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Articles Published in Computing

  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. Griebel, M.. "Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences" Computing. 61 (2). 1998. pp. 151--179.


  3. Gerstner, T. and Griebel, M.. "Dimension-adaptive tensor-product quadrature" Computing. 71 (1). SEP 2003. pp. 65--87.

    We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself. The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low- and moderate-dimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm. The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.


  4. Bungartz, H.J. and Dirnstorfer, S.. "Multivariate quadrature on adaptive sparse grids" Computing. 71 (1). SEP 2003. pp. 89--114.

    In this paper, we study the potential of adaptive sparse grids for multivariate numerical quadrature in the moderate or high dimensional case, i.e. for a number of dimensions beyond three and up to several hundreds. There, conventional methods typically suffer from the curse of dimension or are unsatisfactory with respect to accuracy. Our sparse grid approach, based upon a direct higher order discretization on the sparse grid, overcomes this dilemma to some extent, and introduces additional flexibility with respect to both the order of the 1 D quadrature rule applied (in the sense of Smolyak's tensor product decomposition) and the placement of grid points. The presented algorithm is applied to some test problems and compared with other existing methods.