Hierarchical Polynomial-Bases

&Sparse Grids

1/21

grid: Gitter <> сéтка sparse: spärlich, dünn <> рéдкий

2/211. Introduction

Let be a function space and

1.1 A few properties of function spaces

V gffi ,,

)())(()()())(( and xfxfxgxfxgf

A few examples: - Cn(Ω , R) is the space of n times differentiable functions from Rd to R - span{1,x,x2,…,xn }

- span{ fi } is a subspace of V

V

- is a infinite dimensional vector spaceV-

3/21

otherwise

1 ,  1  f       

0

||1  :)(

xixx → Image

)()(:),( yxyx → Image

1.2 The tensor product

Let f and g be two functions, then the tensorproduct is defined bygf

)()(:),)(( ygxfyxgf

So if we have the function φ for example:

the tensor product is :

otherwise: sonst <> в другой случае

4/21

Sometimes we want to measure the “length” of a function. In Cn(Ω ,R) we will look at three different norms:

}||)(sup{|:|||| xxff

1q|)(|:|||| q qq dxxff

2

1d2/1

d

1j

2 ||'||)(:|||| fxxf jEf

1.3 Norms in function spaces

(energy norm)

5/212. The hierarchical basis

On page 3 we have seen a function φ. Now we will define functions, which are closely related to φ:

12,,1i)i2(:)( nnin, xx → Image

These are basis functions of

}12,,1i|{span

}|{:

nin,

i

n

1iin,in

V R

→ Image

2.1 A “simple” function space

6/21

We define:

and get

If we take now these basis functions of Wk

we get the hierarchical basis of Vn

odd: ungerade <> нечётный

→ Image

2.2 A new basis

→ Image

i}odd|{span: ik,k W

Applying the tensor product to these functions,we get a hierarchical basis of higher dimensional spaces Vn,d of dimension d.

knk

n WV

7/21

1/222

k/2ik,

k/q

1/q

qik,

ik,

)('||'||

22||||

21q

2||||

1||||

|||| dxxffE

f

E

For all basis functions φk,i the following equations hold:

equation: Gleichung <> уравнение

8/212.3 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.

)2/)1((1/2)2/)1((1/2)2/(

and

)2/(

kkkik,

ni

n

1kk

n

1koddi

12..1iik,ik,in,

12

1iin

k

n

ififif

if

wuf

→ Example

(function values)

(hierarchical surplus)

surplus: Überschuss <> избыток

9/21

With the help of the integral representation ofthe coefficients

dxxfx )('')(2 i,k)1k(

i,k

we get the following estimates:

2/k

E/2/k ||''||4||||

fw

estimate: Abschätzung <> оценка

2)(suppk3/2)(

ik,

k2ik,

|||''||21/6||

||''||21/2||

ik,

f

f

and from this

10/213. Sparse grids

For multi-indices we define:

jdj1

d

1jj1

jj

def

d1

dd11

max:||and:||

:

)2,,2(:2

),,(

),,(

dj1

d1

d0N,

3.1 Multi-indices

11/21

A d-dimensional grid can be written as a multi-index with mesh sizedm N

mesh: Masche <> петля

)2,,2(2:h d1 --- mm mm

The grid points arem

mm 0 2ihi:)x,,x(:xdd11 ,,i, imim

Now we can assign every xm,i a function

12,,1)2(:)(

with)(:)(

jj

jj

jj

jjjj,

j

d

1j,

mmim

im

iixx

x

xim,

→ Example

3.2 Grids

12/213.3 Curse of dimensionality

ll

lill

WV

iW

n||

)(n

j,

:and

}dj1allforodd,12i1|{span:

The dimension of is

)()2()12(|| dn

dndn)(n

hOOV

)(nV

But as we seen before

curse: Fluch <> проклятие

)(||''||4)d(|||| 2n2/

||2/

1 hOfw

l

m and we get

)(||''||4)(|||| 2n2/

n2/

)(n hOfduf

13/213.4 The “solution”

We search for subspaces Wl where the quotient

cost

benefit

(l)

)l(

c

bis as big as possible

}||max{||:)(2||:)( 2|| 1l

1ll ll wbWc → Image

)n(||''||4)d(||||

)|log|(2||

and:

1d2n2/

n1d

0i2/

)1(n

1dn2

1n

1n

0i

i)1(n

1dn||

)1(n

)opt(n

i

1dn

1d

i1d

1

hOfuf

hhOV

WVV

ll

benefit: Nutzen <> польза

14/21

There exists also an optimal choice of grids for the energy-norm. We get the function space

ll

WV)4d44(log1/5)1dn()4(log1/5||

)E(n

n2

d

1jjl

21

:

and the estimates

)(||''||2)(||||

d/22||

n2/n

E)E(

n

dn)E(n

hOfduf

eV

15/213.5 -complexity

)()()()(

)()()()(

getweIn

1/dE

2/d2/

dE

d/22/

)(n

NONNON

ONON

V

)|log|()(

)|log|()(

)|log|()(

)|log|()(

getweIn

1)(d2

1E

1)(d32

22/

1)-(d2

1E

1)-(d3/22

1/22/

)1(n

NNON

NNON

ON

ON

V

)()(

)()(

getweIn

1E

1E

)E(n

NON

ON

V

16/214. Higher-order polynomials

4.1 Construction

Now we want to generalize the piecewise linearbasis functions to polynomials of arbitrary degree . We use the tensor product:

dd1 ),,(: N pp p

)(:)( j

d

1j

)(pi,l

)( j

jjxx

pil, → Image

arbitrary: beliebig <> любой

with R],[:)(jjjjjj

j

jj li,lli,lj)(pi,l hxhxx

To determine this polynomial we need pj+1 points.For that we have to look at the hierarchical ancestors.

→ Example

ancestor: Vorfahr <> предок

17/21

is now defined as the Lagrangian interpolationpolynomial with the following properties:

)(pi,lj

jj

0)(,1)(jjj

j

jjjj

j

jj li,l)(pi,li,l

)(pi,l hxx

and is zero for the pj-2 next ancestors.

→ Example

)(pi,lj

jj

otherwise0

],[for)(:)( jjjjjj

j

jjj

jj

li,lli,ljj)(pi,l

j)(pi,l

hxhxxxx

This scheme is not correct for the linear basis functions, as they are only piecewise linear and need three definition points.

18/214.2 Estimates

For the basis polynomials we get:

1/2d

1j

l2/2||d/2

E

/q||d/qdq

)(

d)(

j1

1

222

5257.3||||

122117.1||||

117.1||||

l(p)il,

lpil,

pil,

q

d

1j j

)/21(

)!1(

2:)(

jj

p

pp

p

We define a constant-function

19/21The estimates for the hierarchical surplus are:

2)(supp

)(/2||)|(|d/2)(

)()|(|d)(

|||||22)(1/6||

||||2)(1/2||

11

1

il,

1pl1plpil,

1p1plpil,

p

p

fD

fD

with }odd|{span: j)()( iW pil,

pl as before

we get for

2/)(

d

1j

l2)|(|E

)(

2)()|(|d/2d

2)(

)()|(|d)(

||||22)()d(||||

||||2)(1/3117.1||||

||||2)(5585.0||||

j1

1

1

fDw

fDw

fDw

1p1plpl

1p1plpl

1p1plpl

p

p

p

)()( pl

pl Ww

20/21

)1(n

)1,1(n

)(

1dn||

)1,p(n :and1pfor:

1

VVWV

pl

l

For a function out of the order of approximation is given by

)1,p(nV

)(||||

)n(||||1p

nE)1,p(

n

1d1pn2/

)1,p(n

hOuf

hOuf

But as the costs do not change:1||)( 2|||| 1l

lpl

WW

we can define the same as before)opt(

nV

21/214.3 ε- complexities

)|log|()(

)|log|()(

)|log|O()(

)|log|O()(

)1(dp2

p)p(E

)1(d2)p(2

)1p()p(2/

)1d(2

1/p-)p(E

)1d(1p

2p

21p

1-

)p(2/

NNON

NNON

N

N

For we get)1,p(

nV

For we get)E,p(

nV

)()(and)()( p(p)E

1/p)p(E

NONON

The End

Image1

Bild1

Image2Bild2

Bild3

Bild3

Image3

n= 3 n= 1 and n= 2

φ1,1φ3,5

Bild4

Image4

This is an example for a function in V3

Bild5

Image5

nodal point basis

natural hierarchical basis

φ1,1

φ2,1 φ2,3

node: Knoten <> узел

W1

W2

W3

Bild6

Image6

))2/)1(()2/)1(((1/2)2/(,kkk

ik ififif

Example1

)1()( xxxf

l=(3,2)

hl = (1/8,1/4)

Example2

Image7W(1,1) W(2,1)

W(1,2))32()(

)(

)2()(

)16()(

1

1

1

|l|

|l|

|l|

Olc

lb

Olc

Olb

Image8

Example3

0 1

Example4