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Articles written by Babuska, I.M.

  1. Babuska, I.M., Tempone, R., and Zouraris, G.E.. "Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations" SIAM J. Numer. Anal.. 42 (2). 2004. pp. 800--825.

    We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the com- putations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method gener- ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The sec- ond method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h- or p-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.

    Keywords: stochastic elliptic equation ; perturbation estimates ; Karhunen--Loeve expansion ; finite elements ; Monte Carlo method ; $k\times h$-version ; $p\times h$-version ; expected value ; error estimates


  2. Deb, M.K., Babuska, I.M., and Oden, J.T. "Solution of stochastic partial differential equations using Galerkin finite element techniques" Comput. Methods Appl. Mech. Engrg.. vol. 190. 2001. pp. 6359--6372.

    This paper presents a framework for the construction of Galerkin approximations of elliptic boundary-value problems with stochastic input data. A variational formulation is developed which allows, among others, numerical treatment by the finite element method; a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized. The paper develops a foundation for treating stochastic partial differential equations (PDEs) which can be further developed in many directions


  3. "A stochastic collocation method for elliptic partial differential equations with random input data" Babuska, I.M., Nobile, F., and Tempone, R.. Dipartimento di Matematica. 2005.

  4. Babuska, I.M., Tempone, R., and Zouraris, G.E.. "Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation" Computer Methods in Applied Mechanics and Engineering. vol. 194. 2005. pp. 1251--1294.


  5. Babuska, I.M., Nobile, F., and Tempone, R.. "A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data" SIAM J. Numer. Anal.. 45 (3). 2007. pp. 1005--1034.

    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.