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Hierarchical Polynomial-Bases & Sparse Grids 1/21 grid: Gitter <> сéтка sparse: spärlich, dünn <> рéдкий

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2/21 1. Introduction Let be a function space and 1.1 A few properties of function spaces A few examples: - C n (Ω, R) is the space of n times differentiable functions from R d to R - span{1,x,x 2,…,x n } - span{ f i } is a subspace of - is a infinite dimensional vector space -

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3/21 Image 1.2 The tensor product Let f and g be two functions, then the tensor product is defined by So if we have the function φ for example: the tensor product is otherwise: sonst <> в другой случае

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4/21 Sometimes we want to measure the length of a function. In C n (Ω,R) we will look at three different norms: 1.3 Norms in function spaces (energy norm)

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5/21 2. The hierarchical basis On page 3 we have seen a function φ. Now we will define functions, which are closely related to φ: Image These are basis functions of R Image 2.1 A simple function space

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6/21 We define: and get If we take now these basis functions of W k we get the hierarchical basis of V n odd: ungerade <> нечётный Image 2.2 A new basis Image Applying the tensor product to these functions, we get a hierarchical basis of higher dimensional spaces V n,d of dimension d.

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7/21 For all basis functions φ k,i the following equations hold: equation: Gleichung <> уравнение

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8/21 2.3 Approximation Now we want to approximate a function f in C([0,1], R) with f(0) = f(1) = 0 by a function in V n. Example (function values) (hierarchical surplus) surplus: Überschuss <> избыток

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9/21 With the help of the integral representation of the coefficients we get the following estimates: estimate: Abschätzung <> оценка and from this

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10/21 3. Sparse grids For multi-indices we define: 3.1 Multi-indices

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11/21 A d-dimensional grid can be written as a multi- index with mesh size mesh: Masche <> петля The grid points are Now we can assign every x m,i a function Example 3.2 Grids

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12/21 3.3 Curse of dimensionality The dimension of is But as we seen before curse: Fluch <> проклятие and we get

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13/21 3.4 The solution We search for subspaces W l where the quotient is as big as possible Image benefit: Nutzen <> польза

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14/21 There exists also an optimal choice of grids for the energy-norm. We get the function space and the estimates

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15/21 3.5 -complexity

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16/21 4. Higher-order polynomials 4.1 Construction Now we want to generalize the piecewise linear basis functions to polynomials of arbitrary degree. We use the tensor product: Image arbitrary: beliebig <> любой with To determine this polynomial we need p j +1 points. For that we have to look at the hierarchical ancestors. Example ancestor: Vorfahr <> предок

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17/21 is now defined as the Lagrangian interpolation polynomial with the following properties: and is zero for the p j -2 next ancestors. Example This scheme is not correct for the linear basis functions, as they are only piecewise linear and need three definition points.

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18/21 4.2 Estimates For the basis polynomials we get: We define a constant-function

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19/21 The estimates for the hierarchical surplus are: withas before we get for

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20/21 For a function out of the order of approximation is given by But as the costs do not change: we can define the same as before

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21/21 4.3 ε- complexities For we get

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The End

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Image1 Bild1

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Image2 Bild2

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Bild3 Image3 n= 3n= 1 and n= 2 φ 1,1 φ 3,5

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Bild4 Image4 This is an example for a function in V 3

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Bild5 Image5 nodal point basis natural hierarchical basis φ 1,1 φ 2,1 φ 2,3 node: Knoten <> узел W1W2W3W1W2W3

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Bild6 Image6

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Example1

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l=(3,2) h l = (1/8,1/4) Example2

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Image7 W (1,1) W (2,1) W (1,2)

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Image8

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Example3 01

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Example4

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