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| '''Sparse grids''' are numerical techniques to represent, integrate or interpolate high [[dimension]]al functions. They were originally developed by the [[Russia]]n [[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]] and [[Christoph Zenger]].
| | I am Aline and was born on 21 January 1987. My hobbies are Kart racing and Seashell Collecting.<br><br>my page - [http://dicalife.com/free/3383504 Bookbyte Buyback Guidelines] |
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| == [[Curse of dimensionality]] ==
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| The standard way of representing multidimensional functions are tensor or full grids. The number of basis functions or nodes (grid points) that have to be stored and processed [[exponential function|depend exponentially]] on the number of dimensions. Even with today's computational power it is not possible to process functions with more than 4 or 5 dimensions.
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| The curse of dimension is expressed in the order of the integration error that is made by a quadrature of level <math>l</math>, with <math>N_{l}</math> points. The function has regularity <math>r</math>, i.e. is <math>r</math> times differentiable. The number of dimensions is <math>d</math>.
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| <math>|E_l| = O(N_l^{-\frac{r}{d}})</math>
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| == Smolyak's quadrature rule ==
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| Smolyak found a computationally more efficient method of integrating multidimensional functions based on a univariate quadrature rule <math>Q^{(1)}</math>. The <math>d</math>-dimensional Smolyak integral <math>Q^{(d)}</math>of a function <math>f</math> can be written as a recursion formula with the tensor product.
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| <math>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</math>
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| The index to <math>Q</math> is the level of the discretization. A <math>1-d</math> integration on level <math>i</math> is computed by the evaluation of <math>O(2^{i})</math> points. The error estimate for a function of regularity <math>r</math> is:
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| <math>|E_l| = O\left(N_l^{-r}\left(\log N_l\right)^{(d-1)(r+1)}\right)</math>
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| == References ==
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| * [http://www.lrr.in.tum.de/~murarasu/ppopp027s-murarasu.pdf A memory efficient data structure for regular sparse grids]
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| * [http://sparse-grids.de/ Code to generate (and pre-generated) nodes and weights for quadrature]
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| * [http://wissrech.iam.uni-bonn.de/research/projects/zumbusch/fd.html Finite difference scheme on sparse grids]
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| * [http://www.vis.uni-stuttgart.de/ger/research/fields/recent/sparse/ Visualization on sparse grids]
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| * [http://wissrech.iam.uni-bonn.de/research/pub/garcke/kdd.pdf Datamining on sparse grids, J.Garcke, M.Griebel (pdf)]
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| * [http://www.math.tu-berlin.de/~garcke/paper/sparseGridTutorial.pdf Sparse Grid Tutorial, J.Garcke (pdf)]
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| {{Mathanalysis-stub}}
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| [[Category:Numerical analysis]]
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I am Aline and was born on 21 January 1987. My hobbies are Kart racing and Seashell Collecting.
my page - Bookbyte Buyback Guidelines