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Conjugate gradient method for the robin inverse problem associated with the Laplace equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2007年第4期71卷 433-453页

作者： Jin, Bangti Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China

This paper studies a non-linear inverse problem associated with the Laplace equation of identifying the robin coefficient from boundary measurements. A variational formulation of the problem is suggested, thereby transforming it into an optimization problem. Mathematical properties relevant to its numerical computation are established. The optimization problem is solved using the conjugate gradient method in conjunction with the discrepancy principle, and the algorithm is implemented using the boundary element method. Numerical results are presented for several benchmark problems with both exact and noisy data, and the convergence of the algorithm with respect to mesh refinement and decreasing the amount of noise in the data is investigated. Copyright (C) 2006 John Wiley & Sons, Ltd.

关键词： robin inverse problem variational formulation conjugate gradient method boundary element method

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Numerical Solutions to the robin inverse problem with Nonnegativity Constraints

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Journal of Applied Mathematics and Physics 2022年第6期10卷 2015-2025页

作者： Weifu Fang Fu-Rong Lin Department of Mathematics and Statistics Wright State University Dayton USA Department of Mathematics Shantou University Shantou China

We present iterative numerical methods for solving the inverse problem of recovering the nonnegative robin coefficient from partial boundary measurement of the solution to the Laplace equation. Based on the boundary integral equation formulation of the problem, nonnegativity constraints in the form of a penalty term are incorporated conveniently into least-squares iteration schemes for solving the inverse problem. Numerical implementation and examples are presented to illustrate the effectiveness of this strategy in improving recovery results.

关键词： robin inverse problem Ill-Posed problem Laplace Equation Boundary Inte-gral Equation Nonnegativity Constraint Penalty Method

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A DIRECT SOLUTION OF THE robin inverse problem

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JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS 2009年第4期21卷 545-557页

作者： Fang, Weifu Zeng, Suxing Wright State Univ Dept Math & Stat Dayton OH 45435 USA W Virginia Univ Dept Math Morgantown WV 26506 USA

We present a direct, linear boundary integral equation method for the inverse problem of recovering the robin coefficient from a single partial boundary measurement of the solution to the Laplace equation.

关键词： robin inverse problem boundary integral equation Tikhonov regularization

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An inverse problem of robin coefficient identification in parabolic equations with interior degeneracy from terminal observation data

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APPLIED NUMERICAL MATHEMATICS 2025年 212卷 242-253页

作者： Sidi, H. Ould Hendy, A. S. Babatin, M. M. Qiao, L. Zaky, M. A. Inst Super Comptabil & Adm Entreprises ISCAE Dept Methodes Quantitat & Informat Nouakchott Mauritania Ural Fed Univ Inst Nat Sci & Math Computat Math & Comp Sci 19 MiraSt Ekaterinburg 620002 Russia Imam Mohammad Ibn Saud Islamic Univ IMSIU Coll Sci Dept Math & Stat POB 65892 Riyadh 11566 Saudi Arabia Shanxi Univ Sch Math Sci Taiyuan 030006 Shanxi Peoples R China

In this work, we determine the unknown robin coefficient in a degenerate parabolic equation. In inverse analysis, the problem under consideration is nonlinear with an ill-formulated operator and nonlocal. For the stable identification of the unknown robin coefficient, the inverse problem is formulated into a regularised optimization problem. We discuss a variety of practical challenges associated with the problem. The finite element approximation is used to discretize the continuous optimization problem. The convergence and stability analyses are also discussed. Morozov's discrepancy principle is used with the conjugate gradient procedure to construct an iterative scheme. Finally, experiment results are reported to demonstrate the efficiency of the proposed schemes.

关键词： Degenerate diffusion equation robin inverse problem Variation problem Conjugate gradient method

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Nonlinear Cauchy/robin inverse problems solved by an optimal splitting-linearizing method

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INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER 2023年 213卷

作者： Liu, Chein-Shan Chang, Chih-Wen Natl Taiwan Ocean Univ Ctr Excellence Ocean Engn Keelung 202301 Taiwan Natl United Univ Dept Mech Engn Miaoli 36063 Taiwan

In the paper we develop a novel optimal splitting-linearizing method (OSLM) to iteratively solve a non-linear inverse Cauchy problem in a simply-connected domain. The nonlinear term in the nonlinear ellip-tic equation is decomposed at two sides through a splitting parameter, which is then linearized around the value at the previous iteration step. The multiple-scale Pascal-polynomial method together with the OSLM is employed to solve the Cauchy problem, of which the optimal value of the splitting parameter is achieved by minimizing a theoretic merit function. Then, we solve the Cauchy/robin inverse problem of a nonlinear elliptic equation in a doubly-connected domain for recovering the unknown Cauchy data and robin transfer coefficient on an inner boundary. Two-parameter bases are derived to automatically sat-isfy the prescribed Cauchy boundary conditions on the outer boundary. When the solution is convergent after solving a sequence of linear systems, one can retrieve the Cauchy data very accurately. Simultane-ously, the unknown robin transfer coefficient is recovered from a given convective boundary condition on the inner boundary by either a division method or a linear system method. To overcome the ill-posed property of Cauchy/robin problems, the optimal splitting parameter and a scaling factor play the role as regularization parameters. These methods assembled are new techniques to solve the Cauchy/robin inverse problems. Although a few overspecified data are merely given on the outer boundary, the novel method is quite accurate, robust against large noise, and is convergent very fast to find the entire so-lution, the Cauchy data and the robin transfer coefficient. We assess the convergence by the computed order of convergence (COC) of the proposed iterative algorithms.& COPY;2023 Elsevier Ltd. All rights reserved.

关键词： Nonlinear elliptic equation Cauchy robin inverse problem Two-parameter bases Optimal splitting-linearizing method Regularization parameters

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robin反问题的线性互补解法

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高等学校计算数学学报 2018年第2期40卷 125-135页

作者：房喜明肇庆学院数学与统计学院

1引言考虑二维Laplace方程的robin边界问题{△u=0,u∈Ω,?u/?v+pu=g,u∈?Ω=Γ,（1）其中Ω■R2,Γ表示区域Ω的边界,v（向量）表示Γ上的单位外法向量,robin系数p是一个非负函数,其支撑Γ1■Γ,g是给定的函数,其支撑Γ0■Γ,Γ0与Γ1... 详细信息

1引言考虑二维Laplace方程的robin边界问题{△u=0,u∈Ω,?u/?v+pu=g,u∈?Ω=Γ,（1）其中Ω■R2,Γ表示区域Ω的边界,v（向量）表示Γ上的单位外法向量,robin系数p是一个非负函数,其支撑Γ1■Γ,g是给定的函数,其支撑Γ0■Γ,Γ0与Γ1满足Γ0∩Γ1=?.这类微分方程产生于一些实际应用,例如模拟电导体和周围环境之间的稳态热传导模型和半导体中金属和硅的接触面模型等,方程中的u,p,g在不同的环境下代表不同的

关键词： robin inverse problem ill-posed regularization method complementarity

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ANALYSIS OF AN ADAPTIVE FINITE ELEMENT METHOD FOR RECOVERING THE robin COEFFICIENT

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2015年第2期53卷 622-644页

作者： Xu, Yifeng Zou, Jun Shanghai Univ Dept Math Sci Comp Key Lab Shanghai 200234 Peoples R China Shanghai Univ Shanghai Normal Univ Inst Computat Sci E Shanghai 200234 Peoples R China Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China

Based on a new a posteriori error estimator, an adaptive finite element method is proposed for recovering the robin coefficient involved in a diffusion system from some boundary measurement. The a posteriori error estimator cannot be derived for this ill-posed nonlinear inverse problem as was done for the existing a posteriori error estimators for direct problems. Instead, we shall derive the a posteriori error estimator from our convergence analysis of the adaptive algorithm. We prove that the adaptive algorithm guarantees a convergent subsequence of discrete solutions in an energy norm to some exact triplet (the robin coefficient, state and costate variables) determined by the optimality system of the least-squares formulation with Tikhonov regularization for the concerned inverse problem. Some numerical results are also reported to illustrate the performance of the algorithm.

关键词： robin inverse problem a posteriori error estimates adaptive finite element method

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Simultaneous uniqueness for an inverse problem in a time-fractional diffusion equation

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APPLIED MATHEMATICS LETTERS 2020年 109卷

作者： Jing, Xiaohua Peng, Jigen Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Peoples R China Guangzhou Univ Sch Math & Informat Sci Guangzhou 510006 Peoples R China

This article investigates the uniqueness of identifying the fractional-order, potential, and robin coefficient simultaneously in one-dimensional time-fractional diffusion equation with non-homogeneous boundary condition. By using one boundary measurement, we prove that the fractional-order, potential on the entire interval, and robin coefficient are determined simultaneously from asymptotic properties of the Mittag-Leffler function and the Marchenko's uniqueness theorem. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Determination of fractional-order inverse potential problem robin inverse problem Time fractional diffusion equation

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Numerical estimation of the robin coefficient in a stationary diffusion equation

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IMA JOURNAL OF NUMERICAL ANALYSIS 2010年第3期30卷 677-701页

作者： Jin, Bangti Zou, Jun Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China

A finite-element method is proposed for the nonlinear inverse problem of estimating the robin coefficient in a stationary diffusion equation from boundary measurements of the solution and the heat flux. The inverse problem is formulated as an output least squares optimization problem with an appropriate regularization, then the finite-element method is employed to discretize the nonlinear optimization system. Mathematical properties of both the continuous and the discrete optimization problems are investigated. The conjugate gradient method is employed to solve the optimization problem, and an efficient preconditioner via the Sobolev inner product is also suggested. Numerical results for several two-dimensional problems are presented to illustrate the efficiency of the proposed algorithm.

关键词： robin inverse problem conjugate gradient method Sobolev gradient finite-element method stationary diffusion equation

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Determination of robin coefficient in a fractional diffusion problem

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APPLIED MATHEMATICAL MODELLING 2016年第17-18期40卷 7948-7961页

作者： Wei, Ting Wang, Jun-Gang Lanzhou Univ Sch Math & Stat Lanzhou 730000 Peoples R China Northwestern Polytech Univ Dept Appl Math Xian Shaanxi Peoples R China

This paper investigates a nonlinear inverse problem associated with a fractional diffusion equation for identifying a robin coefficient in the boundary conditions from a boundary measurement. The existence and uniqueness of a weak solution for the corresponding direct problem is provided. We formulate the inverse problem into a regularized variational problem and deduce the gradient of the regularization functional based on an adjoint problem. Then the standard conjugate gradient method is employed to solve the variational problem. The numerical results for three examples are presented to illustrate the efficiency of the proposed algorithm. (C) 2016 Elsevier Inc. All rights reserved.

关键词： robin inverse problem Fractional diffusion equation Variational problem Conjugate gradient method

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