Orthogonal Functions Numerical Methods in Geophysics Function Approximation The Problem we are trying to approximate a function f(x) by another function

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


... 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


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


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)()(


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)


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


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


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


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


)()(*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 - ‘+’


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


Fourier series - convergence


-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


Orthogonal functions - Gibb’s phenomenon


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.


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:


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


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()(


Chebyshev collocation points

These extrema are not equidistant (like the Fourier extrema)

nkn

kx extk ,...,3,2,1,0),cos()(

k

x(k)


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


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 )!


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


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


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


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!


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


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


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


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


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

Documents

Orthogonal Functions Numerical Methods in Geophysics Function Approximation The Problem we are trying to approximate a function f(x) by another function