Hierarchical%20Polynomial-Bases%20

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Hierarchical Polynomial-BasesSparse Grids
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grid Gitter ltgt ?é??? sparse spärlich, dünn
ltgt ?é????
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1.1 A few properties of function spaces
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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
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1.2 The tensor product
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Let f and g be two functions, then the
tensor product is defined by
So if we have the function f for example
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the tensor product is
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otherwise sonst ltgt ? ?????? ??????
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1.3 Norms in function spaces
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Sometimes we want to measure the length of a
function. In Cn(O ,R) we will look at three
different norms
(energy norm)
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2.1 A simple function space
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2. The hierarchical basis
On page 3 we have seen a function f. Now we will
define functions, which are closely related to f
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These are basis functions of
R
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2.2 A new basis
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We define and get If we take now these
basis functions of Wk we get the hierarchical
basis of Vn
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Applying the tensor product to these
functions, we get a hierarchical basis of higher
dimensional spaces Vn,d of dimension d.
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odd ungerade ltgt ????????
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For all basis functions fk,i the following
equations hold
equation Gleichung ltgt ?????????
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2.3 Approximation
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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 ???????
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With the help of the integral representation
of the coefficients
we get the following estimates
and from this
estimate Abschätzung ltgt ??????
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3. Sparse grids
3.1 Multi-indices
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For multi-indices we define
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3.2 Grids
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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 ?????
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3.3 Curse of dimensionality
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The dimension of is
But as we seen before
and we get
curse Fluch ltgt ?????????
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3.4 The solution
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We search for subspaces Wl where the quotient
is as big as possible
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benefit Nutzen ltgt ??????
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There exists also an optimal choice of grids for
the energy-norm. We get the function space
and the estimates
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3.5 e-complexity
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4. Higher-order polynomials
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4.1 Construction
Now we want to generalize the piecewise
linear basis functions to polynomials of
arbitrary degree .
We use the tensor product
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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 ??????
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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.
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4.2 Estimates
For the basis polynomials we get
We define a constant-function
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The estimates for the hierarchical surplus are
with
as before
we get for
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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
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4.3 e- complexities
For we get
For we get
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The End
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Bild1
Image1
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Bild2
Image2
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Bild3
Bild3
Image3
n 3
n 1 and n 2
f1,1
f3,5
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Bild4
Image4
This is an example for a function in V3
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Bild5
Image5
natural hierarchical basis
W1 W2 W3
f1,1
f2,1
f2,3
nodal point basis
node Knoten ltgt ????
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Bild6
Image6
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Example1
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Example2
l(3,2) hl (1/8,1/4)
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Image7
W(1,1)
W(2,1)
W(1,2)
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Image8
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Example3
0
1
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Example4