1/21
Hierarchical
Polynomial-Bases
&
Sparse Grids
grid: Gitter <> сéтка
sparse: spärlich, dünn <> рéдкий
1. Introduction
1.1 A few properties of function spaces
V be a function space and fi , f , g  V
- V is a infinite dimensional vector space
Let
- ( f  g )( x)  f ( x)  g ( x) and ( f )( x)   f ( x)
- span{ fi } is a subspace of V
A few examples:
n
- C (Ω , R) is the space of n times
d
differentiable functions from   R to R
2
n
- span{1,x,x ,…,x }
2/21
3/21
1.2 The tensor product
Let f and g be two functions, then the tensor
product f  g is defined by
( f  g )( x, y) : f ( x)  g ( y)
So if we have the function φ for example:
 1 | x |
 ( x) : 
 0
if x   1 , 1
otherwise
the tensor product  :    is
( x, y) :  ( x)   ( y)
otherwise: sonst <> в другой случае
→ Image
→ Image
4/21
1.3 Norms in function spaces
Sometimes we want to measure the “length” of
n
a function. In C (Ω ,R) we will look at three
different norms:
|| f || : sup{| f ( x)| | x  }
|| f ||q : q


| f q ( x) |dx q  1
1/ 2


2
|| f || E :    f ( x ) x j  
 j1

d
(energy norm)
d 1
 || f ' ||2
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:
n, i ( x) :  (2n x  i)
i  1, , 2n  1
→ Image
These are basis functions of
n
Vn : {  i  n,i |  i  R }
i 1
 span { n,i | i  1,,2 n  1}
→ Image
6/21
2.2 A new basis
We define:
and get
Wk : span{ k,i | odd i}
Vn   Wk
kn
→ 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:
||  k,i ||  1
1/q
 2 
  2  k/q
||  k,i ||q  
 q 1
||  k,i || E  2  2 k/2
 || f ||  || f ' || 
2
E

  f ' ( x) dx
2

equation: Gleichung <> уравнение
1/2


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.
2 n 1
n
f  un    i  n,i  
i 1

k 1 i 1.. 2 k 1
i odd
n
k,i
 k,i   wk
k 1
  i  f (i / 2 n ) (function values)
and
 k,i  f (i / 2 )  1/2  f ((i  1) / 2 )  1/2  f ((i  1) / 2 )
k
k
(hierarchical surplus)
surplus: Überschuss <> избыток
k
→ Example
9/21
With the help of the integral representation of
the coefficients
 k, i    2
 ( k 1)

 k, i ( x)  f ' ' ( x) dx
we get the following estimates:
|  k,i |  1/2  2
2k
|  k,i |  1/6  2
 || f ' ' ||
 ( 3/2)k
 || f ' ' |supp( k ,i ) ||2
and from this
||wk || / 2 / E    4  || f ' ' || / 2
k
estimate: Abschätzung <> оценка
3. Sparse grids
3.1 Multi-indices
For multi-indices
,   N 0 we define:
d
    (1  1 ,  ,  d   d )
    (    1 ,  ,    d )

1
d
2 : (2 , , 2 )
def
    1  j  d :  j   j
d
|  |1:   j
j1
and |  | : max  j
1 j d
10/21
11/21
3.2 Grids
A d-dimensional grid can be written as a multiindex m  N d with mesh size
h m : 2-m  (2- m 1 ,  , 2- m d )
The grid points are
→ Example
x m,i :  (x m1 ,i1 , , x md ,id ) :  i  h m 0  i  2m
Now we can assign every xm,i a function
d
m,i (x) :   m , i ( x j ) with
j1
j
j
 m , i ( x j ) :  (2 x j  i j )
mj
j
j
mesh: Masche <> петля
i j  1, , 2  1
mj
12/21
3.3 Curse of dimensionality
Wl : span {l ,i | 1  i  2  1, i j odd for all 1  j  d}
l
()
n
and V
:

Wl
|l|  n

The dimension of Vn(  ) is
()
n
|V
nd
d
n
|  (2  1)  O(2 )  O(h )
n
d
But as we seen before
||wm || / 2   (d)  4
|l|1
 || f ' ' || / 2  O(h )
2
n
and we get
|| f  un(  ) || / 2   (d )  4 n  || f ' ' || / 2  O(hn2 )
curse: Fluch <> проклятие
13/21
3.4 The “solution”
We search for subspaces Wl where the quotient
b(l) benefit

c(l)
cost
is as big as possible
b(l) :   max{|| wl ||2 } → Image
c(l) :| Wl | 2|l 1|1
 V
( opt)
n
 V :
(1)
n

Wl
|l|  n  d 1
and
1
n 1
d 1 i
1
d 1
|V |   2 



O
(
h

|
log
h
|
)

n
2 n
 d 1 
i 0
(1)
n
i
d 1
 n  d  1  n
2
d 1
|| f  u || / 2   (d)   

4

||
f
'
'
||

O
(
h

n
)

/2
n
i 
i 0 
(1)
n
benefit: Nutzen <> польза
14/21
There exists also an optimal choice of grids for
the energy-norm. We get the function space
(E)
n
V
:
|l|1 1/5log2 (
j1
d
lj

4 ) ( n  d 1) 1/5log2 ( 4 n  4d  4 )
Wl
and the estimates
| Vn( E ) |
|| f  u
(E)
n
 2 n  d/2  e d
||E   (d )  2  || f ' ' || / 2  O(h n )
n
15/21
3.5 e-complexity
In Vn(  ) we get
N  / 2 (e )  O(e d/2 )
N E (e )  O(e d )
e  / 2 ( N )  O( N  2/d )
e E ( N )  O( N 1/d )
(1)
n
In V
In V
we get
N E (e )  O(e )
1
we get
N  / 2 (e )  O(e
(E)
n
1/2
 | log 2 e |
3/2(d -1)
)
N E (e )  O(e 1  | log 2 e |(d -1) )
e  / 2 ( N )  O( N  2  | log 2 N |3(d 1) )
e E ( N )  O( N 1  | log 2 N |(d 1) )
e E ( N )  O( N 1 )
16/21
4. Higher-order polynomials
4.1 Construction
Now we want to generalize the piecewise linear
basis functions to polynomials of arbitrary degree
d
p : ( p1 , , pd )  N . We use the tensor product:
d

(p j )
(p )
l,i ( x ) :  l j ,i j ( x j )
with

(p j )
l j ,i j
→ Image
j1
( x j ) : [ xl j ,i j  hl j , xl j ,i j  hl j ]  R
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

(p j )
l j ,i j
is now defined as the Lagrangian interpolation
polynomial with the following properties:

(p j )
l j ,i j
and 
( xl j ,i j )  1, 
(p j )
l j ,i j
(p j )
l j ,i j
( xl j ,i j  hl j )  0
is zero for the pj-2 next ancestors.
(p j )

 l j ,i j ( x j ) for x j  [ xl j ,i j  hl j , xl j ,i j  hl j ]
(p j )
 l j ,i j ( x j ) : 
otherwise
 0
→ Example
This scheme is not correct for the linear basis
functions, as they are only piecewise linear
and need three definition points.
18/21
4.2 Estimates
For the basis polynomials we get:
||  (l,pi ) ||  1.117 d
||  (l,pi ) ||q  1.117 d  2d/q  2 |l|1 /q
q 1
|| 
(p)
l,i
1/2
5
2l j 
|l|1 /2 
||E  3.257     2
   2 
2
 j1

d/2
We define a constant-function
d
p j ( p j1)/2
2
 (p) : 
j1 ( p j  1)!
d
19/21
The estimates for the hierarchical surplus are:
|
(p )
l,i
|  1/2    (p)  2|l(p 1)|1  || D (p 1) f ||
d
|  l,(ip ) |  1/6    (p)  2|l(p 1)|1  2|l|1 /2  || D (p 1) f |supp(l,i ) ||2
d/2
with Wl (p ) : span{l,(pi ) | i j odd} as before
we get for wl(p ) Wl (p )
|| wl(p ) ||  0.5585d   (p)  2 |l(p 1)|1  || D (p 1) f ||
|| wl(p ) ||2  1.117 d  1/3   (p)  2 |l(p 1)|1  || D (p 1) f ||2
d/2
d

2l j 
|l( p 1)|1
(p )
|| wl ||E   (d)   (p)  2
   2  || D (p 1) f || / 2
 j1

20/21
But as the costs do not change:
| Wl (p )|  | Wl |  2|l 1|1
( opt )
the
n
we can define V
Vn( p,1) :

Wl (p )
|l|  n  d 1
same as before
for p  1 and
1
( p ,1)
n
For a function out of V
the order of
approximation is given by
|| f  u
( p ,1)
n
|| f  u
( p ,1)
n
p 1
n
|| / 2  O(h
p 1
n
||E  O(h
)
n
d 1
)
Vn(1,1) : Vn(1)
21/21
4.3 ε- complexities
( p ,1)
we
n
For V
N
(p)
/2
get
(e )  O(e
-
1
p 1
 | log 2 e |
p2
( d 1)
p 1
)
N E( p ) (e )  O(e -1/p  | log 2 e |( d 1) )
e
(p)
/2
( N )  O( N
 ( p 1)
 | log 2 N |
e ( N )  O( N  | log 2 N |
(p)
E
p
( p  2)(d 1)
p(d 1)
)
)
( p,E )
V
For n
we get
N (e )  O(e
(p)
E
1/p
) and e ( N )  O( N )
(p)
E
p
The End
Image1
Bild1
Image2
Bild2
Image3
n= 3
n= 1 and n= 2
φ3,5
φ1,1
Bild3
Bild3
Image4
Bild4
This is an example for a function in V3
Image5
natural hierarchical basis
W1
Bild5
φ1,1
W2
W3
nodal point basis
node: Knoten <> узел
φ2,1
φ2,3
Image6
Bild6
Example1
 k ,i  f (i / 2k )  1/2  ( f ((i  1) / 2k )  f ((i  1) / 2k ))
f ( x)  x  (1  x)
Example2
l=(3,2)
hl = (1/8,1/4)
Image7
b(l )  O (16
|l|1
)
W(1,1)
c(l )  O (2|l|1 )
b(l )
 O (32 |l|1 )
c(l )
W(1,2)
W(2,1)
Image8
Example3
0
1
Example4