121 Hierarchical PolynomialBases Sparse Grids grid Gitter sparse

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

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

3/21 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: → Image the tensor product is → Image otherwise: sonst <> в другой случае

4/21 1. 3 Norms in function spaces Sometimes we want to measure the “length” of a function. In Cn(Ω , R) we will look at three different norms: (energy norm)

5/21 2. The hierarchical basis 2. 1 A “simple” function space 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

6/21 2. 2 A new basis We define: and get → Image If we take now these basis functions of Wk we get the hierarchical basis of Vn Applying the tensor product to these functions, we get a hierarchical basis of higher dimensional spaces Vn, d of dimension d. → Image odd: ungerade <> нечётный

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

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 Vn. (function values) (hierarchical surplus) surplus: Überschuss <> избыток → Example

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

3. Sparse grids 3. 1 Multi-indices For multi-indices we define: 10/21

11/21 3. 2 Grids A d-dimensional grid can be written as a multiindex with mesh size The grid points are → Example Now we can assign every xm, i a function mesh: Masche <> петля

12/21 3. 3 Curse of dimensionality The dimension of is But as we seen before and we get curse: Fluch <> проклятие

13/21 3. 4 The “solution” We search for subspaces Wl where the quotient is as big as possible → Image benefit: Nutzen <> польза

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

3. 5 e-complexity 15/21

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

17/21 is now defined as the Lagrangian interpolation polynomial with the following properties: and is zero for the pj-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.