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  • 1. Babuska, I.
    et al.
    Tempone Olariaga, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Zouraris, Georgios
    Galerkin finite element approximations of stochastic elliptic partial differential equations2004In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 42, no 2, p. 800-825Article in journal (Refereed)
    Abstract [en]

    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 computations 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 generates 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 second 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.

  • 2. Björk, T
    et al.
    Szepessy, Anders
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Tempone Olariaga, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Zourari, Georgios
    Monte Carlo euler approximation if HJM term structure financial models2001In: Stochastic Numerics 2001 at ETH, Zurich, Switzerland. February 19 - 21, 2001, 2001Conference paper (Other academic)
  • 3. Moon, Kyoung-Sook
    et al.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Tempone Olariaga, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Zouraris, Georgios
    Convergence rates for adaptive weak approximation of stochastic differential equations2005In: Stochastic Analysis and Applications, ISSN 0736-2994, E-ISSN 1532-9356, Vol. 23, no 3, p. 511-558Article in journal (Refereed)
    Abstract [en]

    Convergence rates of adaptive algorithms for weak approximations of Ito stochastic differential equations are proved for the Monte Carlo Euler method. Two algorithms based either oil optimal stochastic time steps or optimal deterministic time steps are studied. The analysis of their computational complexity combines the error expansions with a posteriori leading order term introduced in Szepessy et al. [Szepessy, A.. R. Tempone, and G. Zouraris. 2001. Comm. Pare Appl. Math. 54:1169-1214] and ail extension of the convergence results for adaptive algorithms approximating deterministic ordinary differential equations, derived in Moon et al. [Moon, K.-S., A. Szepessy, R. Tempone, and G. Zouraris. 2003. Numer. Malh. 93:99-129]. The main step in the extension is the proof of the almost sure convergence of the error density. Both adaptive alogrithms are proven to stop with asymptotically optimal number of steps up to a problem independent factor defined in the algorithm. Numerical examples illustrate the behavior of the adaptive algorithms, motivating when stochastic and deterministic adaptive time steps are more efficient than constant time steps and when adaptive stochastic steps are more efficient than adaptive deterministic steps.

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