Abstract
We investigate the numerical implementation of the alternating iterative algorithm originally proposed by Kozlov et al. (Comput Math Math Phys 31:45–52) for the Cauchy problem associated with the two-dimensional modified Helmholtz equation using a meshless method. The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The iterative MFS algorithm is tested for Cauchy problems for the two-dimensional modified Helmholtz operator to confirm the numerical convergence, stability and accuracy of the method.
Similar content being viewed by others
References
Beskos DE (1997) Boundary element method in dynamic analysis: part II (1986−1996). ASME Appl Mech Rev 50: 149–197
Burgess G, Maharejin E (1984) A comparison of the boundary element and superposition methods. Comput Struct 19: 697–705
Chen JT, Wong FC (1998) Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition. J Sound Vib 217: 75–95
Chen G, Zhou J (1992) Boundary element methods. Academic Press, London
Chen CW, Young DL, Tsai CC, Murugesan K (2005) The method of fundamental solutions for inverse 2D Stokes problems. Comput Mech 37: 2–14
Cho HA, Golberg MA, Muleshkov AS, Li X (2004) Trefftz methods for time dependent partial differential equations. CMC Comput Mater Cont 1: 1–37
Debye P, Hückel E (1923) The theory of electrolytes. I. Lowering of freezing point and related phenomena. Physikalische Zeitschrift 24: 185–206
DeLillo T, Isakov V, Valdivia N, Wang L (2001) The detection of the source of acoustical noise in two dimensions. SIAM J Appl Math 61: 2104–2121
DeLillo T, Isakov V, Valdivia N, Wang L (2003) The detection of surface vibrations from interior acoustical pressure. Inverse Probl 19: 507–524
Dong CF, Sun FY, Meng BQ (2007) A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium. Eng Anal Bound Elem 31: 75–82
Engl HW, Hanke M, Neubauer A (2000) Regularization of inverse problems. Kluwer, Dordrecht
Fairweather G, Karageorghis A (1998) The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 9: 69–95
Fairweather G, Karageorghis A, Martin PA (2003) The method of fundamental solutions for scattering and radiation problems. Eng Anal Bound Elem 27: 759–769
Golberg MA, Chen CS (1999) The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Golberg MA (eds) Boundary integral methods numerical and mathematical aspects. WIT Press and Computational Mechanics Publications, Boston, pp 105–176
Gorzelańczyk P, Kołodziej A (2008) Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods. Eng Anal Bound Elem 32: 64–75
Hadamard J (1923) Lectures on Cauchy problem in linear partial differential equations. Yale University Press, New Haven
Hall WS, Mao XQ (1995) A boundary element investigation of irregular frequencies in electromagnetic scattering. Eng Anal Bound Elem 16: 245–252
Hansen PC (1998) Rank-deficient and discrete ill-posed problems numerical aspects of linear inversion. SIAM, Philadelphia
Harari I, Barbone PE, Slavutin M, Shalom R (1998) Boundary infinite elements for the Helmholtz equation in exterior domains. Int J Numer Methods Eng 41: 1105–1131
Heise U (1978) Numerical properties of integral equations in which the given boundary values and the sought solutions are defined on different curves. Comput Struct 8: 199–205
Hon YC, Wei T (2004) A fundamental solution method for inverse heat conduction problems. Eng Anal Bound Elem 28: 489–495
Hon YC, Wei T (2005) The method of fundamental solutions for solving multidimensional heat conduction problems. Comput Model Eng Sci 13: 219–228
Jin BT, Marin L (2008) The plane wave method for inverse problems associated with Helmholtz-type equations. Eng Anal Bound Elem 32: 223–240
Jin BT, Zheng Y (2005a) Boundary knot method for some inverse problems associated with the Helmholtz equation. Int J Numer Methods Eng 62: 1636–1651
Jin BT, Zheng Y (2005b) Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation. Eng Anal Bound Elem 29: 925–935
Jin BT, Zheng Y (2006) A meshless method for some inverse problems associated with the Helmholtz equation. Comput Methods Appl Mech Eng 195: 2270–2280
Kozlov VA, Maźya VG, Fomin AV (1991) An iterative method for solving the Cauchy problem for elliptic equations. Comput Math Math Phys 31: 45–52
Kraus AD, Aziz A, Welty J (2001) Extended surface heat transfer. Wiley, New York
Liang J, Subramaniam S (1997) Computation of molecular electrostatics with boundary element methods. Biophys J 73: 1830–1841
Ling L, Takeuchi T (2008) Boundary control for inverse Cauchy problems of the Laplace equations. Comput Model Eng Sci 29: 45–54
Lions J-L, Magenes E (1972) Non-homogeneous boundary value problems and their applications. Springer, Heidelberg
Marin L (2005a) A meshless method for solving the Cauchy problem in three-dimensional elastostatics. Comput Math Appl 50: 73–92
Marin L (2005b) Numerical solutions of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials. Int J Solids Struct 42: 4338–4351
Marin L (2005c) A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations. Appl Math Comput 165: 355–374
Marin L (2008) The method of fundamental solutions for inverse problems associated with the steady-state heat conduction in the presence of sources. Comput Model Eng Sci 30: 99–122
Marin L (2009a) Boundary element-minimal error method for the Cauchy problem associated with Helmholtz-type equations. Comput Mech 44: 205–219
Marin L (2009b) An iterative MFS algorithm for the Cauchy problem associated with the Laplace equation. Comput Model Eng Sci 48: 121–153
Marin L, Lesnic D (2004) The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity. Int J Solids Struct 41: 3425–3438
Marin L, Lesnic D (2005a) The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput Struct 83: 267–278
Marin L, Lesnic D (2005b) The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation. Math Comput Model 42: 261–278
Marin L, Elliott L, Heggs PJ, Ingham DB, Lesnic D, Wen X (2003a) An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation. Comput Methods Appl Mech Eng 192: 709–722
Marin L, Elliott L, Heggs PJ, Ingham DB, Lesnic D, Wen X (2003b) Conjugate gradient-boundary element solution to the Cauchy for Helmholtz-type equations. Comput Mech 31: 367–377
Marin L, Elliott L, Heggs PJ, Ingham DB, Lesnic D, Wen X (2004a) Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation. Int J Numer Methods Eng 60: 1933–1947
Marin L, Elliott L, Heggs PJ, Ingham DB, Lesnic D, Wen X (2004b) BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method. Eng Anal Bound Elem 28: 1025–1034
Mathon R, Johnston RL (1977) The approximate solution of elliptic boundary value problems by fundamental solutions. SIAM J Numer Anal 14: 638–650
Morozov VA (1966) On the solution of functional equations by the method of regularization. Dokl Math 7: 414–417
Numerical Algorithms Group Library Mark 21 (2007) NAG(UK) Ltd, Wilkinson House, Jordan Hill Road, Oxford, UK
Qin HH, Wei T (2009a) Modified regularization method for the Cauchy problem of the Helmholtz equation. Appl Math Model 33: 2334–2348
Qin HH, Wei T (2009b) Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation. Math Comput Simul 80: 352–366
Qin HH, Wei T (2010) Two regularization methods for the Cauchy problems of the Helmholtz equation. Appl Math Model 34: 947–967
Qin HH, Wen DW (2009) Tikhonov type regularization method for the Cauchy problem of the modified Helmholtz equation. Appl Math Comput 203: 617–628
Qin HH, Wei T, Shi R (2009) Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation. J Comput Appl Math 224: 39–53
Shi R, Wei T, Qin HH (2009) A fourth-order modified method for the Cauchy problem of the modified Helmholtz equation. Numer Math Theory Methods Appl 2: 326–340
Shigeta T, Young DL (2009) Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points. J Comput Phys 228: 1903–1915
Tikhonov AN, Arsenin VY (1986) Methods for solving ill-posed problems. Nauka, Moscow
Wahba G (1977) Practical approximate solutions to linear operator equations when the data are noisy. SIAM J Numer Anal 14: 651–667
Wei T, Li YS (2009) An inverse boundary problem for one-dimensional heat equation with a multilayer domain. Eng Anal Bound Elem 33: 225–232
Wei T, Hon YC, Ling L (2007) Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Eng Anal Bound Elem 31: 373–385
Wei T, Qin HH, Shi R (2008) Numerical solution of an inverse 2D Cauchy problem connected with the Helmholtz equation. Inverse Probl 24, art. no. 035003
Xiong XT, Fu CL (2007) Two approximate methods of a Cauchy problem for the Helmholtz equation. Comput Appl Math 26: 285–307
Young DL, Tsai CC, Chen CW, Fan CM (2008) The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation. Comput Math Appl 55: 1189–1200
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Marin, L. An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation. Comput Mech 45, 665–677 (2010). https://doi.org/10.1007/s00466-010-0480-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-010-0480-6