##
**A new investigation into regularization techniques for the method of fundamental solutions.**
*(English)*
Zbl 1210.65198

Summary: We examine different regularization approaches to investigate the solution stability of the method of fundamental solutions (MFS). We compare three regularization methods in conjunction with two different regularization parameters to find the optimal stable MFS scheme. Meanwhile, we investigate the relationship among the condition number, the effective condition number, and the MFS solution accuracy. Numerical results show that the damped singular value decomposition under the parameter choice of the generalized cross-validation performs the best in terms of the MFS stability analysis. We also find that the condition number is a superior criterion to the effective condition number.

### MSC:

65N80 | Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

65F22 | Ill-posedness and regularization problems in numerical linear algebra |

### Keywords:

method of fundamental solutions; regularization technique; regularization parameter; effective condition number; stability; numerical results; singular value decomposition; generalized cross-validation
PDF
BibTeX
XML
Cite

\textit{J. Lin} et al., Math. Comput. Simul. 81, No. 6, 1144--1152 (2011; Zbl 1210.65198)

Full Text:
DOI

### References:

[1] | Banoczi, J. M.; Chiu, N. C.; Cho, G. E.; Ipsen, I. C.F., The lack of influence of the right-hand side on the accuracy of linear system solution, SIAM Journal on Scientific Computing, 20, 203-227 (1998) · Zbl 0914.65047 |

[2] | Chen, J. T.; Chen, I. L.; Lee, Y. T., Eigensolutions of multiply connected membranes using the method of fundamental solutions, Engineering Analysis with Boundary Elements, 29, 166-174 (2005) · Zbl 1182.74249 |

[3] | Chen, C. S.; Cho, H. A.; Golberg, M. A., Some comments on the ill-conditioning of the method of fundamental solutions, Engineering Analysis with Boundary Elements, 30, 405-410 (2006) · Zbl 1187.65136 |

[4] | Carlos Alves, J. S., On the choice of source points in the method of fundamental solutions, Engineering Analysis with Boundary Elements, 33, 1348-1361 (2009) · Zbl 1244.65216 |

[5] | Drombosky, T. W.; Meyer, A. L.; Ling, L., Applicability of the method of fundamental solutions, Engineering Analysis with Boundary Elements, 33, 637-643 (2009) · Zbl 1244.65220 |

[6] | Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics, 9, 69-95 (1998) · Zbl 0922.65074 |

[7] | Golberg, M. A.; Chen, C. S., The method of fundamental solutions for potential Helmholtz and diffusion problems, (Golberg, M. A., Boundary Inetgral Methodsnumerical and Mathematical Aspects (1998), Computational Mechanics Publications), 103-176 · Zbl 0945.65130 |

[8] | Gorzelanczyk, P.; Kolodziej, J. A., Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods, Engineering Analysis with Boundary Elements, 32, 64-75 (2008) · Zbl 1272.74623 |

[9] | Hansen, P. C., Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, 34, 561-580 (1992) · Zbl 0770.65026 |

[10] | Hansen, P. C., Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems, Numerical Algorithms, 6, 1-35 (1994) · Zbl 0789.65029 |

[11] | Hansen, P. C., The truncated SVD as a method for regularization, BIT, 27, 534-553 (1987) · Zbl 0633.65041 |

[12] | Hon, Y. C.; Wei, T., The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES: Computer Modeling in Engineering and Sciences, 7, 119-132 (2005) · Zbl 1114.80004 |

[13] | Hanke, M.; Hansen, P. C., Regularization methods for large-scale problems, Surveys on Mathematics for Industry, 3, 4, 253-315 (1993) · Zbl 0805.65058 |

[14] | Kupradze, V.; Aleksidze, M., The method of functional equations for the approximate solution of certain boundary value problems, Computational Mathematics and Mathematical Physics, 4, 82-126 (1964) · Zbl 0154.17604 |

[15] | Kitagawa, T., On the numerical stability of the method of fundamental solution applied to Dirichlet problem, Japan Journal of Industrial and Applied Mathematics, 5, 123-133 (1988) · Zbl 0644.65060 |

[16] | Liu, C. S., Improving the ill-conditioning of the method of fundamental solutions for 2D laplace equation, CMES: Computer Modeling in Engineering and Sciences, 851, 1-17 (2009) |

[17] | Liu, C. S.; Atluri, S. N., A highly accurate technique for interpolations using very high-order polynomials, and its applications to some ill-posed linear problems, CMES: Computer Modeling in Engineering and Sciences, 43, 253-276 (2009) · Zbl 1232.65021 |

[18] | Li, Z. C.; Huang, H. T.; Huang, J., Effective condition number of the hermite finite element methods for Biharmonic equations, Applied Numerical Mathematics, 58, 1291-1308 (2008) · Zbl 1163.65086 |

[19] | Mathé, P.; Pereverzev, S. V., Regularization of some linear ill-posed problems with discretized random noisy data, Mathematics of Computation, 75, 1913-1929 (2006) · Zbl 1103.62031 |

[20] | Mathé, P.; Pereverzev, S. V., Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods, SIAM Journal on Numerical Analysis, 38, 1999-2021 (2001) · Zbl 1049.65046 |

[21] | Ramachandran, P. A., Method of fundamental solutions: singular value decomposition analysis, Communications in Numerical Methods in Engineering, 18, 789-801 (2002) · Zbl 1016.65095 |

[22] | Rodriguez, G.; Theis, D., An algorithm for estimating the optimal regularization parameter by the L-curve, Rendiconti di Matematica, 25, 69-84 (2005) · Zbl 1072.65058 |

[23] | Reichel, L.; Sadok, H., A new L-curve for ill-posed problems, Journal of Computational and Applied Mathematics, 219, 493-508 (2008) · Zbl 1145.65035 |

[24] | Schaback, R., (Chen, C. S.; Karageorghis, A.; Smyrlis, Y. S., An Adaptive Numerical Solution of MFS Systems. The Method of Fundamental Solutions - A Meshless Method (2008), Dynamic Publishers), 1-27 |

[25] | Tikhonov, A. N.; Arsenin, V. Y., Solutions of Ill-posed problems, SIAM Review, 21, 266-267 (1979) |

[26] | Tikhonov, A. N.; Goncharsky, A. V.; Stepanov, V. V.; Yagola, A. G., Numerical Methods for the Solution of Ill-Posed Problems (1995), Kluwer Academic Publishers: Kluwer Academic Publishers MA, Boston · Zbl 0831.65059 |

[28] | Winkler, J. R., Polynomial basis conversion made stable by truncated singular value decomposition, Applied Mathematical Modelling, 21, 557-568 (1997) · Zbl 0895.65004 |

[29] | Wei, T.; Hon, Y. C.; Ling, L., Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Engineering Analysis with Boundary Elements, 31, 373-385 (2007) · Zbl 1195.65206 |

[30] | Wang, F. Z.; Ling, L.; Chen, W., Effective condition number for boundary knot method, CMC: Computers, Materials and Continua, 12, 57-70 (2009) |

[31] | Wang, F. Z.; Chen, W.; Jiang, X. R., Investigation of regularized techniques for boundary knot method, Communications in Numerical Methods in Engineering (2009) |

[32] | Yang, F. L.; Yan, L.; Wei, T., Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method, Mathematics and Computer in Simulation, 79, 2148-2156 (2009) · Zbl 1161.65363 |

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.