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The Sparse Grid Method
Published on Feb 25, 20079349 Views
The sparse grid method is a special discretization technique, which allows to cope with the curse of dimensionality to some extent. It is based on a hierarchical basis and a sparse tensor product deco
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Chapter list
Sparse Grids00:00
Outline00:02
Partial Differential Equations00:31
Galerkin-Variational Principle05:05
Discretisation07:30
Example for VN in One Dimension09:01
One-dimensional Basis Functions10:04
Basis Functions in More Dimensions10:43
Some Notation12:40
Triangulation Instead of Tensor Product14:14
Approximation Properties15:47
Interpolation with Hierarchical Basis20:52
Hierarchical Difference Spaces23:31
Hierarchical Tensor Product Decomposition24:43
Hierarchical Subspaces Wl for V3,325:33
Hierarchical Basis [Faber:09,Yserentant:86]26:11
Interpolation with Hierarchical Basis27:00
Sobolev-Space H2<br>mix with Domin. Mixed Deriv.28:33
Hierarchical Values l;j are Bounded I30:32
Hierarchical Values l;j are Bounded II31:24
Hierarchical Values l;j are Bounded III33:26
Hier. Compon. Bounded by Size of Support34:26
Hierarchical Subspaces Wl35:48
Hierarchical Subspaces Wl 0136:11
Sparse Grids36:43
Sparse Grids in two and three dimensions38:56
History of Sparse Grids39:24
Some Recent Applications of Sparse Grids40:37
Simple Example in Numerical Integration 10D40:54
How to Compute on a Sparse Grid43:11
Combination Technique of Level 4 in 2d43:54
Telescope Sum Property for Interpolation44:59
Sparse Grid Combination Technique45:15
Generalised Combination Technique47:16
Summary Sparse Grids48:23
Problem Setting for Regression /<br>Classfication49:11
Regularisation Theory50:02
Discretisation51:25