Sparse grid
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Sparse grids are numerical techniques to represent, integrate or interpolate high dimensional functions. They were originally developed by the Russian mathematician Sergey A. Smolyak, a student of Lazar Lyusternik, and are based on a sparse tensor product construction. Computer algorithms for efficient implementations of such grids were later developed by Michael Griebel, Christoph Zenger, and Dirk Pflüger.
Sparse grids are numerical techniques to represent, integrate or interpolate high dimensional functions. They were originally developed by the Russian mathematician Sergey A. Smolyak, a student of Lazar Lyusternik, and are based on a sparse tensor product construction. Computer algorithms for efficient implementations of such grids were later developed by Michael Griebel, Christoph Zenger, and Dirk Pflüger. |E_l| = O(N_l^{-\frac{r}{d}}) Q_l^{(d)} f = \left(\sum_{i=1}^l \left(Q_i^{(1)}-Q_{i-1}^{(1)}\right)\otimes Q_{l-i+1}^{(d-1)}\right)f The index to Q is the level of the discretization. If a 1-dimension integration on level i is computed by the evaluation of O(2^{i}) points, the error estimate for a function of regularity r will be |E_l| = O\left(N_l^{-r}\left(\log N_l\right)^{(d-1)(r+1)}\right)
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