OBSTACLE SHAPE RECONSTRUCTION BY LOCALLY SUPPORTED BASIS FUNCTIONS

- Journal title : Honam Mathematical Journal
- Volume 36, Issue 4, 2014, pp.831-852
- Publisher : The Honam Mathematical Society
- DOI : 10.5831/HMJ.2014.36.4.831

Title & Authors

OBSTACLE SHAPE RECONSTRUCTION BY LOCALLY SUPPORTED BASIS FUNCTIONS

Lee, Ju-Hyun; Kang, Sungkwon;

Lee, Ju-Hyun; Kang, Sungkwon;

Abstract

The obstacle shape reconstruction problem has been known to be difficult to solve since it is highly nonlinear and severely ill-posed. The use of local or locally supported basis functions for the problem has been addressed for many years. However, to the authors` knowledge, any research report on the proper usage of local or locally supported basis functions for the shape reconstruction has not been appeared in the literature due to many difficulties. The aim of this paper is to introduce the general concepts and methodologies for the proper choice and their implementation of locally supported basis functions through the two-dimensional Helmholtz equation. The implementations are based on the complex nonlinear parameter estimation (CNPE) formula and its robust algorithm developed recently by the authors. The basic concepts and ideas are simple. The derivation of the necessary properties needed for the shape reconstructions are elementary. However, the capturing abilities for the local geometry of the obstacle are superior to those by conventional methods, the trial and errors, due to the proper implementation and the CNPE algorithm. Several numerical experiments are performed to show the power of the proposed method. The fundamental ideas and methodologies described in this paper can be applied to many other shape reconstruction problems.

Keywords

inverse scattering;Helmholtz equation;shape reconstruction;locally supported basis functions;complex nonlinear parameter estimation (CNPE);

Language

English

References

1.

H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging (Mathematiques et Application), Springer, 2008.

2.

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer, Berlin, 2006.

3.

F. Cakoni, D. Colton, and P. Monk, The Liner Sampling Method in Inverse Electromagnetic Scattering, SIAM, 2011.

4.

D. Colton and R. Kress, Handbook of Mathematical Methods in Imaging(Scherzer O(ed.)), Chapter 13. Inverse Scattering, Springer, 2011.

5.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (3rd Ed), Springer, Berlin, 2013.

6.

O. Dorn and D. Lesselier, Handbook of Mathematical Methods in Imaging( Scherzer O(ed.)), Chapter 10. Level Set Methods for Structural Invasion and Image Reconstruction, Springer, 2011.

7.

U. Guo, F. Ma, and D. Zhang, An optimization method for acoustic inverse obstacle scattering problems with multiple incident waves, Inverse Problems in Science and Engineering 19 (2011), 461-484.

8.

B. Guzina, F. Cakoni, and C. Bellis, On multi-frequency obstacle reconstruction via linear sampling method, Inverse Problems 26 (2010) 125005(29pp).

9.

O. Ivanyshyn, Shape reconstruction of acoustic obstacles from the modulus of the far field pattern, Inverse Problems and Imaging 1(4) (2007), 609-622.

10.

O. Ivanyshyn and T. Johansson, Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle, Journal of Integral equations and applications 19(3) (2007), 289-308.

11.

O. Ivanyshyn and R. Kress, Nonlinear integral equations in inverse obstacle scattering, Mathematical Methods in Scattering Theory and Biomedical Engineering (D. Fotiadis and C. Massalas(eds.)), World Scientific, (2006), 39-50.

12.

T. Johansson and B. D. Sleeamn, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern, IMA J. Appl. Math. 72 (2007), 96-112.

13.

A. Kirsch, The domain derivative and two applications in inverse scattering theory, Inverse Problems 9 (1993), 81-96.

14.

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.

15.

R. Kress and W. Rundel, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems 21 (2005), 1207-1223.

16.

J. Lee and S. Kang, Complex nonlinear parameter estimation(CNPE) and obstacle shape reconstruction, Computers and Mathematics with Applications 67 (2014), 1631-1642.

17.

R. Pothast, Frechet differentiability of boundary integral operators in inverse acoustic scattering, Inverse Problems 10 (1994), 431-447.

18.

M. Schultz, Spline Analysis, Prentice-Hall, 1973.

19.

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd Ed., Springer, 2002.