A coupled complex boundary method for the Cauchy problem

Cheng, Xiaoliang; Gong, Rongfang; Han, Weimin

doi:10.1080/17415977.2015.1130040

Cited by 14 publications

(10 citation statements)

References 52 publications

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“…Due to the equivalence of Problem 2 and Problem 3, K is compact and the equation ( 13) is ill-posed. Moreover, we can observe the following results about K. Proposition 2.2 Let K be defined in (12). Then (i) K is injective, that is, Ker(K) = {0};…”

Section: A Reformulation Of the Cauchy Problem With The Ccbmmentioning

confidence: 95%

“…Let i = √ −1 be an imaginary unit. Then, without going into detail, the inverse Cauchy problem (1) can be studied through ( [12]).…”

Section: A Reformulation Of the Cauchy Problem With The Ccbmmentioning

confidence: 99%

“…In this paper, a coupled complex boundary method (CCBM) is applied to Problem 1 in order to construct specific K and f . The CCBM was originally considered appropriate for inverse source problems in [11], and was subsequently extended to the inverse Cauchy problem in [12]. Here, we further investigate the properties of CCBM beyond the work of [12].…”

Section: Introductionmentioning

confidence: 99%

“…The CCBM was originally considered appropriate for inverse source problems in [11], and was subsequently extended to the inverse Cauchy problem in [12]. Here, we further investigate the properties of CCBM beyond the work of [12]. We skip the systematic analysis and list some merits of CCBM, as follows:…”

Section: Introductionmentioning

confidence: 99%

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A CCBM-based generalized GKB iterative regularizing algorithm for inverse Cauchy problems

Gong¹,

Wang²,

Huang³

et al. 2022

Preprint

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This paper examines inverse Cauchy problems that are governed by a kind of elliptic partial differential equation. The inverse problems involve recovering the missing data on an inaccessible boundary from the measured data on an accessible boundary, which is severely ill-posed. By using the coupled complex boundary method (CCBM), which integrates both Dirichlet and Neumann data into a single Robin boundary condition, we reformulate the underlying problem into an operator equation. Based on this new formulation, we prove the existence of a unique solution even in cases with noisy data. A Golub-Kahan bidiagonalization (GKB) process together with Givens rotation is employed for iteratively solving the proposed operator equation. The regularizing property of the developed method, called CCBM-GKB, and its convergence rate results are proved under a posteriori stopping rule. Finally, a linear finite element method is used for the numerical realization of CCBM-GKB. Various numerical experiments demonstrate that CCBM-GKB is a kind of accelerated iterative regularization method, as it is much faster than the classic Landweber method.

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Section: A Reformulation Of the Cauchy Problem With The Ccbmmentioning

confidence: 95%

“…Let i = √ −1 be an imaginary unit. Then, without going into detail, the inverse Cauchy problem (1) can be studied through ( [12]).…”

Section: A Reformulation Of the Cauchy Problem With The Ccbmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A CCBM-based generalized GKB iterative regularizing algorithm for inverse Cauchy problems

Gong¹,

Wang²,

Huang³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Then many research works have been conducted to propose different regularization approaches, we can not list all of them here. The most popular one is Tikhonov's regularization method (Tikhonov et al 1977;Feng et al 2011;Cimetiere et al 2001;Cheng et al 2016;Berntsson et al 2017), which transforms the original ill-posed problem into a well posed one by minimizing the L 2 -norm of the solution subjected to constraint equations. Other methods have been proposed to regularize the Cauchy problem, we can mention for instance the alternating method (Berntsson et al 2014;Kozlov et al 1991;Jourhmane and Nachaoui 1999), the universal regularization method (Klibanov 2015(Klibanov , 2006(Klibanov , 2013, the technic of fundamental solution (Chen et al 2019;Fairweather and Karageorghis 1998;Marin and Cipu 2017;Yeih et al 2015), the fixed point approach based on domain decomposition like methods (Chakib et al 2018).…”

Section: Introductionmentioning

confidence: 99%

On numerical resolution of an inverse Cauchy problem modeling the airflow in the bronchial tree

Chakib

Ouaissa

2021

Comp. Appl. Math.

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This paper is devoted to the numerical resolution of an inverse Cauchy problem governed by Stokes equation modeling the airflow in the lungs. It consists in determining the air velocity and pressure on the artificial boundaries of the bronchial tree. This data completion problem is one of the highly ill-posed problems in the Hadamard sense (Hadamard in Lectures on Cauchy's problem in linear partial differential equations. Dover, New York, 1953). This gives great importance to its numerical resolution and in particular to carry out stable numerical approaches, mostly in the case of noisy data. The main idea of this work is to extend some regularizing, stable and fast iterative algorithms for solving this problem based on the domain decomposition approach (Chakib et al. in Inverse Prob 35(1):015008, 2018). We discuss the efficiency and the feasibility of the proposed approach through some numerical tests performed using different domain decomposition algorithms. Finally, we opt for the Robin-Robin algorithm, which showed its performance, for the numerical simulation of the airflow in the bronchial tree configuration.

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Analysis of Lavrentiev-finite element methods for data completion problems

2017

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The variational finite element solution of Cauchy's problem, expressed in the Steklov-Poincaré framework and regularized by the Lavrentiev method, has been introduced and computationally assessed in [Inverse Problems in Science and Engineering, 18, 1063-1086(2011]. The present work concentrates on the numerical analysis of the semi-discrete problem. We perform the mathematical study of the error to rigorously establish the convergence of the global bias-variance error.

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A coupled complex boundary method for the Cauchy problem

Cited by 14 publications

References 52 publications

A CCBM-based generalized GKB iterative regularizing algorithm for inverse Cauchy problems

A CCBM-based generalized GKB iterative regularizing algorithm for inverse Cauchy problems

On numerical resolution of an inverse Cauchy problem modeling the airflow in the bronchial tree

Analysis of Lavrentiev-finite element methods for data completion problems

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