×

zbMATH — the first resource for mathematics

An operator marching method for inverse problems in range-dependent waveguides. (English) Zbl 1194.76209
Summary: For large-scale inverse problems in acoustics and electromagnetics, numerical schemes based on direct methods, e.g. FEM and meshless methods, often result in huge linear systems, and thus are not feasible in terms of computing speed and memory storage. This work proposes the “inverse fundamental operator marching method” based on the Dirichlet-to-Neumann map for solving large-scale inverse boundary-value problems in range-dependent waveguides. Truncated singular value decomposition is employed to solve ill-conditioned linear systems arising in marching process, and the number of propagating modes in the waveguide assumes the role of a regularization parameter. Numerical results show that the method is computationally efficient, highly accurate, stable with respect to data noise for retrieving the propagating part of the starting field. It particularly suits to long-range wave propagation in slowly varying stratified waveguide.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76Q05 Hydro- and aero-acoustics
Software:
OASES
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bai, M.R., Application of BEM (boundary element method)-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries, J. acoust. soc. am., 92, 533-549, (1992)
[2] Wang, Z.; Wu, S.F., Helmholtz equation least-squares method for reconstructing the acoustic pressure field, J. acoust. soc. am., 102, 2020-2032, (1997)
[3] Wu, S.F.; Yu, J., Reconstructing interior acoustic pressure field via Helmholtz equation-least-squares method, J. acoust. soc. am., 104, 2054-2060, (1998)
[4] Delillo, T.; Isakov, V.; Valdivia, N.; Wang, L., The detection of the source of acoustical noise in two dimensions, SIAM J. appl. math., 61, 2104-2121, (2001) · Zbl 0983.35149
[5] Delillo, T.; Isakov, V.; Valdivia, N.; Wang, L., The detection of surface vibrations from interior acoustical pressure, Inverse problems, 19, 507-524, (2003) · Zbl 1033.76053
[6] Marin, L.; Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Wen, X., Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations, Comput. mech., 31, 367-372, (2003) · Zbl 1047.65097
[7] Marin, L.; Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Wen, X., An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comput. methods appl. mech. engrg., 192, 709-722, (2003) · Zbl 1022.78012
[8] Marin, L.; Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Wen, X., Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation, Int. J. numer. methods engrg., 60, 1933-1947, (2004) · Zbl 1062.78015
[9] Marin, L.; Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Wen, X., BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method, Engrg. anal. boundary elem., 28, 1025-1034, (2004) · Zbl 1066.80009
[10] Jin, B.T.; Zheng, Y., A meshless method for some inverse problems associated with the Helmholtz equation, Comput. methods appl. mech. engrg., 195, 2270-2288, (2006) · Zbl 1123.65111
[11] Jin, B.T.; Zheng, Y., Boundary knot method for some inverse problems associated with the Helmholtz equation, Int. J. numer. methods engrg., 62, 1636-1651, (2005) · Zbl 1085.65104
[12] Jin, B.T.; Zheng, Y., Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation, Engrg. anal. bound. elem., 29, 925-935, (2005) · Zbl 1182.65179
[13] Marin, L.; Lesnic, D., The method of fundamental solutions for the Cauchy problem associated with 2D Helmholtz-type equations, Comput. struct., 83, 267-278, (2005) · Zbl 1088.35079
[14] Marin, L., A meshless method for the numerical solution of the Cauchy problem associated with 3D Helmholtz-type equations, Appl. math. comput., 165, 355-374, (2005) · Zbl 1070.65115
[15] Jin, B.T.; Marin, L., The plane wave method for inverse problems associated with Helmholtz-type equations, Engrg. anal. bound. elem., 32, 223-240, (2008) · Zbl 1244.65163
[16] Jensen, F.B.; Kuperman, W.A.; Porter, M.B.; Schmidt, H., Computational Ocean acoustics, (1994), AIP
[17] Fishman, L., One-way wave propagation methods in direct and inverse scalar wave propagation modeling, Radio sci., 28, 865-876, (1993)
[18] Fishman, L.; Gautesen, A.K.; Sun, Z., Uniform high-frequency approximations of the square root Helmholtz operator symbol, Wave motion, 26, 127-161, (1997) · Zbl 0918.35039
[19] Zhu, J.X.; Lu, Y.Y., Validity of one-way models in the weak range dependence limit, J. comput. acoust., 12, 55-66, (2004) · Zbl 1256.76067
[20] Lu, Y.Y.; McLaughlin, J.R., The Riccati method for the Helmholtz equation, J. acoust. soc. am., 100, 1432-1446, (1996)
[21] Lu, Y.Y., One-way large range step methods for Helmholtz waveguides, J. comput. phys., 152, 231-250, (1999) · Zbl 0944.65110
[22] Lu, Y.Y.; Huang, J.; McLauphlin, J.R., Local orthogonal transformation and one-way methods for acoustics waveguides, Wave motion, 34, 193-207, (2001) · Zbl 1163.74396
[23] Zhu, J.X.; Lu, Y.Y., Large range step method for acoustic waveguide with two layer media, Progr. nat. sci., 12, 820-825, (2002) · Zbl 1035.78020
[24] Collins, M.D.; Kuperman, W.A., Inverse problems in Ocean acoustics, Inverse problems, 10, 1023-1040, (1994) · Zbl 0807.35155
[25] Lu, Y.Y.; Zhu, J.X., A local orthogonal transform for acoustic waveguides with an internal interface, J. comput. acoust., 12, 37-53, (2004) · Zbl 1256.76064
[26] Hansen, P.C., Analysis of discrete ill-posed problems by means of the L-curve, SIAM rev., 34, 561-580, (1992) · Zbl 0770.65026
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.