<i>A Posteriori</i> Regularization Parameter Choice Rule for Truncation Method for Identifying the Unknown Source of the Poisson Equation

Li, Xiaoxiao; Li, Dun-Gang

doi:10.1155/2013/590737

Cited by 2 publications

(1 citation statement)

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“…Currently, there have been many other fields of study where the Helmholtz-type equations can be greatly used, such as the influence of the frequency on the stability of Cauchy problems [7], finding the shape of a part of a boundary in [2], regularization of the modified Helmholtz equation in [12] and the problem of identifying source functions in [1,10].…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of the electric field of the Helmholtz equation in 3D

Nguyen¹,

Khoa²,

Minh³

et al. 2014

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In this paper, we rigorously investigate the truncation method for the Cauchy problem of Helmholtz equations which is widely used to model propagation phenomena in physical applications. The method is a well-known approach to the regularization of several types of ill-posed problems, including the model postulated by Regińska and Regiński [14].Under certain specific assumptions, we examine the ill-posedness of the non-homogeneous problem by exploring the representation of solutions based on Fourier mode. Then the so-called regularized solution is established with respect to a frequency bounded by an appropriate regularization parameter. Furthermore, we provide a short analysis of the nonlinear forcing term. The main results show the stability as well as the strong convergence confirmed by the error estimates in L 2 -norm of such regularized solutions. Besides, the regularization parameters are formulated properly. Finally, some illustrative examples are provided to corroborate our qualitative analysis.

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