Chein-Shan Liu¹, Chung-Lun Kuo¹, Chih-Wen Chang^2,*

CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.3, pp. 3189-3208, 2024, DOI:10.32604/cmes.2023.046002 - 11 March 2024

Abstract To solve the Laplacian problems, we adopt a meshless method with the multiquadric radial basis function (MQ-RBF) as a basis whose center is distributed inside a circle with a fictitious radius. A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function. A sample function is interpolated by the MQ-RBF to provide a trial coefficient vector to compute the merit function. We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm. The novel method provides the More >

Chih-Wen Chang^*

CMC-Computers, Materials & Continua, Vol.73, No.1, pp. 433-451, 2022, DOI:10.32604/cmc.2022.027021 - 18 May 2022

Abstract We retrieve unknown nonlinear large space-time dependent forces burdened with the vibrating nonlinear Euler-Bernoulli beams under varied boundary data, comprising two-end fixed, cantilevered, clamped-hinged, and simply supported conditions in this study. Even though some researchers used several schemes to overcome these forward problems of Euler-Bernoulli beams; however, an effective numerical algorithm to solve these inverse problems is still not available. We cope with the homogeneous boundary conditions, initial data, and final time datum for each type of nonlinear beam by employing a variety of boundary shape functions. The unknown nonlinear large external force can be… More >

Yuye Zhang^1,*, Jingli Huang¹, Jiandong Liu¹, Hakeel Ahmed Chohan²

Computer Systems Science and Engineering, Vol.36, No.2, pp. 407-416, 2021, DOI:10.32604/csse.2021.014154 - 05 January 2021

Abstract To implement restoration in a single motion blurred image, the PSF (Point Spread Function) is difficult to estimate and the image deconvolution is ill-posed as a result that a good recovery effect cannot be obtained. Considering that several different PSFs can get joint invertibility to make restoration well-posed, we proposed a motion-blurred image restoration method based on joint invertibility of PSFs by means of computational photography. Firstly, we designed a set of observation device which composed by multiple cameras with the same parameters to shoot the moving target in the same field of view continuously… More >

Ji-Chuan Liu¹, Jun-Gang Wang²

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.5, pp. 437-462, 2014, DOI:10.3970/cmes.2014.097.437

Abstract We consider the determination of heat flux within a body from the Cauchy data. The aim of this paper is to seek an approach to solve the onedimensional heat equation in a bounded domain without initial value. This problem is severely ill-posed and there are few theoretic results. A quasi-reversibility regularization method is used to obtain a regularized solution and convergence estimates are given. For numerical implementation, we apply a method of lines to solve the regularized problem. From numerical results, we can see that the proposed method is reasonable and feasible. 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 >

Chih-Wen Chang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.88, No.2, pp. 65-92, 2012, DOI:10.3970/cmes.2012.088.065

Abstract We utilize a regularized integral equation scheme to resolve the three-dimensional backward advection-dispersion equation (BADE) for identifying the groundwater pollution source identification problems in this research. First, the Fourier series expansion method is employed to estimate the concentration field C(x, y, z, t) at any time t < T. Second, we contemplate a direct regularization by adding an extra term a(x, y, z, 0) to transform a second-kind Fredholm integral equation for C(x, y, z, 0). The termwise separable property of the kernel function permits us to acquire a closed-form regularized solution. In addition, a More >

C. Shi¹, C. Wang¹, T. Wei^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.86, No.2, pp. 71-92, 2012, DOI:10.3970/cmes.2012.086.071

Abstract In this paper, we consider a typical ill-posed inverse heat source problem, that is, we determine a space-dependent heat source term in a multi-dimensional heat equation from a pair of Cauchy data on a part of boundary. By a simple transformation, the inverse heat source problem is changed into a Cauchy problem of a homogenous heat conduction equation. We use the method of fundamental solutions (MFS) coupled with the Tikhonov regularization technique to solve the ill-conditioned linear system of equations resulted from the MFS discretization. The generalized cross-validation rule for determining the regularization parameter is More >

Jiun-Yu Wu^1,2, Chih-Wen Chang³

CMC-Computers, Materials & Continua, Vol.25, No.3, pp. 215-238, 2011, DOI:10.3970/cmc.2011.025.215

Abstract In this paper, we employ the differential quadrature method (DQM) to tackle the inverse heat conduction problem (IHCP) of heat source. These advantages of this numerical approach are that no a priori presumption is made on the functional form of the estimates, and that evaluated heat source can be obtained directly in the calculation process. Seven examples show the effectiveness and accuracy of our algorism in providing excellent estimates of unknown heat source from the given data. We find that the proposed scheme is applicable to the IHCP of heat source. Even though the noise More >

Chein-Shan Liu¹, Chung-Lun Kuo²

CMES-Computer Modeling in Engineering & Sciences, Vol.76, No.2, pp. 109-132, 2011, DOI:10.3970/cmes.2011.076.109

Abstract The Tikhonov method is a famous technique for regularizing ill-posed systems. In this theory a regularization parameter α needs to be determined. Based-on an invariant-manifold defined in the space of (x,t) and from the Tikhonov minimization functional, we can derive an optimal vector driven system of nonlinear ordinary differential equations (ODEs). In the Optimal Vector Driven Algorithm (OVDA), the optimal regularization parameter αk is presented in the iterative solution of x, which means that a dynamical Tikhonov regularization method is involved in the solution of nonlinear ill-posed problem. The OVDA is an extension of the Landweber-Scherzer iterative More >

Linjun Wang¹, Xu Han², Youxiang Xie³

CMES-Computer Modeling in Engineering & Sciences, Vol.82, No.3&4, pp. 179-194, 2011, DOI:10.32604/cmes.2011.082.179

Abstract In this paper, a new homotopy perturbation method (IHPM) is presented and suggested to solve an ill-posed problem of multi-source dynamic loads reconstruction. We propose a stable and reliable modification, and obtain a new regularization method, then employ it to find the exact solution for the multi-source dynamic load identification problem. Also, this present method only needs easy computations rather than successive integrations. Finally, the performances of two numerical examples are given. Comparisons are performed between the original homotopy perturbation method (HPM) and IHPM. The results verify that the present method is very simple and More >