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  • Open Access

    ARTICLE

    An Optimal Multi-Vector Iterative Algorithm in a Krylov Subspace for Solving the Ill-Posed Linear Inverse Problems

    Chein-Shan Liu 1

    CMC-Computers, Materials & Continua, Vol.33, No.2, pp. 175-198, 2013, DOI:10.3970/cmc.2013.033.175

    Abstract An optimal m-vector descent iterative algorithm in a Krylov subspace is developed, of which the m weighting parameters are optimized from a properly defined objective function to accelerate the convergence rate in solving an ill-posed linear problem. The optimal multi-vector iterative algorithm (OMVIA) is convergent fast and accurate, which is verified by numerical tests of several linear inverse problems, including the backward heat conduction problem, the heat source identification problem, the inverse Cauchy problem, and the external force recovery problem. Because the OMVIA has a good filtering effect, the numerical results recovered are quite smooth with small error, even under… More >

  • Open Access

    ARTICLE

    Fast Near-duplicate Image Detection in Riemannian Space by A Novel Hashing Scheme

    Ligang Zheng1,*, Chao Song2

    CMC-Computers, Materials & Continua, Vol.56, No.3, pp. 529-539, 2018, DOI: 10.3970/cmc.2018.03780

    Abstract There is a steep increase in data encoded as symmetric positive definite (SPD) matrix in the past decade. The set of SPD matrices forms a Riemannian manifold that constitutes a half convex cone in the vector space of matrices, which we sometimes call SPD manifold. One of the fundamental problems in the application of SPD manifold is to find the nearest neighbor of a queried SPD matrix. Hashing is a popular method that can be used for the nearest neighbor search. However, hashing cannot be directly applied to SPD manifold due to its non-Euclidean intrinsic geometry. Inspired by the idea… More >

  • Open Access

    ARTICLE

    A Proposal of Nonlinear Formulation of Cell Method for Thermo-Elastostatic Problems

    C. Delprete1, F. Freschi2, M. Repetto2, C. Rosso1

    CMES-Computer Modeling in Engineering & Sciences, Vol.94, No.5, pp. 397-420, 2013, DOI:10.3970/cmes.2013.094.397

    Abstract The growing necessity of accuracy in analyzing engineering problems requires more detailed and sophisticated models. Those models can include multiphysics interactions, that, sometimes, are highly nonlinear and the application of the superposition principle is then not possible. The cell method can be suitably used to study nonlinear multiphysics problems, because its theoretical framework for the physical laws is intrinsically multiphysics. In this way it is possible to take into account the mutual effects between different physics. Within the cell method framework, the coupling terms can be directly formulated in terms of the global variables used for the solution of the… More >

  • Open Access

    ARTICLE

    An Optimal Preconditioner with an Alternate Relaxation Parameter Used to Solve Ill-Posed Linear Problems

    Chein-Shan Liu1

    CMES-Computer Modeling in Engineering & Sciences, Vol.92, No.3, pp. 241-269, 2013, DOI:10.32604/cmes.2013.092.241

    Abstract In order to solve an ill-posed linear problem, we propose an innovative Jacobian type iterative method by presetting a conditioner before the steepest descent direction. The preconditioner is derived from an invariant manifold approach, which includes two parameters α and γ to be determined. When the weighting parameter α is optimized by minimizing a properly defined objective function, the relaxation parameter γ can be derived to accelerate the convergence speed under a switching criterion. When the switch is turned-on, by using the derived value of γ it can pull back the iterative orbit to the fast manifold. It is the… More >

  • Open Access

    ARTICLE

    A new implementation of the numerical manifold method (NMM) for the modeling of non-collinear and intersecting cracks

    Y.C. Cai1,2,3, J. Wu2, S.N. Atluri3

    CMES-Computer Modeling in Engineering & Sciences, Vol.92, No.1, pp. 63-85, 2013, DOI:10.3970/cmes.2013.092.063

    Abstract The numerical manifold method (NMM), based on the finite covers, unifies the continuum analyses and discontinuum analyses without changing a predefined mathematical mesh of the uncracked solid, and has the advantages of being concise in theory as well as being clear in concept. It provides a natural method to analyze complex shaped strong discontinuities as well as weak discontinuities such as multiple cracks, intersecting cracks, and branched cracks. However, the absence of an effective algorithm for cover generation, to date, is still a bottle neck in the research and application in the NMM. To address this issue, a new method… More >

  • Open Access

    ARTICLE

    A New Optimal Scheme for Solving Nonlinear Heat Conduction Problems

    Chih-Wen Chang1,2, Chein-Shan Liu3

    CMES-Computer Modeling in Engineering & Sciences, Vol.88, No.4, pp. 269-292, 2012, DOI:10.3970/cmes.2012.088.269

    Abstract In this article, we utilize an optimal vector driven algorithm (OVDA) to cope with the nonlinear heat conduction problems (HCPs). From this set of nonlinear ordinary differential equations, we propose a purely iterative scheme and the spatial-discretization of finite difference method for revealing the solution vector x, without having to invert the Jacobian matrix D. Furthermore, we introduce three new ideas of bifurcation, attracting set and optimal combination, which are restrained by two parameters g and a. Several numerical instances of nonlinear systems under noise are examined, finding that the OVDA has a fast convergence rate, great computation accuracy and… More >

  • Open Access

    ARTICLE

    Dynamical Newton-Like Methods for Solving Ill-Conditioned Systems of Nonlinear Equations with Applications to Boundary Value Problems

    Cheng-Yu Ku1,2,3,Weichung Yeih1,2, Chein-Shan Liu4

    CMES-Computer Modeling in Engineering & Sciences, Vol.76, No.2, pp. 83-108, 2011, DOI:10.3970/cmes.2011.076.083

    Abstract In this paper, a general dynamical method based on the construction of a scalar homotopy function to transform a vector function of Non-Linear Algebraic Equations (NAEs) into a time-dependent scalar function by introducing a fictitious time-like variable is proposed. With the introduction of a transformation matrix, the proposed general dynamical method can be transformed into several dynamical Newton-like methods including the Dynamical Newton Method (DNM), the Dynamical Jacobian-Inverse Free Method (DJIFM), and the Manifold-Based Exponentially Convergent Algorithm (MBECA). From the general dynamical method, we can also derive the conventional Newton method using a certain fictitious time-like function. The formulation presented… More >

  • Open Access

    ARTICLE

    Computing Prager's Kinematic Hardening Mixed-Control Equations in a Pseudo-Riemann Manifold

    Chein-Shan Liu1

    CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.3, pp. 161-180, 2006, DOI:10.3970/cmes.2006.012.161

    Abstract Materials' internal spacetime may bear certain similarities with the external spacetime of special relativity theory. Previously, it is shown that material hardening and anisotropy may cause the internal spacetime curved. In this paper we announce the third mechanism of mixed-control to cause the curvedness of internal spacetime. To tackle the mixed-control problem for a Prager kinematic hardening material, we demonstrate two new formulations. By using two-integrating factors idea we can derive two Lie type systems in the product space of Mm+1⊗Mn+1. The Lie algebra is a direct sum of so(m,1)so(n,1), and correspondingly the symmetry group is a direct product of… More >

  • Open Access

    ARTICLE

    A Globally Optimal Iterative Algorithm to Solve an Ill-Posed Linear System

    Chein-Shan Liu1

    CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.4, pp. 383-404, 2012, DOI:10.3970/cmes.2012.084.383

    Abstract An iterative algorithm based on the critical descent vector is proposed to solve an ill-posed linear system: Bx = b. We define a future cone in the Minkowski space as an invariant manifold, wherein the discrete dynamics evolves. A critical value αc in the critical descent vector u = αcr + BTr is derived, which renders the largest convergence rate as to be the globally optimal iterative algorithm (GOIA) among all the numerically iterative algorithms with the descent vector having the form u = αr + BTr to solve the ill-posed linear problems. Some numerical examples are used to reveal… More >

  • Open Access

    ARTICLE

    The Concept of Best Vector Used to Solve Ill-Posed Linear Inverse Problems

    Chein-Shan Liu

    CMES-Computer Modeling in Engineering & Sciences, Vol.83, No.5, pp. 499-526, 2012, DOI:10.3970/cmes.2012.083.499

    Abstract The iterative algorithms based on the concept of best vector are proposed to solve an ill-conditioned linear system: Bx-b=0, which might be a discretization of linear inverse problem. In terms of r:=Bx-b and a monotonically increasing positive function Q(t) of a time-like variable t, we define a future cone in the Minkowski space, wherein the discrete dynamics of the proposed algorithm is evolved. We propose two methods to approximate the best vector B-1r, and obtain three iterative algorithms for solving x, which we label them as the steepest-descent and optimal vectors iterative algorithm (SOVIA), the mixed optimal iterative algorithm (MOIA),… More >

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