View
214
Download
1
Category
Preview:
Citation preview
Numerical Methods in Geophysics Orthogonal Functions
Function Approximation
The Problem
we are trying to approximate a function f(x) by another function gn(x) which consists of a sum over N orthogonal functions (x) weighted by some coefficients an.
)()()(0
xaxgxfN
iiiN
Numerical Methods in Geophysics Orthogonal Functions
... and we are looking for optimal functions in a least squares (l2) sense ...
... a good choice for the basis functions (x) are orthogonal functions. What are orthogonal functions? Two functions f and g are said to be
orthogonal in the interval [a,b] if
b
a
dxxgxf 0)()(
How is this related to the more conceivable concept of orthogonal vectors? Let us look at the original definition of integrals:
The Problem
!Min)()()()(
2/12
2
b
a
NN dxxgxfxgxf
Numerical Methods in Geophysics Orthogonal Functions
Orthogonal Functions - Definition
... where x0=a and xN=b, and xi-xi-1=x ...If we interpret f(xi) and g(xi) as the ith components of an N component
vector, then this sum corresponds directly to a scalar product of vectors. The vanishing of the scalar product is the condition for orthogonality of
vectors (or functions).
N
iii
b
aN
xxgxfdxxgxf1
)()(lim)()(
figi
0 ii
iii gfgf
Numerical Methods in Geophysics Orthogonal Functions
Periodic functions
-15 -10 -5 0 5 10 15 200
10
20
30
40
Let us assume we have a piecewise continuous function of the form
)()2( xfxf
2)()2( xxfxf
... we want to approximate this function with a linear combination of 2 periodic functions:
)sin(),cos(),...,2cos(),2cos(),sin(),cos(,1 nxnxxxxx
N
kkkN kxbkxaaxgxf
10 )sin()cos(
2
1)()(
Numerical Methods in Geophysics Orthogonal Functions
Orthogonality of Periodic functions
... are these functions orthogonal ?
0,00)sin()cos(
0
0,,0
)sin()sin(
0
02
0
)cos()cos(
kjdxkxjx
kj
kjkj
dxkxjx
kj
kj
kj
dxkxjx
... YES, and these relations are valid for any interval of length 2.Now we know that this is an orthogonal basis, but how can we obtain the
coefficients for the basis functions?
from minimising f(x)-g(x)
Numerical Methods in Geophysics Orthogonal Functions
Fourier coefficients
optimal functions g(x) are given if
0)()(!Min)()(22
xfxgorxfxg nan k
leading to
... with the definition of g(x) we get ...
dxxfkxbkxaaa
xfxga
N
kkk
kn
k
2
10
2)()sin()cos(
2
1)()(
2
Nkdxkxxfb
Nkdxkxxfa
kxbkxaaxg
k
k
N
kkkN
,...,2,1,)sin()(1
,...,1,0,)cos()(1
with)sin()cos(2
1)(
10
Numerical Methods in Geophysics Orthogonal Functions
Fourier approximation of |x|
... Example ...
.. and for n<4 g(x) looks like
leads to the Fourier Serie
...
5
)5cos(
3
)3cos(
1
)cos(4
2
1)(
222
xxxxg
xxxf ,)(
-20 -15 -10 -5 0 5 10 15 200
1
2
3
4
Numerical Methods in Geophysics Orthogonal Functions
Fourier approximation of x2
... another Example ...
20,)( 2 xxxf
.. and for N<11, g(x) looks like
leads to the Fourier Serie
N
kN kx
kkx
kxg
12
2
)sin(4
)cos(4
3
4)(
-10 -5 0 5 10 15-10
0
10
20
30
40
Numerical Methods in Geophysics Orthogonal Functions
Fourier - discrete functions
iN
xi2
.. the so-defined Fourier polynomial is the unique interpolating function to the function f(xj) with N=2m
it turns out that in this particular case the coefficients are given by
,...3,2,1,)sin()(2
,...2,1,0,)cos()(2
1
*
1
*
kkxxfN
b
kkxxfN
a
N
jjj
N
jjj
k
k
)cos(2
1)sin()cos(
2
1)( *
1
1
****0 kxakxbkxaaxg m
m
km kk
... what happens if we know our function f(x) only at the points
Numerical Methods in Geophysics Orthogonal Functions
)()(*iim xfxg
Fourier - collocation points
... with the important property that ...
... in our previous examples ...
-10 -5 0 5 100
0.5
1
1.5
2
2.5
3
3.5
f(x)=|x| => f(x) - blue ; g(x) - red; xi - ‘+’
Numerical Methods in Geophysics Orthogonal Functions
Fourier series - convergence
f(x)=x2 => f(x) - blue ; g(x) - red; xi - ‘+’
-10 -5 0 5 10 15-10
0
10
20
30
40
50 N = 4
-10 -5 0 5 10 15-10
0
10
20
30
40
50 N = 8
Numerical Methods in Geophysics Orthogonal Functions
Fourier series - convergence
f(x)=x2 => f(x) - blue ; g(x) - red; xi - ‘+’
-10 -5 0 5 10 15-10
0
10
20
30
40
50 N = 16
-10 -5 0 5 10 15-10
0
10
20
30
40
50 N = 32
Numerical Methods in Geophysics Orthogonal Functions
Orthogonal functions - Gibb’s phenomenon
f(x)=x2 => f(x) - blue ; g(x) - red; xi - ‘+’
0 0.5 1 1.5-6
-4
-2
0
2
4
6
N = 32
0 0.5 1 1.5-6
-4
-2
0
2
4
6
N = 16
0 0.5 1 1.5-6
-4
-2
0
2
4
6
N = 64
0 0.5 1 1.5-6
-4
-2
0
2
4
6
N = 128
0 0.5 1 1.5-6
-4
-2
0
2
4
6
N = 256
The overshoot for equi-spaced Fourier
interpolations is 14% of the step height.
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev polynomials
We have seen that Fourier series are excellent for interpolating (and differentiating) periodic functions defined on a regularly spaced grid. In many circumstances physical phenomena which are not periodic (in space) and occur in a limited area. This quest leads to the use of Chebyshev polynomials.
We depart by observing that cos(n) can be expressed by a polynomial in cos():
1cos8cos8)4cos(
cos3cos4)3cos(
1cos2)2cos(
24
3
2
... which leads us to the definition:
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev polynomials - definition
NnxxxTTn nn ],1,1[),cos(),())(cos()cos(
... for the Chebyshev polynomials Tn(x). Note that because of x=cos() they are defined in the interval [-1,1] (which - however - can be extended to ). The first polynomials are
0
244
33
22
1
0
and]1,1[for1)(
where188)(
34)(
12)(
)(
1)(
NnxxT
xxxT
xxxT
xxT
xxT
xT
n
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev polynomials - Graphical
The first ten polynomials look like [0, -1]
The n-th polynomial has extrema with values 1 or -1 at
0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
1
x
T_
n(x)
nkn
kx extk ,...,3,2,1,0),cos()(
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev collocation points
These extrema are not equidistant (like the Fourier extrema)
nkn
kx extk ,...,3,2,1,0),cos()(
k
x(k)
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev polynomials - interpolation
... we are now faced with the same problem as with the Fourier series. We want to approximate a function f(x), this time not a
periodical function but a function which is defined between [-1,1]. We are looking for gn(x)
)()(2
1)()(
100 xTcxTcxgxf
n
kkkn
... and we are faced with the problem, how we can determine the coefficients ck. Again we obtain this by finding the extremum
(minimum)
01
)()(2
1
1
2
x
dxxfxg
c nk
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev polynomials - interpolation
... to obtain ...
nkx
dxxTxfc kk ,...,2,1,0,
1)()(
2 1
12
... surprisingly these coefficients can be calculated with FFT techniques, noting that
nkdkfck ,...,2,1,0,cos)(cos2
0
... and the fact that f(cos) is a 2-periodic function ...
nkdkfck ,...,2,1,0,cos)(cos1
... which means that the coefficients ck are the Fourier coefficients ak of the periodic function F()=f(cos )!
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev - discrete functions
iN
xi
cos
... leading to the polynomial ...
in this particular case the coefficients are given by
2/,...2,1,0,)cos()(cos2
1
* NkkfN
cN
jjjk
m
kkkm xTcTcxg
1
**0
* )(2
1)( 0
... what happens if we know our function f(x) only at the points
... with the property
N0,1,2,...,jj/N)cos(xat)()( j* xfxgm
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev - collocation points - |x|
f(x)=|x| => f(x) - blue ; gn(x) - red; xi - ‘+’
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 N = 8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 N = 16
8 points
16 points
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev - collocation points - |x|
f(x)=|x| => f(x) - blue ; gn(x) - red; xi - ‘+’
32 points
128 points
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 N = 32
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 N = 128
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev - collocation points - x2
f(x)=x2 => f(x) - blue ; gn(x) - red; xi - ‘+’
8 points
64 points
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2 N = 8
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2 N = 64
The interpolating function gn(x) was shifted by a small amount to be visible at all!
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev vs. Fourier - numerical
f(x)=x2 => f(x) - blue ; gN(x) - red; xi - ‘+’
This graph speaks for itself ! Gibb’s phenomenon with Chebyshev?
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1 N = 16
0 2 4 6-5
0
5
10
15
20
25
30
35
N = 16
Chebyshev Fourier
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev vs. Fourier - Gibb’s
f(x)=sign(x-) => f(x) - blue ; gN(x) - red; xi - ‘+’
Gibb’s phenomenon with Chebyshev? YES!
Chebyshev Fourier
-1 -0.5 0 0.5 1-1.5
-1
-0.5
0
0.5
1
1.5 N = 16
0 2 4 6-1.5
-1
-0.5
0
0.5
1
1.5 N = 16
Numerical Methods in Geophysics Orthogonal Functions
Chebyshev vs. Fourier - Gibb’s
f(x)=sign(x-) => f(x) - blue ; gN(x) - red; xi - ‘+’
Chebyshev Fourier
-1 -0.5 0 0.5 1-1.5
-1
-0.5
0
0.5
1
1.5 N = 64
0 2 4 6 8-1.5
-1
-0.5
0
0.5
1
1.5 N = 64
Numerical Methods in Geophysics Orthogonal Functions
Fourier vs. Chebyshev
Fourier Chebyshev
iN
xi2
iN
xi
cos
periodic functions limited area [-1,1]
)sin(),cos( nxnx
cos
),cos()(
x
nxTn
)cos(2
1
)sin()cos(
2
1)(
*
1
1
**
**0
kxa
kxbkxa
axg
m
m
k
m
kk
m
kkkm xTcTcxg
1
**0
* )(2
1)( 0
collocation points
domain
basis functions
interpolating function
Numerical Methods in Geophysics Orthogonal Functions
Fourier vs. Chebyshev (cont’d)
Fourier Chebyshev
coefficients
some properties
N
jjj
N
jjj
kxxfN
b
kxxfN
a
k
k
1
*
1
*
)sin()(2
)cos()(2
N
jjj kf
Nck
1
* )cos()(cos2
• Gibb’s phenomenon for discontinuous functions
• Efficient calculation via FFT
• infinite domain through periodicity
• limited area calculations
• grid densification at boundaries
• coefficients via FFT
• excellent convergence at boundaries
• Gibb’s phenomenon
Recommended