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  • Open Access


    Optimal Shape Factor and Fictitious Radius in the MQ-RBF: Solving Ill-Posed Laplacian Problems

    Chein-Shan Liu1, Chung-Lun Kuo1, Chih-Wen Chang2,*

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

    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 >

  • Open Access


    A Meshless Method for Retrieving Nonlinear Large External Forces on Euler-Bernoulli Beams

    Chih-Wen Chang*

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

    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 >

  • Open Access


    Motion-Blurred Image Restoration Based on Joint Invertibility of PSFs

    Yuye Zhang1,*, 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

    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 >

  • Open Access


    A moving modified Trefftz method for inverse Laplace problems in two dimensional multiply-connected domain

    C.-L. Kuo, C.-S. Liu

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.11, No.3, pp. 85-86, 2009, DOI:10.3970/icces.2009.011.085

    Abstract In this paper, the inverse problems in a multiply connected domain governed by the Laplace equation have been investigated numerically by the developed moving modified Trefftz method. When solving the direct Laplace problem with the conventional Trefftz method, one may treat the ill-posed linear algebraic equations because the solution is obtained by expanding the diverging series; while when the inverse Laplace problem is encountered, it is more difficult to treat the more seriously ill-posed behaviors because the incomplete boundary data, and its solution, if exists, does not depend on the given boundary data continuously. Even… More >

  • Open Access


    The Lie-Group Shooting Method for Quasi-Boundary Regularization of Backward Heat Conduction Problems

    Chih-Wen Chang1, Chein-Shan Liu2, Jiang-Ren Chang1

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.3, No.2, pp. 69-80, 2007, DOI:10.3970/icces.2007.003.069

    Abstract By using a quasi-boundary regularization we can formulate a two-point boundary value problem of the backward heat conduction equation. The ill-posed problem is analyzed by using the semi-discretization numerical schemes. Then, the resulting ordinary differential equations in the discretized space are numerically integrated towards the time direction by the Lie-group shooting method to find the unknown initial conditions. The key point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then, by imposing G(T) = G(r) we can seek the missing More >

  • Open Access


    A Differential Quadrature Method for Multi-Dimensional Inverse Heat Conduction Problem of Heat Source

    Jiun-Yu Wu1,2, 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 >

  • Open Access


    Cauchy Problem for the Heat Equation in a Bounded Domain Without Initial Value

    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 >

  • Open Access


    The Generalized Tikhonov Regularization Method for High Order Numerical Derivatives

    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 >

  • Open Access


    A Regularized Integral Equation Scheme for Three-Dimensional Groundwater Pollution Source Identification Problems

    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 >

  • Open Access


    A Dynamical Tikhonov Regularization Method for Solving Nonlinear Ill-Posed Problems

    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 >

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