An Iterative Regularization Method for Identifying the Source Term in a Second Order Differential Equation

Zouyed, Fairouz; Djemoui, Sebti

doi:10.1155/2015/713403

Cited by 8 publications

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References 16 publications

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“…[9,25]) has to be employed in order to obtain some stable approximations for the source function. For ψ ≡ 1 in [0, τ ], an iterative regularization method has been proposed in [38] to obtain stable approximations for the identification of the spatially dependent source function. For the general source term of the separable form that is considered in this paper, it seems the work on the regularization aspect is in the initial stage.…”

Section: Introductionmentioning

confidence: 99%

Identification of Matrix Diffusion Coefficient in a Parabolic PDE

Mondal

Nair

2021

Computational Methods in Applied Mathematics

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An inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE is considered. Following the idea of natural linearization, considered by Cao and Pereverzev (2006), the nonlinear inverse problem is transformed into a problem of solving an operator equation where the operator involved is linear. Solving the linear operator equation turns out to be an ill-posed problem. The method of Tikhonov regularization is employed for obtaining stable approximations and its finite-dimensional analysis is done based on the Galerkin method, for which an orthogonal projection on the space of matrices with entries from L 2 ⁢ ( Ω ) L^{2}(\Omega) is defined. Since the error estimates in Tikhonov regularization method rely heavily on the adjoint operator, an explicit representation of adjoint of the linear operator involved is obtained. For choosing the regularizing parameter, the adaptive technique is employed in order to obtain order optimal rate of convergence. For the relaxed noisy data, we describe a procedure for obtaining a smoothed version so as to obtain the error estimates. Numerical experiments are carried out for a few illustrative examples.

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Section: Introductionmentioning

confidence: 99%