Abstract
Sparse grids have turned out to be a very efficient discretization scheme that, to some extent, breaks the curse of dimensionality and, therefore, is especially well-suited for higher dimensional scenarios. Besides the classical sparse grid application, the numerical solution of partial differential equations, sparse grids have been used for various topics such as Fourier transform, image compression, numerical quadrature, or data mining, so far. In this paper, we summarize and assess recent results concerning the application of sparse grids to integrate functions of higher dimensionality, the focus being on the explicit and adaptive use of higher order basis polynomials.
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References
Achatz, S.: Higher order sparse grids methods for elliptic partial differential equations with variable contraints. Computing 71(1), 1–15 (2003)
Bonk, T.: A new algorithm for multi-dimensional adaptive numerical quadrature. In: Hackbusch, W., Wittum, G. (eds.) Adaptive Methods – Algorithms, Theory, and Applications. NNFM, vol. 46, pp. 54–68. Vieweg (1994)
Bungartz, H.-J.: A multigrid algorithm for higher order finite elements on sparse grids. ETNA 6, 63–77 (1997)
Bungartz, H.-J.: Finite Elements of Higher Order on Sparse Grids. Shaker, Aachen (1998)
Bungartz, H.-J., Dirnstorfer, S.: Multivariate quadrature on adaptive sparse grids. Computing 71(1), 89–114 (2003)
Bungartz, H.-J., Griebel, M.: A note on the complexity of solving Poisson’s equation for spaces of bounded mixed derivatives. J. Complex. 15, 167–199 (1999)
Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numerica (2004)
Bungartz, H.-J., Griebel, M., Rüde, U.: Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems. Comput. Meth. Appl. Mech. Eng. 116, 243–252 (1994)
Caflisch, R., Morokoff, W., Owen, A.: Valuation of mortgage backed securities using brownian bridges to reduce effective dimension. J. Comput. Finance 1 (1997)
Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numerical Algorithms 18, 209–232 (1998)
Gerstner, T., Griebel, M.: Dimension-adaptive tensor-product quadrature. Computing 71, 65–87 (2003)
Kern, T.: Dünngittertechniken zur hochdimensionalen numerischen Quadratur. Master’s thesis, Universität Stuttgart (2003)
Morokoff, W., Caflisch, R.: Quasi-monte carlo integration. J. Comp. Phys. 122, 218–230 (1995)
Niederreiter, H.: Random Number Generation and Quasi-Monte-Carlo Methods. SIAM, Philadelphia (1992)
Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numer. Math. 75(1), 79–98 (1996)
Smolyak, S.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240–243 (1963)
Zenger, C.: Sparse grids. In: Hackbusch, W. (ed.) Parallel Algorithms for Partial Differential Equations. NNFM, vol. 31, Vieweg (1991)
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Bungartz, HJ., Dirnstorfer, S. (2004). Higher Order Quadrature on Sparse Grids. In: Bubak, M., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds) Computational Science - ICCS 2004. ICCS 2004. Lecture Notes in Computer Science, vol 3039. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25944-2_52
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DOI: https://doi.org/10.1007/978-3-540-25944-2_52
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