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The Inverse Medium Problem

  • David Colton
  • Rainer Kress
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 93)

Abstract

We now turn our attention to the problem of reconstructing the refractive index from a knowledge of the far field pattern of the scattered acoustic or electromagnetic wave. We shall call this problem the inverse medium problem. We first consider the case of acoustic waves and the use of the Lippmann–Schwinger equation to reformulate the acoustic inverse medium problem as a problem in constrained optimization. Included here is a brief discussion of the use of the Born approximation to linearize the problem. We then proceed to the proof of a uniqueness theorem for the acoustic inverse medium problem. Our uniqueness result is then followed by a discussion of decomposition methods for solving the inverse medium problem for acoustic waves and the use of sampling methods and transmission eigenvalues to obtain qualitative estimates on the refractive index. We conclude by examining the use of decomposition methods to solve the inverse medium problem for electromagnetic waves followed by some numerical examples illustrating the use of decomposition methods to solve the inverse medium problem for acoustic waves.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • David Colton
    • 1
  • Rainer Kress
    • 2
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Institut für Numerische und Angewandte MathematikGeorg-August-Universität GöttingenGöttingenGermany

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