The no-response approach and its relation to non-iterative methods for the inverse scattering
- 59 Downloads
This paper addresses the inverse obstacle scattering problem. In the recent years several non-iterative methods have been proposed to reconstruct obstacles (penetrable or impenetrable) from near or far field measurements. In the chronological order, we cite among others the linear sampling method, the factorization method, the probe method and the singular sources method. These methods use differently the measurements to detect the unknown obstacle and they require the use of many incident fields (i.e. the full or a part of the far field map). More recently, two other approaches have been added. They are the no-response test and the range test. Both of them use few incident fields to detect some informations about the scatterer. All the mentioned methods are based on building functions depending on some parameter. These functions share the property that their behaviors with respect to the parameter change drastically. The surface of the obstacle is located at most in the interface where these functions become large. The goal of this work is to investigate the relation between some of the non-iterative reconstruction schemes regarding the convergence issue. A given method is said to be convergent if it reconstructs a part or the entire obstacle by using few or many incident fields respectively. For simplicity we consider the obstacle reconstruction problem from far field data for the Helmholtz equation.
KeywordsIndicator Function Response Test Inverse Scattering Helmholtz Equation Dirichlet Eigenvalue
Unable to display preview. Download preview PDF.
- 2.Ammari H., Kang H. (2004): Reconstruction of Small Inhomogeneities from Boundary Measurments. Springer, Berlin Heidelberg New YorkGoogle Scholar
- 10.Isakov, V.: Inverse Problems for Partial Differential Equations. Springer Series in Applied Math. Science, vol. 127. Springer, Berlin Heidelberg New York (1998)Google Scholar
- 14.Nakamura, G., Potthast, R., Sini, M.: Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test. Accepted for publication by Comm. PDE. Available on the page web http://eprints.math.sci.hokudai.ac.jp/Google Scholar
- 15.Nakamura, G., Potthast, R., Sini, M.: A comparative study between some non-iterative methods for the inverse scattering. In: Ammari, H., Kang, H. (eds.) Inverse Problems, Multi-Scale Analysis, and Homogenization. Proceedings of the Workshop in Seoul, 2005. Contemporary Mathematics Volume, American Mathematical Society (To appear, 2006)Google Scholar
- 16.Nečas J. (1967): Les méthodes directes en théorie des équations élliptiques. Academia, PragueGoogle Scholar
- 17.Potthast, R.: On the convergence of the no-response test (preprint)Google Scholar
- 18.Potthast, R.: Point sources and multipoles in inverse scattering theory, vol. 427 of Chapman-Hall/CRC, Research Notes in Mathematics. Chapman-Hall/CRC, Boca Raton (2001)Google Scholar
- 20.Potthast, R., Schulz, J.: A multiwave rangetest for obstacle reconstruction with unknown physical properties (preprint)Google Scholar