Article
Mathematics, Applied
Gaurav Mittal, Ankik Kumar Giri
Summary: In this paper, we propose an extension of the classical steepest descent method to solve non-smooth nonlinear ill-posed problems. The extension incorporates the Bouligand sub-derivative of the forward mapping. The convergence analysis of the proposed method is studied assuming boundedness of the Bouligand subderivative and a modified tangential cone condition. Additionally, the strongly convergent regularizing nature of the proposed method is discussed, and numerical simulations are provided to demonstrate its practicality and compare with existing methods in the literature.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics
Hassan K. Ibrahim Al-Mahdawi, Hussein Alkattan, Mostafa Abotaleb, Ammar Kadi, El-Sayed M. El-kenawy
Summary: The Landweber iteration method is a popular approach for solving linear discrete ill-posed problems. This paper presents a new version of the method that utilizes a polar decomposition to improve the speed and accuracy of the iteration process. Convergence and analysis were conducted to validate the usability of the new method, which was compared to the classical Landweber method. A numerical experiment demonstrated the effectiveness of the new method in solving an inverse boundary value problem of the heat equation (IBVP).
Article
Thermodynamics
Shibin Wan, Kun Wang, Peng Xu, Yajin Huang
Summary: In this paper, a single neural adaptive PID (SNA-PID) inverse method is proposed to estimate the thermal boundary condition. Compared with the traditional PID inverse method, SNA-PID can adaptively adjust the weights of PID parameters, thus improving the tuning effect and anti-interference ability.
INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER
(2022)
Article
Engineering, Multidisciplinary
Lin Qiu, Ji Lin, Fajie Wang, Qing-Hua Qin, Chein-Shan Liu
Summary: This method proposes a simple and effective way to solve inverse heat source problems in functionally graded materials, eliminating the need for mesh generation, numerical integration, iteration, regularization, and fundamental solutions. The heat source problems are solved directly by calculating a linear matrix system, making it easy to program and implement.
APPLIED MATHEMATICAL MODELLING
(2021)
Article
Mathematics, Applied
Dinh-Nho Hao, Thuy T. Le, Loc H. Nguyen
Summary: This article introduces a new technique for computing numerical solutions to the nonlinear inverse heat conduction problem. By truncating the Fourier series and employing the Runge-Kutta method, the high-dimensional problem is converted into a 1D problem, addressing the nonlinearity and lack of partial derivative data.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Thermodynamics
Andrzej Frackowiak, Agnieszka Wroblewska, Michal Cialkowski
Summary: This paper presents a concept of solving the inverse heat conduction problem using Trefftz functions and provides two examples to validate the effectiveness of the method.
INTERNATIONAL JOURNAL OF THERMAL SCIENCES
(2022)
Article
Acoustics
Haniye Dehestani, Yadollah Ordokhani, Mohsen Razzaghi
Summary: In this article, a newly modified Bessel wavelet method is proposed to solve fractional variational problems. The operational matrices of integration, derivative, and dual with coefficient are introduced, with detailed error analysis. The efficiency and accuracy of the method are illustrated through testing the behavior of approximate solutions in examples.
JOURNAL OF VIBRATION AND CONTROL
(2021)
Article
Mathematics, Applied
Yuxin Xia, Bo Han, Ruixue Gu
Summary: In this paper, an accelerated homotopy-perturbation-Kaczmarz iteration method based on sequential subspace optimization is proposed for solving nonlinear systems of inverse problems. The method iteratively projects the initial value onto stripes controlled by the search direction, forward operator, and noise level to expedite convergence. Convergence and regularization analysis are provided under general assumptions, and numerical examples demonstrate the effectiveness of reconstructing solutions and acceleration effects of the method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Thermodynamics
Farzaneh Safari
Summary: We propose a meshless method to solve the inverse heat problem in multi-dimensional situations using correcting functions. The scheme reformulates the problem as an operator equation subject to boundary conditions. The ill-conditioned problem is solved using trigonometric basis functions on noisy boundary data. We demonstrate the practical performance of the scheme with different parameter values and provide acceptable estimates for the local error.
INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER
(2023)
Article
Mathematics
Hassan K. Ibrahim Al-Mahdawi, Mostafa Abotaleb, Hussein Alkattan, Al-Mahdawi Zena Tareq, Amr Badr, Ammar Kadi
Summary: This paper discusses and solves the inverse problems for the boundary value and initial value in a heat equation. By reformulating the problems as integral equations and discretizing them, an approximation solution is obtained using the Landweber-type iterative method and the V-cycle multigrid method.
Article
Engineering, Electrical & Electronic
Jose O. Vargas, Ricardo Adriano
Summary: A subspace-based conjugate-gradient method (S-CGM) is proposed in this article to improve the performance of the linearized CGM. By retrieving the deterministic part of the variational-induced current, the S-CGM estimates the total electric field more accurately, resulting in faster convergence speed and higher accuracy.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
(2022)
Article
Engineering, Electrical & Electronic
Tiantian Yin, Li Pan, Xudong Chen
Summary: This article proposes an iterative method called the subspace-based distorted-Rytov iterative method (S-DRIM) for solving inverse scattering problems. The method utilizes the subspace-Rytov approximation (SRA) and updates the parameters of the background medium in each iteration of S-DRIM. Through simulations, the approximation errors of both the Rytov Approximation (RA) and SRA are analyzed and compared in various background media. The performance of S-DRIM is verified using both synthetic and experimental data, showing that it outperforms the non-iterative SRA inversion method for mild scatterer reconstructions and has a smaller reconstruction error in optimization processes compared to the distorted-Rytov iterative method (DRIM).
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
(2023)
Article
Automation & Control Systems
Martin P. Neuenhofen, Eric C. Kerrigan
Summary: We propose a modified augmented Lagrangian method (ALM) to minimize constrained optimization problems with large quadratic penalties of inconsistent equality constraints. This modification addresses the issue of ALM's failure to converge when the equality constraints are inconsistent. The modified ALM demonstrates faster convergence in minimizing certain quadratic penalty augmented functions compared to the quadratic penalty method (QPM).
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
(2023)
Article
Engineering, Electrical & Electronic
Tao Shan, Zhichao Lin, Xiaoqian Song, Maokun Li, Fan Yang, Shenheng Xu
Summary: In this article, the authors propose the neural Born iterative method (NeuralBIM) for solving 2-D inverse scattering problems (ISPs) by using the physics-informed supervised residual learning (PhiSRL). NeuralBIM employs independent convolutional neural networks (CNNs) to learn the alternate update rules for two candidate solutions. The article presents two schemes: supervised NeuralBIM, trained with knowledge of total fields and contrasts, and unsupervised NeuralBIM, guided by a physics-embedded objective function. Numerical and experimental results confirm the effectiveness of NeuralBIM.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
(2023)
Article
Thermodynamics
Geyong Cao, Bo Yu, Leilei Chen, Weian Yao
Summary: This paper proposes a precise integration multi-patch isogeometric dual reciprocity boundary element method (IG-DRBEM) for the non-Fourier transient heat transfer problem in functionally graded materials (FGMs), and establishes a stochastic analysis of multidimensional material uncertainty using hyperbolic truncated adaptive sparse polynomial chaos expansion (PCE) based on deterministic theory. The method addresses the challenges posed by the higher-order derivative term in the non-Fourier heat transfer equation and utilizes dual reciprocity method (DRM) and precise integration method (PIM) to improve stability and accuracy. It also introduces a sparse PCE strategy with hyperbolic truncation to overcome the dimensional disaster difficulty of traditional PCE. The proposed method is validated to be accurate and efficient, expanding the application scope of IG-DRBEM.
INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER
(2023)
Article
Mathematics, Applied
Chein-Shan Liu, Botong Li
JOURNAL OF INVERSE AND ILL-POSED PROBLEMS
(2020)
Article
Thermodynamics
Chein-Shan Liu, Lin Qiu, Ji Lin
NUMERICAL HEAT TRANSFER PART B-FUNDAMENTALS
(2020)
Article
Mathematics, Applied
Zhuo-Jia Fu, Li-Wen Yang, Qiang Xi, Chein-Shan Liu
Summary: This paper presents a method to solve anomalous heat conduction problems under functionally graded materials using SBM, DRM, and Laplace transformation technique. It achieves high accuracy by combining these methods and avoiding the impact of time step on computational efficiency. The transient heat conduction equation with Caputo time fractional derivative is used to describe the phenomena, demonstrating the effectiveness of the proposed method through numerical examples and comparisons with analytical solutions and COMSOL simulations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics
Chein-Shan Liu, Tsung-Lin Lee
Summary: The paper proves that two-step fourth-order optimal iterative schemes of the same class share a common feature and develops a new family of fourth-order optimal iterative schemes.
JOURNAL OF MATHEMATICS
(2021)
Article
Thermodynamics
Chein-Shan Liu, Jiang-Ren Chang
Summary: This paper presents a method for dealing with the nonlocal boundary conditions problem of nonlinear heat equations, using nonlocal boundary shape functions and novel techniques to quickly and accurately solve the problem.
NUMERICAL HEAT TRANSFER PART B-FUNDAMENTALS
(2021)
Article
Mathematics
Chein-Shan Liu, Yung-Wei Chen
Summary: A new analytic method has been developed to improve the Lindstedt-Poincare method for strongly nonlinear oscillators by introducing a linearization technique in the nonlinear differential equation. This method provides accurate higher order solutions and is significant for solving strongly nonlinear oscillators with large amplitudes.
Article
Thermodynamics
Chein-Shan Liu, Chih-Wen Chang
Summary: In this article, a solution is provided for a nonlinear parabolic type partial differential equation (PDE) with non-separated and nonlocal conditions. The use of a nonlocal boundary shape function (NLBSF) and a novel splitting-linearizing technique allows for fast and accurate solutions to the nonlocal and nonlinear parabolic equation.
NUMERICAL HEAT TRANSFER PART B-FUNDAMENTALS
(2022)
Article
Multidisciplinary Sciences
Chein-Shan Liu, Chih-Wen Chang, Yung-Wei Chen, Yen-Shen Chang
Summary: In this paper, we determine the period of an n-dimensional nonlinear dynamical system using a derived formula in an (n + 1)-dimensional augmented space. We propose a boundary shape function method (BSFM) and iterative algorithms to solve periodic problems with given or unknown boundary values. The numerical examples demonstrate the advantages of the BSFM in terms of convergence speed, accuracy, and stability compared to the shooting method.
Article
Mathematics
Chein-Shan Liu, Essam R. El-Zahar, Chih-Wen Chang
Summary: In this paper, a mth-order asymptotic-numerical method is developed to solve a second-order singularly perturbed problem with variable coefficients. The method decomposes the solutions into two independent sub-problems and couples them through a left-end boundary condition. Unlike traditional asymptotic solutions, this method performs asymptotic series solution in the original coordinates, leading to better results.
Article
Mathematics
Chein-Shan Liu, Chih-Wen Chang, Yung-Wei Chen, Jian-Hung Shen
Summary: This paper presents a numerical method for solving non-homogeneous wave problems with nonlocal boundary conditions, and validates the effectiveness and stability of the method through numerical tests.
Article
Mathematics
Chein-Shan Liu, Jiang-Ren Chang, Jian-Hung Shen, Yung-Wei Chen
Summary: In this paper, the general Sturm-Liouville problem is transformed into two canonical forms with different boundary conditions. A boundary shape function method is proposed to solve these problems. By using normalization conditions for eigenfunctions and solving the eigenvalue curve, eigenvalues and eigenfunctions with desired accuracy can be obtained.
Article
Mathematics
Chein-Shan Liu, Chung-Lun Kuo, Chih-Wen Chang
Summary: Researchers have developed a simple method for solving linear and nonlinear eigenvalue problems. By solving a nonhomogeneous system and using a normalization condition on the eigen-equation, the method ensures the uniqueness of the eigenvector. By introducing a merit function, the method can obtain precise eigenvalues.
Article
Mathematics
Chein-Shan Liu, Essam R. El-Zahar, Chih-Wen Chang
Summary: This paper proposes a dynamical approach to determine the optimal values of parameters in iterative methods for solving linear equation systems. The new methods provide an alternative and proper choice of parameter values for accelerating convergence speed without knowing the theoretical optimal values. Numerical testings were used to assess the performance of the dynamic optimal methods.
Article
Engineering, Mechanical
Chein-Shan Liu, Chung-Lun Kuo, Chih-Wen Chang
Summary: This article introduces methods for solving the free vibration problem of multi-degree mechanical structures by linearizing it to a generalized eigenvalue problem or using iterative detection methods for quadratic eigenvalue problem. The study uses nonhomogeneous linear systems and projected eigen-equation to obtain response curves and eigenvectors, thereby saving computational cost.
