An iterative MFS algorithm for the Cauchy problem associated with the Laplace equation

CMES - Computer Modeling in Engineering and Sciences

Volumn 48, Issue 2, 2009, Pages 121-152

  Marin, Liviu ^a  

^a INSTITUTE OF SOLID MECHANICS   (Romania)

Author keywords

Cauchy Problem; Inverse Problem; Iterative Method of Fundamental Solutions (MFS); Laplace Equation; Regularization

Indexed keywords

CAUCHY PROBLEMS; EXACT SOLUTION; GENERALIZED CROSS VALIDATION; ITERATIVE ALGORITHM; ITERATIVE METHOD OF FUNDAMENTAL SOLUTIONS (MFS); ITERATIVE PROCEDURES; LAPLACE OPERATOR; MESH-LESS METHODS; METHOD OF FUNDAMENTAL SOLUTIONS; NUMERICAL CONVERGENCE; NUMERICAL IMPLEMENTATION; NUMERICAL PROCEDURES; OPTIMAL VALUES; REGULARIZATION; REGULARIZATION PARAMETERS; STOPPING CRITERIA; TIKHONOV REGULARIZATION METHOD; TWO-DIMENSIONAL GEOMETRY;

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; INTEGRODIFFERENTIAL EQUATIONS; INVERSE PROBLEMS; LAPLACE EQUATION; LAPLACE TRANSFORMS; MATHEMATICAL OPERATORS; SIGNAL FILTERING AND PREDICTION; TWO DIMENSIONAL;

ITERATIVE METHODS;

EID: 77449122653     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (66)

1. Andrieux, S.; Baranger, T.; Ben Abda, A. (2006): Solving Cauchy problems by minimizing an energy-like functional. Inverse Problems, vol. 22, pp. 115-133.
2. Ang, D. D.; Nghia, H.; Tam, N. C. (1998): Regularized solutions of a Cauchy problem for the Laplace equation in an irregular layer: A three-dimensional model. Acta Mathematica Vietnamica, vol. 23, pp. 65-74.
3. Bourgeois, L. (2005): A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation. Inverse Problems, vol. 21, pp. 1087-1104.
4. Burgess, G.; Maharejin, E. (1984): A comparison of the boundary element and superposition methods. Computers & Structures, vol. 19, pp. 697-705
5. Chen, C. W.; Young, D. L.; Tsai, C. C.; Murugesan, K. (2005): The method of fundamental solutions forinverse 2D Stokes problems. ComputationalMechanics, vol. 37, pp. 2-14.
6. Cheng, J.; Hon, Y. C.; Wei, T.; Yamamoto, M. (2001): Numerical computation of a Cauchy problem for Laplace's equation. ZAMM- ZeitschriftfürAngewandte Mathematik undMechanik, vol. 81, pp. 665-674.
7. Cho, H. A.; Golberg, M. A.; Muleshkov, A. S.; Li, X. (2004): Trefftz methods for time dependentpartial differential equations. CMC: Computers, Materials, and Continua, vol. 1, pp. 1-37.
8. Cimetière, A,; Delvare, F.; Jaoua, M.; Pons, F. (2001): Solution of the Cauchy problem using iterated Tikhonov regularization. Inverse Problems, vol. 17, pp. 553-570.
9. Cimetière, A,; Delvare, F.; Jaoua, M.; Pons, F. (2002): An inversion method for harmonic functions reconstruction. International Journal of Thermal Sciences, vol. 41, pp. 509-516.
10. Delvare, F.; Cimetière, A. (2008): A first order method for the Cauchy problem for the Laplace equation using BEM. Computational Mechanics, vol. 41, pp. 789-796.
11. Delvare, F.; Cimetière, A.; Pons, F. (2002): An iterative boundary element method for Cauchy inverse problems. ComputationalMechanics, vol. 28, pp. 291-302.
12. Dong, C. F.; Sun, F. Y.; Meng, B. Q. (2007): A method of fundamental solu-tions for inverse heat conduction problems in an anisotropic medium. Engineering Analysis with Boundary Elements, vol. 31, pp. 75-82.
13. Engl, H. W.; Hanke, M.; Neubauer, A. (2000): Regularization ofInverse Prob-lems. Kluwer Academic, Dordrecht.
14. Fairweather, G.; Karageorghis, A.(1998): The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics, vol. 9, pp. 69-95.
15. Fairweather, G.; Karageorghis, A.; Martin, P. A. (2003): The method of fundamental solutions for scattering and radiation problems. Engineering Analysis with Boundary Elements, vol. 27, pp. 759-769.
16. Golberg, M. A.; Chen, C. S. (1999): The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: M. A. Golberg (ed.) Boundary Integral Methods: Numerical and Mathematical Aspects, WIT Press and Compu-tational Mechanics Publications, Boston, pp. 105-176.
17. Gorzelańczyk, P.; Kołodziej, A. (2008): Some remarks concerning the shape of the shape contour with application of the method of fundamental solutions to elastic torsion of prismatic rods. Engineering Analysis with Boundary Elements, vol. 32, pp. 64-75.
18. Hadamard, J. (1923): Lectures on Cauchy Problem in Linear Partial Differential Equations. Yale University Press, New Haven.
19. Hansen, P. C. (1998): Rank-Deficient and Discrete Ill-Posed Problems: Numeri-calAspects ofLinearInversion. SIAM, Philadelphia.
20. Hào, D. N.; Lesnic, D. (2000): The Cauchy problem for Laplace's equation via the conjugate gradient method. IMA Journal of Applied Mathematics, vol. 65, pp. 199-217.
21. Heise, U. (1978): Numerical properties of integral equations in which the given boundary values and the sought solutions are defined on different curves. Comput-ers & Structures, vol. 8, pp. 199-205.
22. Hon, Y. C.; Wei, T. (2001): Backus-Gilbert algorithm for the Cauchy problem of the Laplace equation. Inverse Problems, vol. 17, pp. 261-271.
23. Hon, Y. C.; Wei, T. (2002): A meshless computational method for solving inverse heat conduction problem. International Series on Advances in Boundary Elements, vol. 13, pp. 135-144.
24. Hon, Y. C.; Wei, T. (2003): A meshless scheme for solving inverse problems of Laplace equation. In: Y. C. Hon, M. Yamamoto, J. Cheng and J. L. Lee (eds.) Re-cent Developments in Theories and Numerics. International Conference on Inverse Problems, World Scientific, Hong Kong, pp. 291-300.
25. Hon, Y. C.; Wei, T. (2004): A fundamental solution method for inverse heat conduction problems. Engineering Analysis with Boundary Elements, vol. 28, pp.
26. Hon, Y. C.; Wei, T. (2005): The method of fundamental solutions for solving multidimensional heat conduction problems. CMES: ComputerModeling in Engi-neering & Sciences, vol. 13, pp. 219-228.
27. Jin, B. T.; Zheng, Y. (2006): A meshless method for some inverse problems associated with the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering, vol. 195, pp. 2270-2280.
28. Jourhmane, M.; Lesnic, D.; Mera, N. S. (2004): Relaxation procedures for an iterative algorithm for solving the Cauchy problem for the Laplace equation. Engineering Analysis with Boundary Elements, vol. 28, pp. 655-665.
29. Klibanov, M.; Santosa, F. (1991): A computational quasi-reversibility method for Cauchy problems for Laplace's equation. SIAM Journal on Applied Mathematics, vol. 51, pp. 1653-1675.
30. Kozlov, V. A.; Maz'ya, V. G.; Fomin, A. V. (1991): An iterative method for solving the Cauchy problem for elliptic equations. Computational Mathematics andMathematicalPhysics, vol. 31, pp. 45-52.
31. Kunisch, K.; Zou, J. (1998): Iterative choices of regularization parameters in linear inverse problems. Inverse Problems, vol. 14, pp. 1247-1264.
32. Lesnic, D.; Elliott, L.; Ingham, D. B. (1997): An iterative boundary element method for solving numerically the Cauchy problem for the Lapace equation. En-gineering Analysis with Boundary Elements, vol. 20, pp. 123-133.
33. Ling, L.; Takeuchi, T. (2008): Boundary control for inverse Cauchy problems of the Laplace equations. CMES: ComputerModeling in Engineering & Sciences, vol. 29, pp. 45-54.
34. Lions, J.-L.; Magenes, E. (1972): Non-Homogeneous Boundary Value Problems and Their Applications. Springer Verlag, Heidelberg.
35. Liu, C.-S. (2008a): Solving an inverse Sturm-Liouville problem by a Lie-group method. Boundary Value Problems, vol. 2008, Article ID 749865.
36. Liu, C.-S. (2008b): Identifying time-dependent damping and stiffness functions by a simple and yet accurate method. Journal ofSound and Vibration, vol. 318, pp. 148-165.
37. Liu, C.-S. (2008c): A Lie-group shooting method for simultaneously estimating the time-dependent damping and stiffness coefficients. CMES: ComputerModeling in Engineering & Sciences, vol. 27, pp. 137-149.
38. Liu, C.-S. (2008d): A time-marching algorithm for solving non-linear obstacle problems with the aid of an NCP-function. CMC: Computers, Materials & Con-tinua, vol. 8, pp. 53-65.
39. Liu, C.-S. (2008e): A fictitious time integration method for two-dimensional quasilinear elliptic boundary value problems. CMES: ComputerModeling in Engineering & Sciences, vol. 33, pp. 179-198.
40. Liu, C.-S. (2008f): A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation. Engineering Analysis with Boundary Elements, vol. 32, pp. 778-785.
41. Liu, C.-S.; Atluri, S. N. (2008a): A novel time integration method for solving a large system of non-linear algebraic equations. CMES: Computer Modeling in Engineering & Sciences, vol. 31, pp. 71-83.
42. Liu, C.-S.; Atluri, S. N. (2008b): A novel fictitious time integration method for solving the discretized inverse Sturm-Liouville problems, for specified eigenvalues. CMES: ComputerModeling in Engineering & Sciences, vol. 36, pp. 261-285.
43. Liu, C.-S.; Atluri, S. N. (2009): A fictitious time integration method for the nu-merical solution of the Fredholm integral equation and for numerical differentiation of noisy data, and its relation with the filter theory. CMES: Computer Modeling in Engineering & Sciences, vol. 41, pp. 243-261.
44. Liu, C.-S.; Chang, J. R.; Chang, K. H.; Chen, Y. W. (2008): Simultaneously estimating the time-dependent damping and stiffness coefficients with the aid of vibrational data. CMC: Computers, Materials & Continua, vol. 7, pp. 97-107.
45. Marin, L. (2005a): A meshless method for solving the Cauchy problem in three-dimensional elastostatics. Computers & Mathematics with Applications, vol. 50, pp. 73-92.
46. Marin, L. (2005b): Numerical solutions of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials. International Jour-nal ofSolids and Structures, vol. 42, pp. 4338-4351.
47. Marin, L. (2005c): A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations. Applied Mathematics and Computation, vol. 165, pp. 355-374.
48. Marin, L. (2008): The method of fundamental solutions for inverse problems associated with the steady-state heat conduction in the presence of sources. CMES: ComputerModeling in Engineering & Sciences, vol. 30, pp. 99-122.
49. Marin, L.; Lesnic, D. (2004): The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity. International Journal of Solids andStructures, vol. 41, pp. 3425-3438.
50. Marin, L.; Lesnic, D. (2005a): The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Com-puters & Structures, vol. 83, pp. 267-278.
51. Marin, L.; Lesnic, D. (2005b): The method of fundamental solutions for in-verse boundary value problems associated with the two-dimentional biharmonic equation. Mathematical and Computers in Modelling, vol. 42, pp. 261-278.
52. Marin, L.; Elliott, L.; Ingham, D. B.; Lesnic; D. (2001): Boundary element method for the Cauchy problem in linear elasticity. Engineering Analysis with Boundary Elements, vol. 25, pp. 783-793.
53. Marin, L.; Elliott, L.; Heggs, P. J.; Ingham, D. B.; Lesnic; D.; Wen, X. (2003): An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering, vol. 192, pp. 709-722.
54. Mathon, R.; Johnston, R. L. (1977): The approximate solution ofelliptic bound-ary value problems by fundamental solutions. SIAM Journal on Numerical Analy-sis, vol. 14, pp. 638-650.
55. Mera, N. S. (2005): The method of fundamental solutions for the backward heat conduction problem. Inverse Problems in Science and Engineering, vol. 13, pp. 79-98.
56. Mera, N. S.; Elliott, L.; Ingham, D. B.; Lesnic; D. (2000): The boundary element solution of the Cauchy steady heat conduction problem in an anisotropic medium. International JournalforNumerical Methods in Engineering, vol. 49, pp. 481-499.
57. Morozov, V. A. (1966): On the solution offunctional equations by the method of regularization. Doklady Mathematics, vol. 7, pp. 414-417.
58. Numerical Algorithms Group Library Mark 21 (2007). NAG(UK) Ltd, Wilkinson House, Jordan Hill Road, Oxford, UK.
59. Reinhardt, H. J.; Han, H.; Hào, D. N. (1999): Stability and regularization of a discrete approximation to the Cauchy problem for Laplace's equation. SIAM Journal on Numerical Analysis, vol. 36, pp. 890-905.
60. Shigeta, T.; Young, D. L. (2009): Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singularpoints. JournalofComputationalPhysics, vol. 228, pp. 1903-1915.
61. Tikhonov, A. N.; Arsenin, V. Y. (1986): Methodsfor Solving Ill-PosedProblems. Nauka, Moscow.
62. Wahba, G. (1977): Practical approximate solutions to linear operator equations when the data are noisy. SIAM Journal on Numerical Analysis, vol. 14, pp. 651-667.
63. Wei, T.; Hon, Y. C.; Ling, L. (2007): Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Engineering Analysis with Boundary Elements, vol. 31, 373-385.
64. Wei, T.; Li, Y. S. (2009): An inverse boundary problem for one-dimensional heat equation with a multilayer domain. Engineering Analysis with Boundary Elements, vol. 33, pp. 225-232.
65. Wei, T.; Zhou, D. Y. (2009): Convergence analysis for the Cauchy problem of Laplace's equation by aregularized method offundamental solutions. Advances in ComputationalMathematics, inpress, DOI 10.1007/s10444-009-9134-7.
66. Young, D. L.; Tsai, C. C.; Chen, C. W.; Fan, C. M. (2008): The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation. Computers and Mathematics with Applications, vol. 55, pp. 1189-1200.