# zbMATH — the first resource for mathematics

Boundary knot method for some inverse problems associated with the Helmholtz equation. (English) Zbl 1085.65104
Summary: The boundary knot method is an inherently meshless, integration-free, boundary-type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill-posed Cauchy problem. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the $$L$$-curve method.
Numerical results are presented for both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems.

##### MSC:
 65N21 Numerical methods for inverse problems for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35R30 Inverse problems for PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
Full Text:
##### References:
 [1] Bai, Journal of the Acoustical Society of America 92 pp 533– (1992) [2] Fredderick, Inverse Problems 12 pp 139– (1996) [3] Wang, Journal of the Acoustical Society of America 102 pp 2020– (1997) [4] Wu, Journal of the Acoustical Society of America 104 pp 2054– (1998) [5] Isakov, Inverse Problems 18 pp 1147– (2002) [6] Marin, Computer Methods in Applied Mechanics and Engineering 192 pp 709– (2003) [7] Gryzin, Journal of Computational Physics 184 pp 122– (2003) [8] Kansa, Computers and Mathematics with Applications 19 pp 147– (1990) · Zbl 0692.76003 [9] , . The method of fundamental solution for potential, Helmholtz and diffusion equation. In Boundary Integral Methods–Numerical and Mathematical Aspects, (ed.). Computational Mechanics Publications: Southampton, U.K., 1998; 103-176. [10] , . New advances in dual reciprocity and boundary-only RBF methods. In Proceeding of BEM Technique Conference, Tanaka M (ed.), vol. 10. Tokyo, Japan, 2000; 17-22. [11] Chen, International Journal of Nonlinear Sciences and Numerical Simulation 1 pp 145– (2000) · Zbl 0954.65084 [12] Chen, Computers and Mathematics with Applications 43 pp 379– (2002) [13] Wood, International Communications in Heat and Mass Transfer 22 pp 99– (1995) [14] . Recent research at Cambridge on radial basis functions. In New Developments in Approximation Theory (Durtmund, 1998), , , , (eds), International Series in Numerical Mathematics, vol. 132. Birkhäuser: Basel, 1999; 215-232. · Zbl 0958.41501 [15] Chen, Computer Methods in Applied Mechanics and Engineering 192 pp 1859– (2003) [16] Hon, International Journal for Numerical Methods in Engineering 56 pp 1931– (2003) [17] Chen, Engineering Analysis with Boundary Elements 26 pp 489– (2002) [18] Kansa, Computers and Mathematics with Applications 39 pp 123– (2000) [19] Hansen, BIT 27 pp 533– (1987) [20] Engl, Surveys on Mathematics for Industry 3 pp 71– (1993) [21] . Methods for Solving Incorrectly Posed Problems. Springer: New York, 1984. [22] Hansen, SIAM Review 34 pp 561– (1992) [23] Hansen, SIAM Journal of Scientific Computing 14 pp 1487– (1993) [24] Golub, Technometrics 21 pp 215– (1979) [25] , . Solving Least Squares Problems. Prentice-Hall: Englewood Cliffs, 1974. · Zbl 0860.65028 [26] , . A new approach for free vibration analysis using boundary elements. In Boundary Element Methods in Engineering, (ed.). Springer: Berlin, 1982; 312-326. [27] Hanke, Surveys on Mathematics for Industry 3 pp 253– (1993) [28] Beatson, Advances in Computational Mathematics 11 pp 253– (1999)
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.