# An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation

- 177 Downloads
- 18 Citations

## 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.

## Keywords

Modified Helmholtz equation Inverse problem Cauchy problem Iterative method of fundamental solutions (MFS) Regularization## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Beskos DE (1997) Boundary element method in dynamic analysis: part II (1986−1996). ASME Appl Mech Rev 50: 149–197CrossRefGoogle Scholar
- 2.Burgess G, Maharejin E (1984) A comparison of the boundary element and superposition methods. Comput Struct 19: 697–705zbMATHCrossRefGoogle Scholar
- 3.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: 75–95CrossRefGoogle Scholar
- 4.Chen G, Zhou J (1992) Boundary element methods. Academic Press, LondonzbMATHGoogle Scholar
- 5.Chen CW, Young DL, Tsai CC, Murugesan K (2005) The method of fundamental solutions for inverse 2D Stokes problems. Comput Mech 37: 2–14zbMATHCrossRefMathSciNetGoogle Scholar
- 6.Cho HA, Golberg MA, Muleshkov AS, Li X (2004) Trefftz methods for time dependent partial differential equations. CMC Comput Mater Cont 1: 1–37Google Scholar
- 7.Debye P, Hückel E (1923) The theory of electrolytes. I. Lowering of freezing point and related phenomena. Physikalische Zeitschrift 24: 185–206Google Scholar
- 8.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: 2104–2121zbMATHCrossRefMathSciNetGoogle Scholar
- 9.DeLillo T, Isakov V, Valdivia N, Wang L (2003) The detection of surface vibrations from interior acoustical pressure. Inverse Probl 19: 507–524zbMATHCrossRefMathSciNetGoogle Scholar
- 10.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: 75–82CrossRefGoogle Scholar
- 11.Engl HW, Hanke M, Neubauer A (2000) Regularization of inverse problems. Kluwer, DordrechtGoogle Scholar
- 12.Fairweather G, Karageorghis A (1998) The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 9: 69–95zbMATHCrossRefMathSciNetGoogle Scholar
- 13.Fairweather G, Karageorghis A, Martin PA (2003) The method of fundamental solutions for scattering and radiation problems. Eng Anal Bound Elem 27: 759–769zbMATHCrossRefGoogle Scholar
- 14.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 105–176Google Scholar
- 15.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: 64–75CrossRefGoogle Scholar
- 16.Hadamard J (1923) Lectures on Cauchy problem in linear partial differential equations. Yale University Press, New HavenzbMATHGoogle Scholar
- 17.Hall WS, Mao XQ (1995) A boundary element investigation of irregular frequencies in electromagnetic scattering. Eng Anal Bound Elem 16: 245–252CrossRefGoogle Scholar
- 18.Hansen PC (1998) Rank-deficient and discrete ill-posed problems numerical aspects of linear inversion. SIAM, PhiladelphiaGoogle Scholar
- 19.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: 1105–1131zbMATHCrossRefGoogle Scholar
- 20.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: 199–205zbMATHCrossRefMathSciNetGoogle Scholar
- 21.Hon YC, Wei T (2004) A fundamental solution method for inverse heat conduction problems. Eng Anal Bound Elem 28: 489–495zbMATHCrossRefGoogle Scholar
- 22.Hon YC, Wei T (2005) The method of fundamental solutions for solving multidimensional heat conduction problems. Comput Model Eng Sci 13: 219–228Google Scholar
- 23.Jin BT, Marin L (2008) The plane wave method for inverse problems associated with Helmholtz-type equations. Eng Anal Bound Elem 32: 223–240CrossRefGoogle Scholar
- 24.Jin BT, Zheng Y (2005a) Boundary knot method for some inverse problems associated with the Helmholtz equation. Int J Numer Methods Eng 62: 1636–1651zbMATHCrossRefMathSciNetGoogle Scholar
- 25.Jin BT, Zheng Y (2005b) Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation. Eng Anal Bound Elem 29: 925–935CrossRefGoogle Scholar
- 26.Jin BT, Zheng Y (2006) A meshless method for some inverse problems associated with the Helmholtz equation. Comput Methods Appl Mech Eng 195: 2270–2280zbMATHCrossRefMathSciNetGoogle Scholar
- 27.Kozlov VA, Maźya VG, Fomin AV (1991) An iterative method for solving the Cauchy problem for elliptic equations. Comput Math Math Phys 31: 45–52Google Scholar
- 28.Kraus AD, Aziz A, Welty J (2001) Extended surface heat transfer. Wiley, New YorkGoogle Scholar
- 29.Liang J, Subramaniam S (1997) Computation of molecular electrostatics with boundary element methods. Biophys J 73: 1830–1841CrossRefGoogle Scholar
- 30.Ling L, Takeuchi T (2008) Boundary control for inverse Cauchy problems of the Laplace equations. Comput Model Eng Sci 29: 45–54MathSciNetGoogle Scholar
- 31.Lions J-L, Magenes E (1972) Non-homogeneous boundary value problems and their applications. Springer, HeidelbergGoogle Scholar
- 32.Marin L (2005a) A meshless method for solving the Cauchy problem in three-dimensional elastostatics. Comput Math Appl 50: 73–92zbMATHCrossRefMathSciNetGoogle Scholar
- 33.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: 4338–4351zbMATHCrossRefMathSciNetGoogle Scholar
- 34.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: 355–374zbMATHCrossRefMathSciNetGoogle Scholar
- 35.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: 99–122MathSciNetGoogle Scholar
- 36.Marin L (2009a) Boundary element-minimal error method for the Cauchy problem associated with Helmholtz-type equations. Comput Mech 44: 205–219zbMATHCrossRefMathSciNetGoogle Scholar
- 37.Marin L (2009b) An iterative MFS algorithm for the Cauchy problem associated with the Laplace equation. Comput Model Eng Sci 48: 121–153Google Scholar
- 38.Marin L, Lesnic D (2004) The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity. Int J Solids Struct 41: 3425–3438zbMATHCrossRefGoogle Scholar
- 39.Marin L, Lesnic D (2005a) The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput Struct 83: 267–278CrossRefMathSciNetGoogle Scholar
- 40.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: 261–278zbMATHCrossRefMathSciNetGoogle Scholar
- 41.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: 709–722zbMATHCrossRefMathSciNetGoogle Scholar
- 42.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: 367–377zbMATHMathSciNetGoogle Scholar
- 43.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: 1933–1947zbMATHCrossRefMathSciNetGoogle Scholar
- 44.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: 1025–1034zbMATHCrossRefGoogle Scholar
- 45.Mathon R, Johnston RL (1977) The approximate solution of elliptic boundary value problems by fundamental solutions. SIAM J Numer Anal 14: 638–650zbMATHCrossRefMathSciNetGoogle Scholar
- 46.Morozov VA (1966) On the solution of functional equations by the method of regularization. Dokl Math 7: 414–417zbMATHGoogle Scholar
- 47.Numerical Algorithms Group Library Mark 21 (2007) NAG(UK) Ltd, Wilkinson House, Jordan Hill Road, Oxford, UKGoogle Scholar
- 48.Qin HH, Wei T (2009a) Modified regularization method for the Cauchy problem of the Helmholtz equation. Appl Math Model 33: 2334–2348CrossRefMathSciNetGoogle Scholar
- 49.Qin HH, Wei T (2009b) Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation. Math Comput Simul 80: 352–366zbMATHCrossRefMathSciNetGoogle Scholar
- 50.Qin HH, Wei T (2010) Two regularization methods for the Cauchy problems of the Helmholtz equation. Appl Math Model 34: 947–967CrossRefGoogle Scholar
- 51.Qin HH, Wen DW (2009) Tikhonov type regularization method for the Cauchy problem of the modified Helmholtz equation. Appl Math Comput 203: 617–628CrossRefMathSciNetGoogle Scholar
- 52.Qin HH, Wei T, Shi R (2009) Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation. J Comput Appl Math 224: 39–53zbMATHCrossRefMathSciNetGoogle Scholar
- 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: 326–340Google Scholar
- 54.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: 1903–1915zbMATHCrossRefMathSciNetGoogle Scholar
- 55.Tikhonov AN, Arsenin VY (1986) Methods for solving ill-posed problems. Nauka, MoscowGoogle Scholar
- 56.Wahba G (1977) Practical approximate solutions to linear operator equations when the data are noisy. SIAM J Numer Anal 14: 651–667zbMATHCrossRefMathSciNetGoogle Scholar
- 57.Wei T, Li YS (2009) An inverse boundary problem for one-dimensional heat equation with a multilayer domain. Eng Anal Bound Elem 33: 225–232CrossRefMathSciNetGoogle Scholar
- 58.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: 373–385CrossRefGoogle Scholar
- 59.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. 035003Google Scholar
- 60.Xiong XT, Fu CL (2007) Two approximate methods of a Cauchy problem for the Helmholtz equation. Comput Appl Math 26: 285–307CrossRefMathSciNetGoogle Scholar
- 61.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: 1189–1200zbMATHCrossRefMathSciNetGoogle Scholar