Boundary element-minimal error method for the Cauchy problem associated with Helmholtz-type equations

Computational Mechanics

Volumn 44, Issue 2, 2009, Pages 205-219

  Marin, Liviu ^a  

^a INSTITUTE OF SOLID MECHANICS   (Romania)

Author keywords

Boundary element method (BEM); Cauchy problem; Helmholtz type equations; Inverse problem; Iterative algorithms; Regularization

Indexed keywords

BOUNDARY ELEMENT METHOD; HELMHOLTZ EQUATION; ITERATIVE METHODS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; SAILING VESSELS;

CAUCHY PROBLEMS; HELMHOLTZ-TYPE EQUATIONS; ITERATIVE ALGORITHM; MODIFIED HELMHOLTZ EQUATIONS; NUMERICAL IMPLEMENTATION; REGULARIZATION; STABLE NUMERICAL SOLUTIONS; STOPPING CRITERIA;

INVERSE PROBLEMS;

EID: 65449154360     PISSN: 01787675     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00466-009-0368-5     Document Type: Article

References (42)

1. G Bastay VA Kozlov B Turesson 2001 Iterative methods for and inverse heat conduction problem J Inverse Ill-Posed Prob 9 375 388
2. DE Beskos 1997 Boundary element method in dynamic analysis, Part II (1986-1996) ASME Appl Mech Rev 50 149 197 (Pubitemid 127666479)
3. Chen G, Zhou J (1992) Boundary element methods. Academic Press, London
4. JT Chen FC Wong 1998 Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition J Sound Vib 217 75 95 (Pubitemid 128398505)
5. T DeLillo V Isakov N Valdivia L Wang 2001 The detection of the source of acoustical noise in two dimensions SIAM J Appl Math 61 2104 2121 (Pubitemid 34125463)
6. T DeLillo V Isakov N Valdivia L Wang 2003 The detection of surface vibrations from interior acoustical pressure Inverse Probl 19 507 524
7. Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems. Kluwer, Boston
8. Hadamard J (1923) Lectures on Cauchy problem in linear partial differential equations. Oxford University Press, London
9. WS Hall XQ Mao 1995 A boundary element investigation of irregular frequencies in electromagnetic scattering Eng Anal Bound Elem 16 245 252
10. Hanke M (1995) Conjugate gradient type methods for ill-posed problems. Longman Scientific and Technical, New York
11. M Hanke 1995 The minimal error conjugate gradient method is a regularization method Proc Am Math Soc 123 3487 3497 (Pubitemid 126013451)
12. DN Háo H-J Reinhardt 1998 Gradient methods for inverse heat conduction problems Inverse Probl Eng 6 177 211
13. DN Háo D Lesnic 2000 The Cauchy problem for Laplace's equation via the conjugate gradient method IMA J Appl Math 65 199 217
14. I Harari PE Barbone M Slavutin R Shalom 1998 Boundary infinite elements for the Helmholtz equation in exterior domains Int J Numer Methods Eng 41 1105 1131 (Pubitemid 128570841)
15. B Jin L Marin 2008 The plane wave method for inverse problems associated with Helmholtz-type equations Eng Anal Bound Elem 32 223 240
16. B Jin Y Zheng 2005 Boundary knot method for some inverse problems associated with the Helmholtz equation Int J Numer Methods Eng 62 1636 1651 (Pubitemid 40427190)
17. B Jin Y Zheng 2005 Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation Eng Anal Bound Elem 29 925 935
18. B Jin Y Zheng 2006 A meshless method for some inverse problems associated with the Helmholtz equation Comput Methods Appl Mech Eng 195 2270 2288 (Pubitemid 43308636)
19. Johansson T (2000) Reconstruction of a Stationary Flow from Boundary Data. Dissertations No. 853, Linköping Studies in Science and Technology
20. T Johansson D Lesnic 2006 A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow Appl Anal 85 1327 1341
21. T Johansson D Lesnic 2006 Reconstruction of a stationary flow from incomplete boundary data using iterative methods Eur J Appl Math 17 651 663
22. T Johansson D Lesnic 2007 An iterative method for the reconstruction of a stationary flow Numer Methods Partial Differ Equ 23 998 1017 (Pubitemid 47381156)
23. JT King 1989 The minimal error conjugate gradient method for ill-posed problems J Optim Theory Appl 60 297 304
24. Kraus AD, Aziz A, Welty J (2001) Extended surface heat transfer. Wiley, New York
25. L Landweber 1951 An iteration formula for Fredholm integral equations of the first kind Am J Math 73 615 624
26. L Marin 2005 A meshless method for the numerical solution of Cauchy problem associated with three-dimensional Helmholtz-type equations Appl Math Comput 165 355 374
27. L Marin 2009 The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity Int J Solids Struct 46 957 974
28. L Marin D Lesnic 2005 The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations Comput Struct 83 267 278 (Pubitemid 41454921)
29. L Marin D Lesnic 2005 Boundary element-Landweber method for the Cauchy problem in linear elasticity IMA J Appl Math 18 817 825
30. L Marin DN Háo D Lesnic 2002 Conjugate gradient-boundary element method for the Cauchy problem in elasticity Quart J Mech Appl Math 55 227 247 (Pubitemid 34658627)
31. L Marin L Elliott PJ Heggs DB Ingham D Lesnic X Wen 2003 An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation Comput Methods Appl Mech Eng 192 709 722 (Pubitemid 36211369)
32. L Marin L Elliott PJ Heggs DB Ingham D Lesnic X Wen 2003 Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations Comput Mech 31 367 377
33. L Marin L Elliott PJ Heggs DB Ingham D Lesnic X Wen 2004 Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation Int J Numer Methods Eng 60 1933 1947 (Pubitemid 38978689)
34. L Marin L Elliott PJ Heggs DB Ingham D Lesnic X Wen 2004 BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method Eng Anal Bound Elem 28 1025 1034
35. L Marin L Elliott PJ Heggs DB Ingham D Lesnic X Wen 2006 Dual reciprocity boundary element method solution of the Cauchy problem for Helmholtz-type equations with variable coefficients J Sound Vib 297 89 105
36. VA Morozov 1966 On the solution of functional equations by the method of regularization Soviet Math Doklady 167 414 417
37. AS Nemirovskii 1986 Regularizing properties of the conjugate gradient method in ill-posed problems USSR Comput Math Math Phys 26 7 16
38. Qin HH, Wen DW (2009) Tikhonov type regularization method for the Cauchy problem of the modified Helmholtz equation. Appl Math Comput (in press)
39. HH Qin T Wei R Shi 2009 Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation J Comput Appl Math 224 39 53
40. Tumakov DN (2006) The Cauchy problem for the Helmholtz equation in a domain with a piecewise-smooth boundary. Mathematical Methods in Electromagnetic Theory, MMET, Conference Proceedings, art. no. 1689818, pp. 446-448
41. Wei T, Qin HH, Shi R (2008) Numerical solution of an inverse 2D Cauchy problem connected with the Helmholtz equation. Inverse Problems, vol 24, art. no. 035003
42. X-T Xiong C-L Fu 2007 Two approximate methods of a Cauchy problem for the Helmholtz equation Comput Appl Math 26 285 307