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Articles Published on Sparse Grids

  1. Novak, E. and Ritter, K.. "High dimensional integration of smooth functions over cubes" Numerische Mathematik. vol. 75. 1996. pp. 79--97.
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  2. "The curse of dimension and a universal method for numerical integration" Multivariate Approximation and Splines. Novak, E. and Ritter, K.. Editted by G. Nurnberger and J. W. Schmidt and G. Walz. Birkhauser. 1997. pp. 177--188.

  3. Novak, E., Ritter, K., Schmitt, R., and Steinbauer, A.. "On an interpolatory method for high dimensional integration" Journal of Computational and Applied Mathematics. vol. 112. 1999. pp. 215--228.


  4. "Uncertainty Modeling using Fuzzy Arithmetic and Sparse Grids" Klimke, A.. Universitt Stuttgart. 2006.

  5. Barthelmann, V., Novak, E., and Ritter, K.. "High dimensional polynomial interpolation on sparse grids" Adv. Comput. Math.. vol. 12. 2000. pp. 273--288.


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


  7. Vasilyev, O.V. and Paolucci, S.. "A fast adaptive wavlet collocation algorithm for multidimensional PDEs" Journal of Computational Physics. vol. 138. 1997. pp. 16--56.


  8. Gerstner, T. and Griebel, M.. "Numerical integration using sparse grids" Numer. Alg.. 18 (3-4). 1998. pp. 209--232.

    We present new and review existing algorithms for the numerical integration of multivariate functions defined over d-dimensional cubes using several variants of the sparse grid method first introduced by Smolyak [49]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suitable one-dimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the one-dimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw-Curtis and Gauss rules in several numerical experiments and applications. For the computation of path integrals further improvements can be obtained by combining generalized Smolyak quadrature with the Brownian bridge construction.


  9. 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.


  10. 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.