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Solving Inverse Problems with Spectral Data

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Surveys on Solution Methods for Inverse Problems

Abstract

We consider a two dimensional membrane. The goal is to find properties of the membrane or properties of a force on the membrane. The data is natural frequencies or mode shape measurements. As a result, the functional relationship between the data and the solution of our inverse problem is both indirect and nonlinear. In this paper we describe three distinct approaches to this problem. In the first approach the data is mode shape level sets and frequencies. Here formulas for approximate solutions are given based on perturbation results. In the second approach the data is frequencies and boundary mode shape measurements; uniqueness results are obtained using the boundary control method. In the third approach the data is frequencies for four boundary value problems. Local existence, uniqueness results are established together with numerical results for approximate solutions.

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Joyce R. McLaughlin

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  1. Joyce R. McLaughlin
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Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, Delaware, USA

    David Colton

  2. Institut für Mathematik, Johannes-Kepler-Universität, Linz, Austria

    Heinz W. Engl

  3. Fachbereich Mathematik, Universität des Saarlandes, Saarbrücken, Federal Republic of Germany

    Alfred K. Louis

  4. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York, USA

    Joyce R. McLaughlin

  5. Department of Mathematics, Texas A & M University, College Station, Texas, USA

    William Rundell

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McLaughlin, J.R. (2000). Solving Inverse Problems with Spectral Data. In: Colton, D., Engl, H.W., Louis, A.K., McLaughlin, J.R., Rundell, W. (eds) Surveys on Solution Methods for Inverse Problems. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6296-5_10

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  • DOI: https://doi.org/10.1007/978-3-7091-6296-5_10

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-83470-1

  • Online ISBN: 978-3-7091-6296-5

  • eBook Packages: Springer Book Archive

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