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Title: Hierarchical%20Polynomial-Bases%20


1
Hierarchical Polynomial-BasesSparse Grids
1/21
grid Gitter ltgt ?é??? sparse spärlich, dünn
ltgt ?é????
2
1.1 A few properties of function spaces
2/21
1. Introduction
Let be a function space and
- is a infinite dimensional vector space
-
  • span fi is a subspace of

A few examples - Cn(O , R) is the space of n
times differentiable functions from
Rd to R - span1,x,x2,,xn
3
1.2 The tensor product
3/21
Let f and g be two functions, then the
tensor product is defined by
So if we have the function f for example
? Image
the tensor product is
? Image
otherwise sonst ltgt ? ?????? ??????
4
1.3 Norms in function spaces
4/21
Sometimes we want to measure the length of a
function. In Cn(O ,R) we will look at three
different norms
(energy norm)
5
2.1 A simple function space
5/21
2. The hierarchical basis
On page 3 we have seen a function f. Now we will
define functions, which are closely related to f
? Image
These are basis functions of
R
? Image
6
2.2 A new basis
6/21
We define and get If we take now these
basis functions of Wk we get the hierarchical
basis of Vn
? Image
Applying the tensor product to these
functions, we get a hierarchical basis of higher
dimensional spaces Vn,d of dimension d.
? Image
odd ungerade ltgt ????????
7
7/21
For all basis functions fk,i the following
equations hold
equation Gleichung ltgt ?????????
8
2.3 Approximation
8/21
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)
? Example
(hierarchical surplus)
surplus Überschuss ltgt ???????
9
9/21
With the help of the integral representation
of the coefficients
we get the following estimates
and from this
estimate Abschätzung ltgt ??????
10
3. Sparse grids
3.1 Multi-indices
10/21
For multi-indices we define
11
3.2 Grids
11/21
A d-dimensional grid can be written as a
multi- index with mesh size
? Example
The grid points are
Now we can assign every xm,i a function
mesh Masche ltgt ?????
12
3.3 Curse of dimensionality
12/21
The dimension of is
But as we seen before
and we get
curse Fluch ltgt ?????????
13
3.4 The solution
13/21
We search for subspaces Wl where the quotient
is as big as possible
? Image
benefit Nutzen ltgt ??????
14
14/21
There exists also an optimal choice of grids for
the energy-norm. We get the function space
and the estimates
15
3.5 e-complexity
15/21
16
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 pj1
points. For that we have to look at the
hierarchical ancestors.
? Example
arbitrary beliebig ltgt ?????
ancestor Vorfahr ltgt ??????
17
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.
18
18/21
4.2 Estimates
For the basis polynomials we get
We define a constant-function
19
19/21
The estimates for the hierarchical surplus are
with
as before
we get for
20
20/21
But as the costs do not change
we can define the same as before
For a function out of the order of
approximation is given by
21
21/21
4.3 e- complexities
For we get
For we get
22
The End
23
Bild1
Image1
24
Bild2
Image2
25
Bild3
Bild3
Image3
n 3
n 1 and n 2
f1,1
f3,5
26
Bild4
Image4
This is an example for a function in V3
27
Bild5
Image5
natural hierarchical basis
W1 W2 W3
f1,1
f2,1
f2,3
nodal point basis
node Knoten ltgt ????
28
Bild6
Image6
29
Example1
30
Example2
l(3,2) hl (1/8,1/4)
31
Image7
W(1,1)
W(2,1)
W(1,2)
32
Image8
33
Example3
0
1
34
Example4
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