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Articles written by Todor, R.A.

  1. Schwab, C. and Todor, R.A.. "Sparse finite elements for stochastic elliptic problems: higher order moments" Computing. 71 (1). 2003. pp. 43--63.

    We define the higher order moments associated to the stochastic solution of an elliptic BVP in D ⊂ Rd with stochastic input data. We prove that the k-th moment solves a deterministic problem in Dk ⊂ Rdk, for which we discuss well-posedness and regularity. We discretize the deterministic k-th moment problem using sparse grids and, exploiting a spline wavelet basis, we propose an efficient algorithm, of logarithmic-linear complexity, for solving the resulting system.


  2. Schwab, C. and Todor, R.A.. "Karhunen-Lo\`eve approximation of random fields by generalized fast multipole methods" Journal of Computational Physics. 217 (1). 2006. pp. 100--122.


  3. Todor, R.A. and Schwab, C.. "Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients" IMA J. of Numer. Anal.. 27 (2). APR 2007. pp. 232--261.

    A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a(x, omega) in a bounded domain D subset of < Ropf >(d) is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x is an element of D) and stochastic (omega is an element of Omega) variables in a(x, omega) via Karhunen-Loeve or Legendre expansions of the diffusion coefficient. Deterministic, approximate solvers are obtained by projection of this problem into a product probability space of finite dimension M and sparse discretizations of the resulting M-dimensional parametric problem. Both Galerkin and collocation approximations are considered. Under regularity assumptions on the fluctuation of a(x, omega) in the deterministic variable x, the convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp. the polynomial degree of the chaos expansion increase simultaneously.


  4. Frauenfelder, P., Schwab, C., and Todor, R.A.. "Finite elements for elliptic problems with stochastic coefficients" 194 (2-5, Sp. Iss. SI). 2005. pp. 205--228.

    We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncertainty) is achieved via a Karhunen-Loeve (KL) expansion. An O(NlogN) algorithm for the computation of the KL eigenvalues is presented, based on a kernel independent fast multipole method (FMM). Truncation of the KL expansion gives an (M, 1) Wiener polynomial chaos (PC) expansion of the stochastic coefficient and is shown to lead to a high dimensional, deterministic boundary value problem (dbvp). Analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r = r(M)) in an (M, r) Wiener PC expansion for the approximate solution. Pointwise error bounds for the FEM approximations of KL eigenpairs, the truncation of the KL expansion and the FE solution to the dbvp are given. Numerical examples show that M depends on the spatial correlation length of the random diffusion coefficient. The variable polynomial degree r in PC-stochastic Galerkin FEM allows to handle KL expansions with M up to 30 and r, up to 10 in moderate time. (C) 2004 Elsevier B.V. All rights reserved.