# zbMATH — the first resource for mathematics

A meshless numerical identification of a sound-hard obstacle. (English) Zbl 1351.74090
Summary: We propose a simple meshless method for detecting a rigid (sound-hard) scatterer embedded in a host acoustic homogeneous medium from scant measurements of the scattered near field. This inverse problem is ill-posed since a solution may not be unique and furthermore, small errors in the input data cause large errors in the output solution. We develop a nonlinear minimization regularized method of fundamental solutions (MFS) for obtaining the numerical solution of the inverse problem in question. Although the MFS is restricted to homogeneous media with constant wavenumber, it is easy to use and simple to implement in higher dimensions. The proposed scheme is tested on several numerical examples and its stability is investigated by inverting measurements contaminated by random noise.

##### MSC:
 74S10 Finite volume methods applied to problems in solid mechanics 74J25 Inverse problems for waves in solid mechanics 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs 76Q05 Hydro- and aero-acoustics
HYBRJ; minpack
Full Text:
##### References:
 [1] Alves, C.J.S., Inverse scattering with spherical incident waves, (), 502-504 · Zbl 0942.35038 [2] Alves, C.J.S., Density results for the Helmholtz equation and the method of fundamental solutions, (), 45-50 [3] Belge, M.; Kilmer, M.E.; Miller, E.L., Efficient determination of multiple regularization parameters in a generalized L-curve framework, Inverse probl, 18, 1161-1183, (2002) · Zbl 1018.65073 [4] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J numer anal, 22, 644-669, (1985) · Zbl 0579.65121 [5] Browder, F.E., Approximations of solutions of partial differential equations, Am J math, 84, 134-160, (1962) · Zbl 0111.09601 [6] Chen, I.L., Using the method of fundamental solutions in conjunction with the degenerate kernel in cylindrical acoustic problems, J chin inst eng, 29, 445-457, (2006) [7] Chen, I.L.; Chen, J.T.; Liang, M.T., Analytical study and numerical experiments for radiation and scattering problems using the CHIEF method, J sound vib, 248, 809-828, (2003) [8] Chen, I.L.; Chen, J.T.; Kuo, S.R.; Liang, M.T., A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method, J acoust soc am, 109, 982-999, (2001) [9] Chen, J.T.; Chen, I.L.; Lee, Y.T., Eigensolutions of multiply connected membranes using the method of fundamental solutions, Eng anal bound elem, 29, 166-174, (2005) · Zbl 1182.74249 [10] Chen, J.T.; Chen, I.L.; Lee, Y.T., Comments on “the boundary point method for the calculation of exterior acoustic radiation problem”, J sound vib, 310, 1167-1169, (2008), [SY Zhang, XZ Chen. J Sound Vib 228; 1999: 761-72] [11] Chen, J.T.; Lin, J.H.; Kuo, S.R.; Chyuan, S.W., Boundary element analysis for the Helmholtz eigenvalue problems with a multiply connected domain, Proc R soc lond A, 457, 2521-2546, (2001) · Zbl 0993.78021 [12] Colton, D.; Kress, R., Inverse acoustic and electromagnetic scattering theory, (1998), Springer-Verlag New York · Zbl 0893.35138 [13] Colton, D.; Kress, R., Using fundamental solutions in inverse scattering, Inverse probl, 22, R49-R66, (2006) · Zbl 1099.35168 [14] Colton, D.; Kress, R., Thirty years and still counting, Inverse probl imaging, 3, 151-153, (2009) · Zbl 1190.35005 [15] Colton, D.; Sleeman, B.D., Uniqueness theorems for the inverse problem of acoustic scattering, IMA J appl math, 31, 253-259, (1983) · Zbl 0539.76086 [16] Fairweather, G.; Karageorghis, A.; Martin, P.A., The method of fundamental solutions for scattering and radiation problems, Eng anal bound elem, 27, 759-769, (2003) · Zbl 1060.76649 [17] Garbow, B.S.; Hillstrom, K.E.; Moré, J.J., MINPACK project, (1980), Argonne National Laboratory [18] Ha-Duong, T.; Jaoua, M.; Menif, F., A modified frozen Newton method to identify a cavity by means of boundary measurements, Math comput simul, 66, 355-366, (2004) · Zbl 1048.78016 [19] Hansen, P.C.; O’Leary, D.P., The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J sci comput, 14, 1487-1503, (1993) · Zbl 0789.65030 [20] Ihlenburg, F., Finite element analysis of acoustic scattering, (1998), Springer New York · Zbl 0908.65091 [21] Ivanyshyn, O., Shape reconstruction of acoustic obstacles from the modulus of the far field pattern, Inverse probl imaging, 1, 609-622, (2007) · Zbl 1194.35502 [22] Karageorghis, A.; Lesnic, D., Detection of cavities using the method of fundamental solutions, Inverse probl sci eng, 17, 803-820, (2009) · Zbl 1175.65130 [23] Karageorghis, A.; Lesnic, D., Application of the MFS to inverse obstacle scattering problems, Eng anal bound elem, 35, 631-632, (2011) · Zbl 1259.76046 [24] Karageorghis, A.; Lesnic, D.; Marin, L., A survey of applications of the MFS to inverse problems, Inverse probl sci eng, 19, 309-336, (2011) · Zbl 1220.65157 [25] Karageorghis, A.; Lesnic, D.; Marin, L., The MFS for inverse geometric inverse problems, (), 191-216 [26] Karageorghis A, Lesnic D, Marin L. A moving pseudo-boundary MFS for void detection. Submitted for publication. · Zbl 1268.65161 [27] Kirsch, A., Inverse scattering theory for time-harmonic waves, (), 337-365 · Zbl 1051.78014 [28] Kirsch, A.; Kress, R., An optimization method in inverse acoustic scattering, (), 3-18 [29] Kirsch, A.; Kress, R., Uniqueness in inverse obstacle scattering, Inverse probl, 9, 285-299, (1993) · Zbl 0787.35119 [30] Kołodziej, J.A.; Zieliński, A.P., Boundary collocation techniques and their application in engineering, (2009), WIT Press Southampton [31] Kress R. Integral equation method in inverse acoustic and electromagnetic scattering. In: Ingham DB, Wrobel LC, editors. Boundary integral formulations for inverse analysis; 1997. pp. 67-93. · Zbl 0909.65096 [32] Kress, R., Uniqueness and numerical methods in inverse obstacle scattering, J appl phys: conf ser, 73, (2007), 012003 (16 pages) [33] Kress, R.; Mohsen, A., On the simulation source technique for exterior problems in acoustics, Math meth appl sci, 8, 585-597, (1986) · Zbl 0626.35019 [34] Kress, R.; Zinn, A., On the numerical solution of the three-dimensional inverse obstacle scattering problem, J comput appl math, 42, 49-61, (1992) · Zbl 0774.65092 [35] Meyer, P.S.; Capistran, M.; Chen, Y., On the naturally induced sources for obstacle scattering, Commun comput phys, 1, 974-983, (2006) · Zbl 1114.76063 [36] Na SW, Kallivokas LF. Shape detection in inverse acoustic scattering problems. In: Sixteenth ASCE engineering mechanics conference. Seattle, USA: University of Washington; July 16-18, 2003. [37] Wong, K.Y.; Ling, L., Optimality of the method of fundamental solutions, Eng anal bound elem, 35, 42-46, (2011) · Zbl 1259.65198 [38] Yang, F.L.; Ling, L., On numerical experiments for Cauchy problems of elliptic operators, Eng anal bound elem, 35, 879-882, (2011) · Zbl 1259.65199 [39] Zeng, T.; Li, X.; Ng, M., Alternating minimization method for total variation based wavelet shrinkage model, Commun comput phys, 8, 976-994, (2010) · Zbl 1364.65123 [40] Zhang, S.; Jin, J., Computation of special functions, (1996), John Wiley New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.