The plane wave method for inverse problems associated with Helmholtz-type equations

Engineering Analysis with Boundary Elements

Volumn 32, Issue 3, 2008, Pages 223-240

  Jin, Bangti ^a   Marin, Liviu ^b  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b UNIVERSITY OF NOTTINGHAM   (United Kingdom)

Author keywords

Helmholtz type equations; Inverse problems; Meshfree meshless method; Plane wave method; Regularization; Ridge function

Indexed keywords

BOUNDARY VALUE PROBLEMS; HELMHOLTZ EQUATION; LINEAR ALGEBRA; NUMERICAL METHODS; SINGULAR VALUE DECOMPOSITION; THREE DIMENSIONAL;

HELMHOLTZ TYPE EQUATIONS; MESHFREE METHODS; MESHLESS METHODS; OPTIMAL TRUNCATION NUMBERS; PLANE WAVE METHODS; REGULARIZATION; RIDGE FUNCTIONS;

INVERSE PROBLEMS;

EID: 38349148294     PISSN: 09557997     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.enganabound.2007.08.005     Document Type: Article

References (56)

1. Wood A.S., Tupholme G.E., Bhatti M.I.H., and Heggs P.J. Steady-state heat transfer through extended plane surfaces. Int Commun Heat Mass Transfer 22 (1995) 99-109
2. Lax P.D., and Phillips R.S. Scattering theory (1967), Academic Press, New York
3. Jones D.S. Methods in electromagnetic wave propagation (1979), Oxford University Press, New York
4. Colton D., and Kress R. Integral equation methods in scattering theory (1983), Wiley, New York
5. Chen J.T. Recent development of dual BEM in acoustic problems. Comput Methods Appl Mech Eng 188 (2000) 833-845
6. Pike ER, Sabatier P, editors. Scattering. London: Academic Press; 2001.
7. Harari I. A survey of finite element methods for time-harmonic acoustics. Comput Methods Appl Mech Eng 195 (2006) 1594-1607
8. John F. Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Commun Pure Appl Math 13 (1960) 551-585
9. Hrycak T., and Isakov V. Increased stability in the continuation of solutions to the Helmholtz equation. Inverse Prob 20 (2004) 697-712
10. Reginska T., and Reginska K. Approximate solution of a Cauchy problem for the Helmholtz equation. Inverse Probl 22 (2006) 975-989
11. Knightly G.H., and St. Mary D.F. Stable marching schemes based on elliptic models of wave propagation. J Acoust Soc Am 93 (1993) 1866-1872
12. Bai M.R. Application of BEM-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries. J Acoust Soc Am 92 (1992) 533-549
13. Wang Z., and Wu S.F. Helmholtz equation least-squares method for reconstructing the acoustic pressure field. J Acoust Soc Am 102 (1997) 2020-2032
14. Wu S.F., and Yu J. Reconstructing interior acoustic pressure fields via Helmholtz equation-least-squares method. J Acoust Soc Am 104 (1998) 2054-2060
15. Isakov V., and Wu S.F. On theory and application of the Helmholtz equation least squares method in inverse acoustics. Inverse Probl 18 (2002) 1147-1159
16. Semenova T., and Wu S.F. On the choice of expansion functions in the Helmholtz equation least-squares method. J Acoust Soc Am 117 (2005) 701-710
17. DeLillo T., Isakov V., Valdivia N., and Wang L. The detection of the source of acoustical noise in two dimensions. SIAM J Appl Math 61 (2001) 2104-2121
18. Natterer F., and Wübbeling F. A propagation-backpropagation method for ultrasound tomography. Inverse Probl 11 (1995) 1225-1232
19. Natterer F., and Wübbeling F. Marching schemes for inverse acoustic scattering problems. Numer Math 100 (2005) 697-710
20. Jin B., and Zheng Y. Boundary knot method for some inverse problems associated with the Helmholtz equation. Int J Numer Methods Eng 62 (2005) 1636-1651
21. Jin B., and Zheng Y. A meshless method for some inverse problems associated with the Helmholtz equation. Comput Methods Appl Mech Eng 195 (2006) 2270-2288
22. Marin L., and Lesnic D. The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput Struct 83 (2005) 267-278
23. Marin L. A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations. Appl Math Comput 165 (2005) 355-374
24. Wei T., Hon Y.C., and Ling L. Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Eng Anal Boundary Elem 31 (2007) 373-385
25. Marin L., Elliott L., Heggs P.J., Ingham D.B., Lesnic D., and Wen X. An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation. Comput Methods Appl Mech Eng 192 (2003) 709-722
26. Marin L., Elliott L., Heggs P.J., Ingham D.B., Lesnic D., and Wen X. Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations. Comput Mech 31 (2003) 367-377
27. Marin L., Elliott L., Heggs P.J., Ingham D.B., Lesnic D., and Wen X. Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation. Int J Numer Methods Eng 60 (2004) 1933-1947
28. Kozlov V.A., Mazỳa V.G., and Fomin A.V. An iterative method for solving the Cauchy problem for elliptic equations. USSR Comput Math Math Phys 31 (1992) 45-52
29. Griebel M, Schweitzer MA, editors. Meshfree methods for partial differential equations. Berlin, New York: Springer; 2003.
30. Liu G.R. Meshfree methods: Moving beyond the finite element method (2003), CRC Press, Boca Raton, FL
31. Fairweather G., and Karageorghis A. The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 9 (1998) 69-95
32. Golberg M.A., and Chen C.S. The method of fundamental solution for potential Helmholtz and diffusion problems. In: Golberg M.A. (Ed). Boundary integral methods-numerical and mathematical aspects (1998), Computational Mechanics Publications, Southampton 103-176
33. Chen W., and Tanaka M. A meshless, integration-free, and boundary-only RBF technique. Comput Math Appl 43 (2002) 379-391
34. Hon Y.C., and Chen W. Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems under complicated geometry. Int J Numer Methods Eng 56 (2003) 1931-1948
35. Pinkus A. Approximating by ridge functions. In: Le Mehaute A., Rabut C., and Schumaker L.L. (Eds). Surface fitting and multiresolution methods (1997), Vanderbilt University Press, Nashville, TN 279-292
36. John F. Plane waves and spherical means applied to partial differential equations (1955), Interscience Publishers, New York
37. Heller E.J. Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits. Phys Rev Lett 53 (1984) 1515-1518
38. Li B., Robnik M., and Hu B. Relevance of chaos in numerical solutions of quantum billiards. Phys Rev E 57 (1998) 4095-4105
39. Sakoda K. Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice. J Opt Soc Am B 14 (1997) 1961-1966
40. Alves CJS, Valtchev SS. Numerical simulation of acoustic wave scattering using a meshfree plane waves method. In: International Workshop on Meshfree Methods 2003, URL: 〈http://www.math.ist.utl.pt/meshfree/svilen.pdf〉; 2003.
41. Alves C.J.S., and Valtchev S.S. Numerical comparison of two meshfree methods for acoustic wave scattering. Eng Anal Boundary Elem 29 (2005) 371-382
42. Li X. On solving boundary value problems of modified Helmholtz equations by plane wave functions. J Comput Appl Math 195 (2006) 66-82
43. Abramowitz M, Stegun IA, editors. Handbook of mathematical functions, with formulas, graphs, and mathematical tables. New York: Dover; 1965.
44. Kroó A. On approximation by ridge functions. Constr Approximation 13 (1997) 447-460
45. Barron A.R. Universal approximation bounds for superposition of a sigmoidal function. IEEE Trans Inform Theory 39 (1993) 930-945
46. Petrushev P.P. Approximation by ridge functions and neural networks. SIAM J Math Anal 30 (1998) 155-180
47. Hansen P.C. Rank-deficient and discrete ill-posed problems: Numerical aspects of linear inversion (1998), SIAM, Philadelphia
48. Hansen P.C., and O'Leary D.P. The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J Sci Comput 14 (1993) 1487-1503
49. Ling L. Matlab program pointonsphere.m., URL: 〈http://www.mathworks.com/matlabcentral〉.
50. Galperin E.A., and Zheng Q. Solution and control of PDE via global optimization methods. Comput Math Appl 25 (1993) 103-118
51. Golub GH, Van Loan CF, Matrix computations, 3rd ed. Baltimore: Johns Hopkins University Press; 1996.
52. Shu C., and Xue H. Solution of Helmholtz equation by differential quadrature method. Comput Methods Appl Mech Eng 175 (1999) 203-212
53. Karageorghis A. Modified methods of fundamental solutions for harmonic and biharmonic problems with boundary singularities. Numer Methods Partial Differential Equations 8 (1992) 1-19
54. Li Z.C., Lu T.T., Hu H.Y., and Cheng A.H.-D. Trefftz and collocation methods (2005), WIT Press, Southampton
55. Vergini E., and Saraceno M. Calculation by scaling of highly excited states of billiards. Phys Rev E 52 (1995) 2204-2207
56. Barnett A.H., Cohen D., and Heller E.J. Deformations and dilations of chaotic billiards, dissipation rate, and quasi-orthogonality of the boundary wavefunctions. Phys Rev Lett 85 (2000) 1412-1415