Novel Algorithm for Transmission Eigenvalue Problems. J Sci Comput 64, 83–108 (2015). https://doi.org/10.1007/s10915-014-9923-0

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  • DOI: https://doi.org/10.1007/s10915-014-9923-0

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