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Raul Tempone - One of the best experts on this subject based on the ideXlab platform.

  • a stochastic collocation method for elliptic partial differential equations with random input data
    Siam Review, 2010
    Co-Authors: Ivo Babuska, Fabio Nobile, Raul Tempone
    Abstract:

    This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in Space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the Probability Space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “Probability error” with respect to the number of Gauss points in each direction of the Probability Space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.

  • a stochastic collocation method for elliptic partial differential equations with random input data
    SIAM Journal on Numerical Analysis, 2007
    Co-Authors: Ivo Babus Caron, Fabio Nobile, Raul Tempone
    Abstract:

    In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in Space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the Probability Space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuscka, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “Probability error” with respect to the number of Gauss points in each direction in the Probability Space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

Gianluca Iaccarino - One of the best experts on this subject based on the ideXlab platform.

  • a least squares approximation of partial differential equations with high dimensional random inputs
    Journal of Computational Physics, 2009
    Co-Authors: Alireza Doostan, Gianluca Iaccarino
    Abstract:

    Uncertainty quantification schemes based on stochastic Galerkin projections, with global or local basis functions, and also stochastic collocation methods in their conventional form, suffer from the so called curse of dimensionality: the associated computational cost grows exponentially as a function of the number of random variables defining the underlying Probability Space of the problem. In this paper, to overcome the curse of dimensionality, a low-rank separated approximation of the solution of a stochastic partial differential (SPDE) with high-dimensional random input data is obtained using an alternating least-squares (ALS) scheme. It will be shown that, in theory, the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying Probability Space of the system. For the case of an elliptic SPDE, an a priori error analysis of the algorithm is derived. Finally, different aspects of the proposed methodology are explored through its application to some numerical experiments.

Ivo Babuska - One of the best experts on this subject based on the ideXlab platform.

  • a stochastic collocation method for elliptic partial differential equations with random input data
    Siam Review, 2010
    Co-Authors: Ivo Babuska, Fabio Nobile, Raul Tempone
    Abstract:

    This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in Space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the Probability Space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “Probability error” with respect to the number of Gauss points in each direction of the Probability Space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.

Yiming Zhao - One of the best experts on this subject based on the ideXlab platform.

Fabio Nobile - One of the best experts on this subject based on the ideXlab platform.

  • a stochastic collocation method for elliptic partial differential equations with random input data
    Siam Review, 2010
    Co-Authors: Ivo Babuska, Fabio Nobile, Raul Tempone
    Abstract:

    This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in Space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the Probability Space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “Probability error” with respect to the number of Gauss points in each direction of the Probability Space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.

  • a stochastic collocation method for elliptic partial differential equations with random input data
    SIAM Journal on Numerical Analysis, 2007
    Co-Authors: Ivo Babus Caron, Fabio Nobile, Raul Tempone
    Abstract:

    In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in Space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the Probability Space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuscka, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “Probability error” with respect to the number of Gauss points in each direction in the Probability Space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.