Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-18T17:07:59.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Meshless Collocation Method for Inverse Source Identification Problems

Published online by Cambridge University Press:  29 May 2015

Fuzhang Wang  and
Zhaoxing Ma
Fuzhang Wang*
Affiliation:
College of Mathematics, Huaibei Normal University, Huaibei 235000, China
Zhaoxing Ma
Affiliation:
School of Electrical Engineering and Automation, Jiangsu Normal University, Xuzhou 221116, China
*
*Corresponding author. Email: wangfuzhang1984@163.com (F. Z. Wang), 82563433@qq.com (Z. X. Ma)
Get access

Abstract

A novel meshless scheme is proposed for inverse source identification problems of Helmholtz-type equations. It is formulated by the non-singular general solutions of the Helmholtz-type equations augmented with radial basis functions. Under this meshless scheme, we can determine smooth source terms from partially accessible boundary measurements with accurate results. Numerical examples are presented to verify validity and accuracy of the present scheme. It is demonstrated that the present scheme is simple, accurate, stable and computationally efficient for inverse smooth source identification problems.

Keywords

65M1078A48Meshless methodHelmholtz equationsource function
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 4 , August 2015 , pp. 496 - 509
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alves, C. J. S. and Antunes, R. R. S., The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates, int. J. Numer. Methods Eng., 77 (2009), pp. 177–194.Google Scholar
[2]Chen, C. S., Hokwon, A. C. and Golberg, M. A., Some comments on the ill-conditioning of the method of fundamental solutions, Eng. Anal. Boundary Elements, 30 (2006), pp. 405–410.Google Scholar
[3]Chen, C. S., Lee, S. W. and Huang, C. S., The method of particular solutions using Chebyshev polynomial based functions, int. J. Comput. Methods, 4 (2007), pp. 15–32.Google Scholar
[4]Chen, W. and Hon, Y. C., Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz and convection-diffusion problems, Comput. Methods Appl. Mech. Eng., 192 (2003), pp. 1859–1875.Google Scholar
[5]Chen, W. and Tanaka, M., A meshless, integration-free, and boundary-only RBF technique, Comput. Math. Appl., 43 (2002), pp. 379–391.Google Scholar
[6]Cho, H. A., Golberg, M. A., Muleshkov, A. S. and Li, X., Trefftz methods for time dependent partial differential equations, Comput. Materials Continua, 1 (2004), pp. 1–37.Google Scholar
[7]Drombosky, T. W., Meyer, A. L. and Ling, L., Applicability of the method of fundamental solutions, Eng. Anal. Boundary Elements, 33 (2009), pp. 637–643.Google Scholar
[8]El, B. A., Inverse source problem in an anisotropic medium by boundary measurements, inverse Problems, 21 (2005), pp. 1487–1506.Google Scholar
[9]Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9(1–2) (1998), pp. 69–95.Google Scholar
[10]Farcas, A., Dual reciprocity boundary element methods for solving inverse problems: applications to inverse source and boundary value problems for the poisson equation, Saarbrucken: Lambert Academic Publishing, 2010.Google Scholar
[11]Fasshauer, G. E. and Zhang, J. G., On choosing “optimal” shape parameters for RBF approximation, Numer. Algorithms, 45 (2007), pp. 345–368.Google Scholar
[12]Fornberg, B. and Wright, G., Stable compuatation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl., 48 (2004), pp. 853–867.Google Scholar
[13]Golub, G. H., Hansen, P. C. and Leary, D. P. O., Tikhonov regularization and total least squares, SIAM J. Matrix Anal. Appl., 21(1) (1999), pp. 185–194.Google Scholar
[14]Gu, Y., Chen, W. and Fu, Z. J., Singular boundary method for inverse heat conduction problems in general anisotropic media, inverse Problems Sci. Eng., 22 (2013), pp. 889–909.Google Scholar
[15]Gu, Y., Chen, W. and He, X. Q., Improved singular boundary method for elasticity problems, Comput. Struct., 135 (2014), pp. 73–82.Google Scholar
[16]Hansen, P. C., Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algorithms, 6 (1994), 1–35.CrossRefGoogle Scholar
[17]Huang, C. S., Lee, C. F. and Cheng, A. H. D., Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method, Eng. Anal. Boundary Elements, 31 (2007), pp. 614–623.CrossRefGoogle Scholar
[18]Jin, B. T. and Marin, L., The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction, int. J. Numer. Methods Eng., 69 (2007), pp. 1570–1589.CrossRefGoogle Scholar
[19]Kagawa, Y., Sun, Y. H. and Matsumoto, O., Inverse solution for Poisson equations using DRM boundary element models–identification of space charge distribution, inverse Problems Sci. Eng., 1 (1995), pp. 247–265.CrossRefGoogle Scholar
[20]Kansa, E. J., Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamicsłI surface approximations and partial derivative estimates, Comput. Math. Appl., 19(8–9) (1990), pp. 127–145.Google Scholar
[21]Karageorghis, A., Lesnic, D. and Marin, L., A survey of applications of the MFS to inverse problems, inverse Problems Sci. Eng., 19 (2011), pp. 309–336.Google Scholar
[22]Kita, E. and Kamiya, N., Trefftz method: an overview, Adv. Eng. Software, 24(1–3) (1995), pp. 3–12.Google Scholar
[23]Ling, L., Hon, Y. C. and Yamamoto, M., Inverse source identification for Poisson equation, inverse Problems Sci. Eng., 13 (2005), pp. 433–447.Google Scholar
[24]Martinus, F., Herrin, D. W. and Han, D. J., Identification of an aeroacoustic source using the inverse boundary element method, Noise Control Eng. J., 58 (2010), pp. 83–92.CrossRefGoogle Scholar
[25]Nara, T. and Ando, S., A projective method for an inverse source problem of the Poisson equation, inverse Problems, 19 (2003), pp. 355–369.Google Scholar
[26]Nowak, A. J. and Neves, A. C., The Multiple Reciprocity Boundary Element Method, Computational Mechanic Publications, Southampton, 1994.Google Scholar
[27]Ohnaka, K. and Uosaki, K., Boundary element approach for identification of point forces of distributed parameter systems, int. J. Control, 49 (1989), pp. 119–127.Google Scholar
[28]Partridge, P. W., Brebbia, C. A. and Wrobel, L. W., The Dual Reciprocity Boundary Element Method, Computational Mechanic Publication, Southampton, 1992.Google Scholar
[29]Poullikkas, A., Karageorghis, A. and Georgiou, G., Method of fundamental solutions for harmonic and biharmonic boundary value problems, Comput. Mech., 21 (1998), pp. 416–423.Google Scholar
[30]Schuhmacher, A., Hald, J., Rasmussen, K. B. and Hansen, P. C., Sound source reconstruction using inverse boundary element calculations, J. Acoustical Soc. Amer., 1 (2003), pp. 114–127.Google Scholar
[31]Trlep, M., Hamler, A. and Hribernik, B., The use of DRM for inverse problems of Poisson’s equation, IEEE Trans. Magn., 36 (2000), pp. 1649–1652.Google Scholar
[32]Wang, F. Z., Applicability of the boundary particle method, Comput. Model. Eng. Sci., 80(3) (2011), pp. 201–217.Google Scholar
[33]Wang, F. Z., Numerical simulation of acoustic problems with high wavenumbers, Appl. Math. Inform. Sci., 9(2) (2015), pp. 1–4.Google Scholar
[34]Wang, F. Z., Chen, W. and Jiang, X. R., Investigation of regularized techniques for boundary knot method, int. J. Numer. Methods Biomedical Eng., 26(12) (2010), pp. 1868–1877.Google Scholar
[35]Wang, F. Z., Chen, W. and Ling, L., Combinations of the method of fundamental solutions for general inverse source identification problems, Appl. Math. Comput., 219(3) (2012), pp. 1173–1182.Google Scholar
[36]Wang, F. W., Ling, L. and Chen, W., Effective condition number for boundary knot method, Comput. Materials Continua, 12 (2009), pp. 57–70.Google Scholar
[37]Wei, T., Hon, Y. C. and Ling, L., Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Eng. Anal. Boundary Elements, 31 (2007), pp. 373–385.Google Scholar
[38]Wen, P. H. and Chen, C. S., The method of particular solutions for solving scalar wave equations, int. J. Numer. Methods Eng., 26 (2010), pp. 1878–1889.Google Scholar
[39]Wertz, J., Kansa, E. J. and Ling, L., The role of the multiquadric shape parameters in solving elliptic partial differential equations, Comput. Math. Appl., 51 (2006), pp. 1335–1348.Google Scholar
[40]Zhang, J. Y. and Wang, F. Z., Boundary knot method: an overview and some novel approaches, Comput. Model. Eng. Sci., 88(2) (2012), pp. 141–154.Google Scholar