| 1. |
- Babuska, Ivo, et al.
(författare)
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A stochastic collocation method for elliptic partial differential equations with random input data
- 2007
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 45:3, s. 1005-1034
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Tidskriftsartikel (refereegranskat)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. Babuska, 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.
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| 2. |
- Babuška, Ivo, et al.
(författare)
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On roundoff error growth in elliptic problems
- 2018
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Ingår i: ACM Transactions on Mathematical Software. - : Association for Computing Machinery (ACM). - 0098-3500 .- 1557-7295. ; 44:3
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Tidskriftsartikel (refereegranskat)abstract
- Large-scale linear systems arise in finite-difference and finite-element discretizations of elliptic problems. With increasing computer performance, ever larger systems are solved using direct methods. How large can such systems be without roundoff compromising accuracy? Here we model roundoff dynamics in standard LU and LDLT decompositions with respect to problem size N. For the one-dimensional (1D) Poisson equation with Dirichlet boundary conditions on an equidistant grid, we show that the relative error in the factorized matrix grows like O(ϵN) if roundoffs are modeled as independent, expectation zero random variables. With bias, the growth rate changes to O(ϵN). Subsequent back substitution results in typical error growths of O(ϵNN) and O(ϵN2), respectively. Error growth is governed by the dynamics of the computational process and by the structure of the boundary conditions rather than by the condition number. Computational results are demonstrated in several examples, including a few fourth-order 1D problems and second-order 2D problems, showing that error accumulation depends strongly on the solution method. Thus, the same LU solver may exhibit different growth rates for the same 2D Poisson problem, depending on whether the five-point or nine-point FDM operator is used.
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| 3. |
- Hills, Richard G., et al.
(författare)
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Validation Challenge Workshop
- 2008
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 197:29-32, s. 2375-2380
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Tidskriftsartikel (refereegranskat)abstract
- This special issue presents the results of the Sandia organized Model Validation Challenge Workshop, held May 2006. The workshop brought together researchers from different fields to present various approaches to model validation, and focused on the methodological elements of model validation rather than on model building. Three problems were defined in the disciplines of structural statics, structural dynamics, and heat transfer, all with a uniform structure. The workshop was specifically designed to investigate the relative merits of different approaches to hierarchal model validation through application to these problems. This paper describes a hierarchal approach in the challenge problems, presents the uniform conceptual framework that was used for the challenge problem definitions, and provides an overview of the organization of this special issue.
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