Recovering the initial condition of parabolic equations from lateral Cauchy data via the quasi-reversibility method

Li, Qitong; Nguyen, Loc H.

doi:10.1080/17415977.2019.1643850

Cited by 15 publications

(14 citation statements)

References 33 publications

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“…Although convergences at N → ∞ are not proven in many cases, numerical results are usually good ones, see, e.g. [10] for the attenuated tomography with complete data, [14] for the 2D version of the Gelfand-Levitan method, [26] for the inverse problem of finding initial condition of heat equation, [30] for the inverse source problem for the Helmholtz equation, and [18,21,22] for the convexification.…”

Section: Convergence Rate Of Regularized Solutions Lipschitz Stabilimentioning

confidence: 99%

On an Inverse Source Problem for the Full Radiative Transfer Equation with Incomplete Data

Smirnov¹,

Klibanov²,

Nguyen³

2019

SIAM J. Sci. Comput.

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A new numerical method to solve an inverse source problem for the radiative transfer equation involving the absorption and scattering terms, with incomplete data, is proposed. No restrictive assumption on those absorption and scattering coefficients is imposed. The original inverse source problem is reduced to boundary value problem for a system of coupled partial differential equations of the first order. The unknown source function is not a part of this system. Next, we write this system in the fully discrete form of finite differences. That discrete problem is solved via the quasi-reversibility method. We prove the existence and uniqueness of the regularized solution. Especially, we prove the convergence of regularized solutions to the exact one as the noise level in the data tends to zero via a new discrete Carleman estimate. Numerical simulations demonstrate good performance of this method even when the data is highly noisy.

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Section: Convergence Rate Of Regularized Solutions Lipschitz Stabilimentioning

confidence: 99%

On an Inverse Source Problem for the Full Radiative Transfer Equation with Incomplete Data

Smirnov¹,

Klibanov²,

Nguyen³

2019

SIAM J. Sci. Comput.

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“…The assumption about the well-approximation in Theorem 3.1 is verified numerically in some recent works of our research group. This verification for elliptic equation can be found in [35] and the one for parabolic equation is in [31]. In those papers, the basis {Ψ n } ∞ n=1 is taken from [24].…”

Section: A Lipschitz Estimate Based On a Truncation Of The Fourier Sementioning

confidence: 99%

“…Another related problem is the inverse problem of reconstructing the initial condition for parabolic equation. This problem is very important and interesting, see [31,36,33,27,38] for theoretical results and numerical methods. In the current paper, we introduce the following approach to solve Problem 1.1.…”

Section: Introductionmentioning

confidence: 99%

A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem

Nguyen

2019

Journal of Inverse and Ill-Posed Problems

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AbstractTwo main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

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“…Recently, the problem to find the source term as in this manuscript was solved by the quasi-reversibility method, see [22]. This method can be used to solve inverse source problem for nonlinear parabolic equations, the difference of this manuscript and that paper is the observation data [21].…”

Section: Introductionmentioning

confidence: 99%

Identification of the Source for Full Parabolic Equations

Umbricht

2021

Mathematical Modelling and Analysis

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In this work, we consider the problem of identifying the time independent source for full parabolic equations in Rn from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a Hölder type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.

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Recovering the initial condition of parabolic equations from lateral Cauchy data via the quasi-reversibility method

Cited by 15 publications

References 33 publications

On an Inverse Source Problem for the Full Radiative Transfer Equation with Incomplete Data

On an Inverse Source Problem for the Full Radiative Transfer Equation with Incomplete Data

A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem

Identification of the Source for Full Parabolic Equations

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