M. J. Huntul^1,*, Taki-Eddine Oussaeif²

Computer Systems Science and Engineering, Vol.40, No.3, pp. 1109-1126, 2022, DOI:10.32604/csse.2022.020175

Abstract In this paper, we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition. We obtain sufficient conditions for the unique solvability of the inverse problem. The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For example, seismology, medicine, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation… More >

M.J. Huntul^*

Computer Systems Science and Engineering, Vol.39, No.3, pp. 415-429, 2021, DOI:10.32604/csse.2021.017924

Abstract The aim of this paper is to find the time-dependent term numerically in a two-dimensional heat equation using initial and Neumann boundary conditions and nonlocal integrals as over-determination conditions. This is a very interesting and challenging nonlinear inverse coefficient problem with important applications in various fields ranging from radioactive decay, melting or cooling processes, electronic chips, acoustics and geophysics to medicine. Unique solvability theorems of these inverse problems are supplied. However, since the problems are still ill-posed (a small modification in the input data can lead to bigger impact on the ultimate result in the output solution) the solution needs… More >

Yihua Wang¹, Zhenhuan Zhou¹, Hao Xu^1,*, Shuai Li², Zhanjun Wu¹

Structural Durability & Health Monitoring, Vol.15, No.2, pp. 85-101, 2021, DOI:10.32604/sdhm.2021.014256

Abstract For practical engineering structures, it is usually difficult to measure external load distribution in a direct manner, which makes inverse load identification important. Specifically, load identification is a typical inverse problem, for which the models (e.g., response matrix) are often ill-posed, resulting in degraded accuracy and impaired noise immunity of load identification. This study aims at identifying external loads in a stiffened plate structure, through comparing the effectiveness of different methods for parameter selection in regulation problems, including the Generalized Cross Validation (GCV) method, the Ordinary Cross Validation method and the truncated singular value decomposition method. With demonstrated high accuracy,… More >

M. J. Huntul^*

CMC-Computers, Materials & Continua, Vol.67, No.3, pp. 3681-3699, 2021, DOI:10.32604/cmc.2021.016036

Abstract The inverse problem of reconstructing the time-dependent thermal conductivity and free boundary coefficients along with the temperature in a two-dimensional parabolic equation with initial and boundary conditions and additional measurements is, for the first time, numerically investigated. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For instance, steel annealing, vacuum-arc welding, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. From literature we already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the… More >

Y.C. Hon¹, T. Wei²

CMES-Computer Modeling in Engineering & Sciences, Vol.7, No.2, pp. 119-132, 2005, DOI:10.3970/cmes.2005.007.119

Abstract We propose in this paper an effective meshless and integration-free method for the numerical solution of multidimensional inverse heat conduction problems. Due to the use of fundamental solutions as basis functions, the method leads to a global approximation scheme in both the spatial and time domains. To tackle the ill-conditioning problem of the resultant linear system of equations, we apply the Tikhonov regularization method based on the generalized cross-validation criterion for choosing the regularization parameter to obtain a stable approximation to the solution. The effectiveness of the algorithm is illustrated by several numerical two- and three-dimensional examples. More >

Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.29, No.2, pp. 103-128, 2012, DOI:10.3970/cmc.2012.029.103

Abstract In order to solve ill-posed linear inverse problems, we modify the Tikhonov regularization method by proposing three different preconditioners, such that the resultant linear systems are equivalent to the original one, without dropping out the regularized term on the right-hand side. As a consequence, the new regularization methods can retain both the regularization effect and the accuracy of solution. The preconditioned coefficient matrix is arranged to be equilibrated or diagonally dominated to derive the optimal scales in the introduced preconditioning matrix. Then we apply the iterative scheme to find the solution of ill-posed linear inverse problem. Two theorems are proved… More >

Xu Liu¹, Guojian Shao¹, Xingxing Yue^2,*, Qingbin Yang³, Jingbo Su⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.117, No.2, pp. 189-211, 2018, DOI:10.31614/cmes.2018.03947

Abstract This paper aims to apply a virtual boundary element method (VBEM) to solve the inverse problems of three-dimensional heat conduction in orthotropic media. This method avoids the singular integrations in the conventional boundary element method, and can be treated as a potential approach for solving the inverse problems of the heat conduction owing to the boundary-only discretization and semi-analytical algorithm. When the VBEM is applied to the inverse problems, the numerical instability may occur if a virtual boundary is not properly chosen. The method encounters a highly ill-conditioned matrix for the larger distance between the physical boundary and the virtual… More >

Akpofure E. Taigbenu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.102, No.4, pp. 271-289, 2014, DOI:10.3970/cmes.2014.102.271

Abstract The solutions to inverse heat conduction problems (IHCPs) are provided in this paper by the Green element method (GEM), incorporating the logarithmic fundamental solution of the Laplace operator (Formulation 1) and the timedependent fundamental solution of the diffusion differential operator (Formulation 2). The IHCPs addressed relate to transient problems of the recovery of the temperature, heat flux and heat source in 2-D homogeneous domains. For each formulation, the global coefficient matrix is over-determined and ill-conditioned, requiring a solution strategy that involves the least square method with matrix decomposition by the singular value decomposition (SVD) method, and regularization by the Tikhonov… More >

Christina Babenko¹, Roman Chapko², B. Tomas Johansson³

CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.5, pp. 299-317, 2014, DOI:10.3970/cmes.2014.101.299

Abstract We consider the numerical solution of the Laplace equations in planar bounded domains with corners for two types of boundary conditions. The first one is the mixed boundary value problem (Dirichlet-Neumann), which is reduced, via a single-layer potential ansatz, to a system of well-posed boundary integral equations. The second one is the Cauchy problem having Dirichlet and Neumann data given on a part of the boundary of the solution domain. This problem is similarly transformed into a system of ill-posed boundary integral equations. For both systems, to numerically solve them, a mesh grading transformation is employed together with trigonometric quadrature… More >

F. Yang¹, C.L. Fu², X.X. Li¹

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.1, pp. 19-29, 2014, DOI:10.3970/cmes.2014.100.019

Abstract Numerical differentiation is a classical ill-posed problem. The generalized Tikhonov regularization method is proposed to solve this problem. The error estimates are obtained for a priori and a posteriori parameter choice rules, respectively. Numerical examples are presented to illustrate the validity and effectiveness of this method. More >