On the reconstruction of low-frequency moments in acoustic scattering

Author:
A. Charalambopoulos

Journal:
Quart. Appl. Math. **70** (2012), 311-343

MSC (2010):
Primary 35R30, 76Q05; Secondary 35J05

DOI:
https://doi.org/10.1090/S0033-569X-2012-01264-X

Published electronically:
February 29, 2012

MathSciNet review:
2953106

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Abstract | References | Similar Articles | Additional Information

Abstract: The inverse scattering method via low-frequency moments was introduced several years ago. The specific structure of moments permitted the construction of a linear inversion algorithm that was based on the assumption that the moments were known, at least theoretically. The present work goes deeper and aims at providing a systematic method to reconstruct these moments from measurements. This turns out to be a demanding inverse problem by itself, serving to establish a realistic implementation for the underlying inversion method. It is proved herein how to determine a specific large class of moments. In addition it is proved that not all the moments are able to be determined purely from the set of data. A demanding integral equation methodology is produced to estimate the large class of the remaining moments that are not directly accessible from measurements.

**1.**Charalambopoulos, A.,*Inverse scattering for an acoustically soft scatterer in the low-frequency region*, Int. J. Engrg. Sci.,**33**, 4, (1995), 599-609. MR**1314301 (95m:76059)****2.**Charalambopoulos, A., Kiriaki, K.,*A method for solving the inverse elastic scattering problem via low-frequency moments*, Wave Motion,**18**(1993), 213-226. MR**1256478 (95f:73032)****3.**Charalambopoulos, A., Dassios, G.,*Inverse scattering via low-frequency moments*, J. Math. Phys.,**33**, (1992), 4201-4216. MR**1191780 (93j:35051)****4.**Charalambopoulos, A.,*The reconstruction of the surface of scatterers with continuous curvature via low-frequency moments*, IMA Journal of Applied Mathematics,**54**, (1995), 171-201. MR**1334459 (96c:35192)****5.**Apostolopoulos, T., Dassios, G.*A parallel algorithm for solving the inverse scattering moment problem*, Journal of Computational and Applied Mathematics,**42**, (1992), 63-77. MR**1181581****6.**Charalambopoulos, A.,*An analytic algorithm for shape reconstruction from low-frequency moments*, accepted for publication in Journal of Mathematical Physics, 2011.**7.**Billingham, J., King, A.C.,*Wave Motion*, Cambridge University Press, 2000. MR**1811404 (2001k:35001)****8.**Kleinman, R.E.,*The Rayleigh region*, PROC. IEEE,**53**, (1965), 848-856.**9.**Kleinman, R.E.,*The Dirichlet problem for the Helmholtz Equation*, Arch. Rat. Mech. Analysis,**18**, (1965), 205-229. MR**0172602 (30:2821)****10.**Kleinman, R.E.,*Far-Field scattering at low-frequencies*, Appl. Sci. Res.,**18**, (1967), 1-8.**11.**Muller, C.,*Radiation Patterns and Radiation Fields*, J. Rat. Mech. and Anal.,**4**, (1955), 235-245. MR**0069026 (16:978b)****12.**Charalambopoulos, A., Kiriaki, K.,*Characterization of Functions as Radiation Patterns in Linear Elasticity*, Mathematical Methods in the Applied Sciences,**15**, (1992), 547-558. MR**1184322 (93j:73030)****13.**Kirsch, A.,*Characterization of the shape of a scattering obstacle using the spectra data of the far field operator*, Inverse Problems,**15**, (1998), 413-419. MR**1662460 (99k:35193)****14.**Groetsch, C.W.,*Stable Approximate Evaluation of Unbounded Operators*, Springer-Verlag, 2007. MR**2268011 (2008a:47022)****15.**Morse, Philip M., Feshbach, Herman,*Methods of Theoretical Physics*, 2 Volumes, McGraw-Hill Book Co., 1953. MR**0059774 (15:583h)****16.**Colton, D., Kress, R.,*Integral equation methods in scattering theory*, John Wiley and Sons, 1983. MR**700400 (85d:35001)****17.**Nedelec, J.C.,*Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems*, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, 2000. MR**1822275 (2002c:35003)**

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Additional Information

**A. Charalambopoulos**

Affiliation:
Department of Materials Science and Engineering, The University of Ioannina, 45110 Greece

DOI:
https://doi.org/10.1090/S0033-569X-2012-01264-X

Received by editor(s):
September 23, 2010

Published electronically:
February 29, 2012

Article copyright:
© Copyright 2012
Brown University