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A Least Squares Functional for Solving Inverse Sturm-Liouville Problems

  • Norbert Röhrl
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 529)

Abstract

We present a method to numerically solve the Sturm-Liouville inverse problem using least squares following (Röhrl, 2005, 2006). We show its merits by computing potential and boundary conditions from two sequences of spectral data in several examples. Finally we prove theorems which show why this approach works particularly well.

Keywords

Inverse Problem Conjugate Gradient Algorithm Exponential Convergence Inverse Spectral Problem Separate Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. G. Borg. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math., 78:1–96, 1946.MathSciNetzbMATHCrossRefGoogle Scholar
  2. B. M. Brown, V. S. Samko, I. W. Knowles, and M. Marletta. Inverse spectral problem for the Sturm-Liouville equation. Inverse Problems, 19(1):235–252, 2003. ISSN 0266-5611.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Björn E. J. Dahlberg and Eugene Trubowitz. The inverse Sturm-Liouville problem. III. Comm. Pure Appl. Math., 37(2):255–267, 1984. ISSN 0010-3640.MathSciNetzbMATHCrossRefGoogle Scholar
  4. I. M. Gel’fand and B. M. Levitan. On the determination of a differential equation from its spectral function. Amer. Math. Soc. Transl. (2), 1: 253–304, 1955.MathSciNetGoogle Scholar
  5. E. L. Isaacson and E. Trubowitz. The inverse Sturm-Liouville problem. I. Comm. Pure Appl. Math., 36(6):767–783, 1983. ISSN 0010-3640.MathSciNetzbMATHCrossRefGoogle Scholar
  6. N. Levinson. The inverse Sturm-Liouville problem. Mat. Tidsskr. B., 1949: 25–30, 1949.MathSciNetGoogle Scholar
  7. B. M. Levitan. Inverse Sturm-Liouville problems. VSP, Zeist, 1987. ISBN 90-6764-055-7. Translated from the Russian by O. Efimov.zbMATHGoogle Scholar
  8. B. D. Lowe, M. Pilant, and W. Rundell. The recovery of potentials from finite spectral data. SIAM J. Math. Anal., 23(2):482–504, 1992. ISSN 0036-1410.MathSciNetzbMATHCrossRefGoogle Scholar
  9. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical recipes in C. Cambridge University Press, Cambridge, second edition, 1992. ISBN 0-521-43108-5. URL http://www.nrbook.com/a/bookcpdf.php.zbMATHGoogle Scholar
  10. Jürgen Pöschel and Eugene Trubowitz. Inverse spectral theory, volume 130 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1987. ISBN 0-12-563040-9.Google Scholar
  11. W. Rundell and P. E. Sacks. Reconstruction techniques for classical inverse Sturm-Liouville problems. Math. Comp., 58(197):161–183, 1992. ISSN 0025-5718.MathSciNetzbMATHCrossRefGoogle Scholar
  12. N. Röhrl. A least squares functional for solving inverse Sturm-Liouville problems. Inverse Problems, 21:2009–2017, 2005. ISSN 0266-5611.MathSciNetzbMATHCrossRefGoogle Scholar
  13. N. Röhrl. Recovering boundary conditions in inverse Sturm-Liouville problems. In Recent advances in differential equations and mathematical physics, volume 412 of Contemp. Math., pages 263–270. Amer. Math. Soc., Providence, RI, 2006.Google Scholar
  14. E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Part I. Second Edition. Clarendon Press, Oxford, 1962.Google Scholar

Copyright information

© CISM, Udine 2011

Authors and Affiliations

  • Norbert Röhrl
    • 1
  1. 1.Department of Mathematics and PhysicsUniversity of StuttgartStuttgartGermany

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