Positive Definite Matrix

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C. Soize - One of the best experts on this subject based on the ideXlab platform.

  • Non‐Gaussian Positive‐Definite Matrix‐valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries
    International Journal for Numerical Methods in Engineering, 2011
    Co-Authors: Johann Guilleminot, C. Soize
    Abstract:

    This paper is devoted to the construction of a class of prior stochastic models for non-Gaussian Positive-Definite Matrix-valued random fields. The proposed class allows the variances of selected random eigenvalues to be specified and exhibits a larger number of parameters than the other classes previously derived within a nonparametric framework. Having recourse to a particular characterization of material symmetry classes, we then propose a mechanical interpretation of the constraints and subsequently show that the probabilistic model may allow prescribing higher statistical fluctuations in given directions. Such stochastic fields turn out to be especially suitable for experimental identification under material symmetry uncertainties, as well as for the development of computational multi-scale approaches where the randomness induced by fine-scale features may be taken into account. We further present a possible strategy for inverse identification, relying on the sequential solving of least-square optimization problems. An application is finally provided. Copyright (c) 2011 John Wiley & Sons, Ltd.

  • Non-Gaussian Positive-Definite Matrix-valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries
    International Journal for Numerical Methods in Engineering, 2011
    Co-Authors: Johann Guilleminot, C. Soize
    Abstract:

    This paper is devoted to the construction of a class of prior stochastic models for non-Gaussian Positive-Definite Matrix-valued random fields. The proposed class allows the variances of selected random eigenvalues to be specified and exhibits a larger number of parameters than the other classes previously derived within a nonparametric framework. Having recourse to a particular characterization of material symmetry classes, we then propose a mechanical interpretation of the constraints and subsequently show that the probabilistic model may allow prescribing higher statistical fluctuations in given directions. Such stochastic fields turn out to be especially suitable for experimental identification under material symmetry uncertainties, as well as for the development of computational multi-scale approaches where the randomness induced by fine-scale features may be taken into account. We further present a possible strategy for inverse identification, relying on the sequential solving of least-square optimization problems. An application is finally provided. Copyright (c) 2011 John Wiley & Sons, Ltd.

  • Non-Gaussian Positive-Definite Matrix-valued random fields for elliptic stochastic partial differential operators
    Computer Methods in Applied Mechanics and Engineering, 2006
    Co-Authors: C. Soize
    Abstract:

    International audienceThis paper deals with the construction of a class of non-Gaussian Positive-Definite Matrix-valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled. The properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given. Such a Matrix-valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters. The class of random fields which can be approximated is presented and their experimental identification is analyzed. An example is given in three-dimensional linear elasticity for which the fourth-order elasticity tensor-valued random field is constructed for a random non-homogeneous anisotropic elastic material

  • Non-Gaussian Positive-Definite Matrix-valued random fields for elliptic stochastic partial differential operators
    Computer Methods in Applied Mechanics and Engineering, 2006
    Co-Authors: C. Soize
    Abstract:

    This paper deals with the construction of a class of non-Gaussian Positive-Definite Matrix-valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled. The properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given. Such a Matrix-valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters. The class of random fields which can be approximated is presented and their experimental identification is analyzed. An example is given in three-dimensional linear elasticity for which the fourth-order elasticity tensor-valued random field is constructed for a random non-homogeneous anisotropic elastic material.

  • Non Gaussian Matrix-valued random fields for nonparametric probabilistic modeling of elliptic stochastic partial differential operators
    2004
    Co-Authors: C. Soize
    Abstract:

    This paper deals with the construction of a non Gaussian Positive-Definite Matrix-valued random field whose mathematical properties allow elliptic stochastic partial differential operators to be modeled. Such a Matrix- valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters. The non Gaussian Positive-Definite Matrix-valued random field presented in this paper allows such a probabilistic model of the fourth-order tensor-valued random field to be constructed and depends only of 4 scalar parameters: three spatial correlation lengths and one parameter allowing the level of the random fluctuations to be controlled. Such a model can directly be used in the stochastic finite element method.

Nobuo Yamashita - One of the best experts on this subject based on the ideXlab platform.

  • Analysis of Sparse Quasi-Newton Updates with Positive Definite Matrix Completion
    Journal of the Operations Research Society of China, 2014
    Co-Authors: Yu-hong Dai, Nobuo Yamashita
    Abstract:

    Based on the idea of maximum determinant Positive Definite Matrix completion, Yamashita (Math Prog 115(1):1–30, 2008) proposed a new sparse quasi-Newton update, called MCQN, for unconstrained optimization problems with sparse Hessian structures. In exchange of the relaxation of the secant equation, the MCQN update avoids solving difficult subproblems and overcomes the ill-conditioning of approximate Hessian matrices. However, local and superlinear convergence results were only established for the MCQN update with the DFP method. In this paper, we extend the convergence result to the MCQN update with the whole Broyden’s convex family. Numerical results are also reported, which suggest some efficient ways of choosing the parameter in the MCQN update the Broyden’s family.

  • Convergence analysis of sparse quasi-Newton updates with Positive Definite Matrix completion for two-dimensional functions
    Numerical Algebra Control & Optimization, 2011
    Co-Authors: Yu-hong Dai, Nobuo Yamashita
    Abstract:

    In this paper, we briefly review the extensions of quasi-Newton methods for large-scale optimization. Specially, based on the idea of maximum determinant Positive Definite Matrix completion, Yamashita (2008) proposed a new sparse quasi-Newton update, called MCQN, for unconstrained optimization problems with sparse Hessian structures. In exchange of the relaxation of the secant equation, the MCQN update avoids solving difficult subproblems and overcomes the ill-conditioning of approximate Hessian matrices. A global convergence analysis is given in this paper for the MCQN update with Broyden's convex family assuming that the objective function is uniformly convex and its dimension is only two.   This paper is dedicated to Professor Masao Fukushima on occasion of his 60th birthday.

  • Sparse quasi-Newton updates with Positive Definite Matrix completion
    Mathematical Programming, 2008
    Co-Authors: Nobuo Yamashita
    Abstract:

    Quasi-Newton methods are powerful techniques for solving unconstrained minimization problems. Variable metric methods, which include the BFGS and DFP methods, generate dense Positive Definite approximations and, therefore, are not applicable to large-scale problems. To overcome this difficulty, a sparse quasi-Newton update with Positive Definite Matrix completion that exploits the sparsity pattern $$E :=\{(i, j)\;|\; (\nabla^2 f(x))_{ij} \neq 0\,{\rm for\,some}\, x \in R^n\}$$ of the Hessian is proposed. The proposed method first calculates a partial approximate Hessian $$H^{QN}_{ij}, (i, j) \in F$$ , where $$F \supseteq E$$ , using an existing quasi-Newton update formula such as the BFGS or DFP methods. Next, a full Matrix H _ k +1, which is a maximum-determinant Positive Definite Matrix completion of $$H^{QN}_{ij}, (i, j) \in F$$ , is obtained. If the sparsity pattern E (or its extension F ) has a property related to a chordal graph, then the Matrix H _ k +1 can be expressed as products of some sparse matrices. The time and space requirements of the proposed method are lower than those of the BFGS or the DFP methods. In particular, when the Hessian Matrix is tridiagonal, the complexities become O ( n ). The proposed method is shown to have superlinear convergence under the usual assumptions.

Eun Heui Kim - One of the best experts on this subject based on the ideXlab platform.

Johann Guilleminot - One of the best experts on this subject based on the ideXlab platform.

  • Non‐Gaussian Positive‐Definite Matrix‐valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries
    International Journal for Numerical Methods in Engineering, 2011
    Co-Authors: Johann Guilleminot, C. Soize
    Abstract:

    This paper is devoted to the construction of a class of prior stochastic models for non-Gaussian Positive-Definite Matrix-valued random fields. The proposed class allows the variances of selected random eigenvalues to be specified and exhibits a larger number of parameters than the other classes previously derived within a nonparametric framework. Having recourse to a particular characterization of material symmetry classes, we then propose a mechanical interpretation of the constraints and subsequently show that the probabilistic model may allow prescribing higher statistical fluctuations in given directions. Such stochastic fields turn out to be especially suitable for experimental identification under material symmetry uncertainties, as well as for the development of computational multi-scale approaches where the randomness induced by fine-scale features may be taken into account. We further present a possible strategy for inverse identification, relying on the sequential solving of least-square optimization problems. An application is finally provided. Copyright (c) 2011 John Wiley & Sons, Ltd.

  • Non-Gaussian Positive-Definite Matrix-valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries
    International Journal for Numerical Methods in Engineering, 2011
    Co-Authors: Johann Guilleminot, C. Soize
    Abstract:

    This paper is devoted to the construction of a class of prior stochastic models for non-Gaussian Positive-Definite Matrix-valued random fields. The proposed class allows the variances of selected random eigenvalues to be specified and exhibits a larger number of parameters than the other classes previously derived within a nonparametric framework. Having recourse to a particular characterization of material symmetry classes, we then propose a mechanical interpretation of the constraints and subsequently show that the probabilistic model may allow prescribing higher statistical fluctuations in given directions. Such stochastic fields turn out to be especially suitable for experimental identification under material symmetry uncertainties, as well as for the development of computational multi-scale approaches where the randomness induced by fine-scale features may be taken into account. We further present a possible strategy for inverse identification, relying on the sequential solving of least-square optimization problems. An application is finally provided. Copyright (c) 2011 John Wiley & Sons, Ltd.

Jae Heon Yun - One of the best experts on this subject based on the ideXlab platform.

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