Article
Mathematics, Applied
Jin Wen, Zhuan-Xia Liu, Shan-Shan Wang
Summary: The present paper focuses on recovering the source term and the initial data simultaneously for a time-fractional diffusion equation using additional temperature data at fixed times t = Ti and t = T2. The uniqueness of the direct problem is first proven, followed by the application of a non-stationary iterative Tikhonov regularization method to solve the inverse problem and the proposal of a finite dimensional approximation algorithm. Three numerical examples in both one-dimensional and two-dimensional cases are provided to demonstrate the effectiveness and feasibility of the proposed method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Xiong-bin Yan, Zheng-qiang Zhang, Ting Wei
Summary: The main purpose of this paper is to simultaneously identify a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation. The existence and uniqueness of the solution for the direct problem are proved using the fixed point theorem, and the uniqueness of the inverse problem is a direct result of the stability estimate. Additionally, a non-stationary iterative Tikhonov regularization method is used to recover the time dependent potential coefficient and source term, and an alternating minimization method is applied to solve the minimization problem.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
M. Thamban Nair, Devika Shylaja
Summary: This paper discusses the numerical approximation of the biharmonic inverse source problem with finite-dimensional measurement data, presenting a unified framework that covers both conforming and nonconforming finite element methods (FEMs). The inverse problem is analyzed through the forward problem, with error estimates derived for the forward solution in an abstract setup applicable to both conforming and Morley nonconforming FEMs. Due to the ill-posed nature of the inverse problem, Tikhonov regularization is employed to obtain a stable approximate solution, and error estimates are established for the regularized solution under different regularization schemes. Numerical results that verify the theoretical findings are also provided.
Article
Physics, Multidisciplinary
Smina Djennadi, Nabil Shawagfeh, Mustafa Inc, M. S. Osman, J. F. Gomez-Aguilar, Omar Abu Arqub
Summary: This research examines an inverse source problem for a fractional diffusion equation containing the Atangana-Baleanu-Caputo fractional derivative, obtaining an explicit solution set through expansion method and overdetermination condition. Due to the ill-posed nature of the problem, Tikhonov regularization method is applied to stabilize the solution, with a focus on two parameter choice rules. A simulation example is used to validate the presented theoretical results.
Article
Mathematics, Applied
Fan Yang, Ying Cao, Xiao-Xiao Li
Summary: This paper considers the inverse problem of identifying the unknown source in the time-fractional diffusion equation with Caputo-Hadamard derivative. The problem is proved to be ill-posed, and two regularization methods are used to solve it. The error estimates of the fractional Landweber iterative regularization method and the fractional Tikhonov regularization method under different parameter selection rules are given. Numerical examples are provided to demonstrate the effectiveness of both regularization methods.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Bangti Jin, Xiliang Lu, Qimeng Quan, Zhi Zhou
Summary: In this work, we address the inverse problem of recovering a space-dependent potential coefficient in elliptic/parabolic problems from distributed observation. We establish novel stability estimates and analyze the error of a reconstruction scheme based on the output least-squares formulation with Tikhonov regularization. The analysis includes convergence rates and error estimates in different norms, and is supported by numerical experiments.
Article
Mathematics, Applied
M. Tadi
Summary: This article presents a computational method for the inverse problem of the Helmholtz equation, aiming to recover subsurface material properties based on boundary data. The major improvement of this method is that it eliminates the need for linearizing the working equations, making it simple and efficient. Using just one set of data is sufficient to obtain a good approximation of the unknown material property, while additional data sets can further improve the quality of the recovered function.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Yun Zhang, Ting Wei, Yuan-Xiang Zhang
Summary: This study introduces a method to recover two initial values for a time-fractional diffusion-wave equation, achieving uniqueness by Laplace transformation and analytic continuation, followed by solving the inverse problem using an iterative Tikhonov regularization method and proposing a finite dimensional approximation algorithm for finding good approximations to the initial values. Numerical examples in one- and two-dimensional cases demonstrate the effectiveness of the proposed method.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Chemistry, Analytical
Tiantian Wang, Joel Karel, Pietro Bonizzi, Ralf L. M. Peeters
Summary: The electrocardiogram (ECG) is commonly used in clinical practice to analyze the heart's electrical activity. However, it cannot provide detailed information about abnormal electrical patterns on the heart's surface. Electrocardiographic imaging aims to overcome these limitations by reconstructing the heart's surface potentials from body surface potentials and the geometry of the heart. This study focuses on the regularization parameter and its impact on reconstruction accuracy, evaluates different parameter estimation methods, and explores the use of a fixed parameter value.
Article
Mathematics, Applied
Davide Bianchi, Guanghao Lai, Wenbin Li
Summary: We propose a non-stationary iterated network Tikhonov (iNETT) method for solving ill-posed inverse problems. The iNETT method uses deep neural networks to build a data-driven regularizer and eliminates the need to estimate the optimal regularization parameter. To ensure the theoretical convergence of iNETT, we introduce uniformly convex neural networks as the data-driven regularizer. We provide rigorous theories, detailed algorithms, and concrete examples of convexity and uniform convexity in neural networks, and develop the iNETT algorithm with a rigorous convergence analysis.
Article
Mathematics, Applied
Wenjun Ma, Liangliang Sun
Summary: This paper investigates an inverse potential problem for a semilinear generalized fractional diffusion equation with a time-dependent principal part. The missing time-dependent potential is reconstructed using additional integral measured data, and a modified non-stationary iterative Tikhonov regularization method is proposed to solve the problem. Numerical experiments demonstrate the effectiveness and robustness of the proposed algorithm.
Article
Engineering, Multidisciplinary
Angel A. Ciarbonetti, Sergio Idelsohn, Ruben D. Spies
Summary: This work addresses the problem of determining a nonhomogeneous heat conductivity profile in a steady-state heat conduction boundary-value problem in a bounded domain in Double-struck capital R-n. A method based on variational approach and finite dimensional projection is developed, resulting in a regularized linear equation that can reconstruct the exact solution even when the conductivity can only take two prescribed values.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2023)
Article
Acoustics
Anita Carevic, Ivan Slapnicar, Mohamed Almekkawy
Summary: This article proposes a novel algorithm based on the Tikhonov regularization for choosing a balanced parameter in ultrasound tomography. Compared to other methods, this algorithm provides better image quality in breast cancer detection.
IEEE TRANSACTIONS ON ULTRASONICS FERROELECTRICS AND FREQUENCY CONTROL
(2022)
Article
Mathematics
Mousa J. Huntul, Ibrahim Tekin
Summary: In this article, the simultaneous identification of the time-dependent lowest and source terms in a 2D parabolic equation from additional measurements is studied. Regularization is used to address the ill-posed nature of the problem, and the ADE method is applied for discretization and solving numerically. The results show the efficiency and stability of the ADE method in reconstructing the solution from minimal data.
HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
(2023)
Article
Mathematics, Applied
Zhenping Li, Xiangtuan Xiong, Jun Li, Jiaqi Hou
Summary: This paper deals with the reconstruction problem of aperture in the plane from their diffraction patterns and proposes a quasi-boundary regularization method to stabilize the problem. The method has better approximation than classical methods in theory without noise.
Article
Mathematics, Applied
Xiong Bin Yan, Ting Wei
JOURNAL OF INVERSE AND ILL-POSED PROBLEMS
(2019)
Article
Mathematics, Applied
Liangliang Sun, Yun Zhang, Ting Wei
APPLIED NUMERICAL MATHEMATICS
(2019)
Article
Mathematics, Applied
Suzhen Jiang, Kaifang Liao, Ting Wei
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2020)
Article
Mathematics, Applied
J. Xian, T. Wei
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2019)
Article
Mathematics, Applied
Kaifang Liao, Ting Wei
Article
Mathematics, Applied
Jin Cheng, Yufei Ke, Ting Wei
JOURNAL OF INVERSE AND ILL-POSED PROBLEMS
(2020)
Article
Engineering, Multidisciplinary
Suzhen Jiang, Ting Wei
Summary: This paper deals with a nonlinear inverse problem for recovering a time-dependent potential term in a time fractional diffusion equation from an additional measurement in the form of integral over the space domain. By using the fixed point theorem, the existence, uniqueness, regularity, and stability of the direct problem are proved. The uniqueness of the inverse problem is proved by the property of Caputo fractional derivative, and the Levenberg-Marquardt method is employed for numerical approximation. The feasibility and efficiency of the proposed method are demonstrated through various examples.
INVERSE PROBLEMS IN SCIENCE AND ENGINEERING
(2021)
Article
Mathematics, Applied
Yun Zhang, Ting Wei, Yuan-Xiang Zhang
Summary: This study introduces a method to recover two initial values for a time-fractional diffusion-wave equation, achieving uniqueness by Laplace transformation and analytic continuation, followed by solving the inverse problem using an iterative Tikhonov regularization method and proposing a finite dimensional approximation algorithm for finding good approximations to the initial values. Numerical examples in one- and two-dimensional cases demonstrate the effectiveness of the proposed method.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Ting Wei, Kaifang Liao
Summary: This paper investigates a nonlinear inverse problem of identifying a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation using measured data at a boundary point. The existence, uniqueness, and regularity of the solution for the corresponding direct problem are proven, and a conditional stability estimate is provided for the inverse zeroth-order coefficient problem. The proposed method is shown to be effective through numerical examples in one-dimensional and two-dimensional cases.
APPLICABLE ANALYSIS
(2022)
Article
Mathematics, Applied
Ting Wei, Yuhua Luo
Summary: This paper investigates the problem of identifying a space-dependent source in a time-fractional diffusion-wave equation using final time data. The inverse source problem is formulated using a first kind of Fredholm integral equation and solved using a generalized quasi-boundary value regularization method. The effectiveness and stability of the proposed algorithms are demonstrated through numerical examples.
Article
Mathematics, Applied
Ting Wei, Yun Zhang, Dingqian Gao
Summary: This paper investigates an inverse problem of recovering the zeroth-order coefficient and fractional order simultaneously in a time-fractional reaction-diffusion-wave equation using boundary measurement data. The uniqueness of the considered inverse problem and the Lipschitz continuity of the forward operator are proven. The inverse problem is formulated into a variational problem with Tikhonov-type regularization, and the existence of the minimizer is proved under an a priori choice of regularization parameter. The proposed method is solved using the steepest descent method combined with Nesterov acceleration, and its efficiency and rationality are supported by numerical examples.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Kaifang Liao, Lei Zhang, Ting Wei
Summary: In this article, the inverse problem of determining a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation is considered using a nonlocal condition. The uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem are proven. The Levenberg-Marquardt method is employed to recover the fractional order and the time source term simultaneously, and a finite-dimensional approximation algorithm is established to find a regularized numerical solution. Furthermore, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Numerical results in both one and multi-dimensional spaces are presented to demonstrate the robustness of the proposed algorithm.
JOURNAL OF INVERSE AND ILL-POSED PROBLEMS
(2023)
Article
Mathematics, Applied
Ting Wei, Xiongbin Yan
JOURNAL OF APPLIED ANALYSIS AND COMPUTATION
(2019)
Article
Mathematics, Applied
Kai Fang Liao, Yu Shan Li, Ting Wei
EAST ASIAN JOURNAL ON APPLIED MATHEMATICS
(2019)
