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On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation. (English) Zbl 1185.65171

This paper is near in its problem, assumptions, regularization method and result to that of H.-H. Qin and T. Wei [Math. Comput. Simul. 80, No. 2, 352–366 (2009; Zbl 1185.65172)] who, however, considered the modified Helmholtz equation and also a second regularization method – proving better in the numerical experiments.
These authors obtain \(L_2\) convergence of the regularized solution choosing the regularization parameter in dependence on solution apriori bound and measurement error, but show no numerical results.

MSC:

65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35R25 Ill-posed problems for PDEs

Citations:

Zbl 1185.65172
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Full Text: DOI

References:

[1] Marin, L.; Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Wen, X., An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comput. meth. appl. mech. eng., 192, 709-722, (2003) · Zbl 1022.78012
[2] Marin, L.; Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Wen, X., Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations, Comput. mech., 31, 367-377, (2003) · Zbl 1047.65097
[3] Wei, T.; Hon, Y.C.; Ling, L., Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Eng. anal. bound. elem., 31, 4, 373-385, (2007) · Zbl 1195.65206
[4] Marin, L., A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations, Appl. math. comput., 165, 2, 355-374, (2005) · Zbl 1070.65115
[5] Marin, L.; Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Wen, X., BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method, Eng. anal. bound. elem., 28, 1025-1034, (2004) · Zbl 1066.80009
[6] Marin, L.; Lesnic, D., The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations, Comput. struct., 83, 267-278, (2005) · Zbl 1088.35079
[7] Marin, L.; Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Wen, X., Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation, Int. J. numer. meth. eng., 60, 11, 1933-1947, (2004) · Zbl 1062.78015
[8] Qin, H.H.; Wei, T., Modified regularization method for the Cauchy problem of the Helmholtz equation, Appl. math. modell., 33, 2334-2348, (2009) · Zbl 1185.65203
[9] X.T. Xiong, Regularization theory and algorithm for some inverse problems for parabolic differential equations, Ph.D. Dissertation, Lanzhou University, 2007 (in Chinese)
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