Sampling and Probe Methods – An Algorithmical View

Inverse scattering problems are concerned with the reconstruction of objects and parameter functions from the knowledge of scattered waves. Today, basically three different categories of methods for the treatment of the full nonlinear scattering problem are known: iterative methods, decomposition me...

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Bibliographic Details
Published in:Computing Vol. 75; no. 2-3; pp. 215 - 235
Main Author: Potthast, Roland
Format: Journal Article
Language:English
Published: Wien Springer 01.08.2005
Springer Nature B.V
Subjects:
Algorithms
Classical and quantum physics: mechanics and fields
Exact sciences and technology
General theory of scattering
Helmholtz equations
Inverse problems
Knowledge
Mathematical analysis
Mathematics
Methods
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis. Scientific computation
Partial differential equations
Physics
Radiation
Sampling
Scattering
Sciences and techniques of general use
Studies
Tomography
survey article sampling- and probe methods
fixed wave number
78A56. Inverse scattering
35R30
Decomposition method
Iterative method
Inverse scattering problem
Nonlinear problems
Probe
Numerical analysis
Scientific computation
Factorization method
Tomography
reconstruction of physical parameters
Sampling
Impedance
Shape reconstruction
Wave number
ISSN:0010-485X, 1436-5057
Online Access:Get full text
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Summary:Inverse scattering problems are concerned with the reconstruction of objects and parameter functions from the knowledge of scattered waves. Today, basically three different categories of methods for the treatment of the full nonlinear scattering problem are known: iterative methods, decomposition methods and sampling/probe methods. Sampling and probe methods have been proposed to detect the unknown scatterer for example in cases when the physical properties are not known or not given in a parametrized modell. A number of different approaches have been suggested over the last years. We will give a survey about these methods and explain the main ideas from an algorithmical point of view. We will group and describe the methods and discuss some relations to ideas which have been developed in the framework of impedance tomography. First, we will study the probe method of Ikehata and the singular sources method of Potthast. We show that these methods are closely related and basically form one unit with two different realizations. The second part is concerned with the range test of Potthast, Sylvester and Kusiak, the linear sampling method of Colton and Kirsch and the factorization method of Kirsch. These methods are founded on similar ideas testing the range of operators for reconstruction. In the final part we study the no response test of Luke and Potthast and the enclosure method of Ikehata. We will see that the enclosure method can be considered as a particular choice of the probing function for the no response test. These two methods are closely related and form two extremes for probing a scatterer with specially constructed waves. [PUBLICATION ABSTRACT]
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ISSN:0010-485X
1436-5057
DOI:10.1007/s00607-004-0084-0