wavelets 3

7/29/2019 Wavelets 3

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Applications of Wavelets inNumerical Mathematics

Kees Verhoeven

1. Brief summary

2. Data compression

3. Denoising

4. Preconditioning

5. Adaptive grids

6. Integral equations

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1. Brief Summary

(t): scaling function.For the following 2-scale relation holds

(t) =

k=

pk(2t k), t IR.

(t): mother wavelet.For the following 2-scale relation holds

(t) =

k=

qk(2t k), t IR.

The decomposition for reads

(2tk) =

m=

h2mk(tm)+g2mk(tm), t IR.

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2. Data Compression

We consider a function f

f : [0, 1] IR.

We want to approximate this function by a function v de-fined by

v =

k

ckk,

where {k|k = 1, . . . , N } is a basis for the linear functionspace V.

The quality of the approximation can be expressed in termsof a norm

f v.

An alternative is to expand f periodically. We thereforelook at the Fourier series of f

f(x) =

m=

cme2imx

and approximate this by

v(x) =M

m=M

cme2imx.

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So

V =

e2mix |m = M , . . . , M

,

with dimension N = 2M + 1. The basis functions form anorthonormal system. Therefore

ck = (k, f) =

10

f(x)e2imxdx.

Error estimates

Given f, g : [0, 1] IR with the Fourier expansions

f =

m=

cme2imx, g =

m=

dme2imx.

Then10

f(x)g(x)dx =

m=

cmdm.

So 10

|f(x)|2dx =

m=

|cm|2.

The error then reads

2M := f v2 =

|m|>M

|cm|2.

In many cases properties of f lead to an error estimate ofthe type

M CM, C, > 0.

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Example

Given

f(x) = x(x 1

2)(x 1), x [0, 1].

We can derive

cm =3

4i3m3.

The error M can therefore be estimated via

2M =9

86

m=M+1

m6 9

86

M

y6dy =9

406M5.

M L2-error

9406 M

5

10 0.426 104 0.484 104

20 0.803 105 0.855 105

40 0.147 105 0.151 105

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How about localized functions?

It seems sensible to approximate a localized function withbasis functions which also have compact support.

Stepfunctions:

k =

h1

2 , x [(k 1)h,kh),

0, else.

Note that:

h = N1 and k = 1

more efficient for function evaluations

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Comparison

We consider the following function.

The error of the approximation using the Fourier series atM = 64, is approximately 0.001. The approximation using

first order splines is 0.01 withN

= 2M

+ 1 = 129.We would like to use localized basis functions only wherethe function to be approximated behaves like a localizedfunction.

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Therefore, we would like basis functions which all have the

same shape but different scales. Then, if we have

v =N

k=1

ckk

and |cj| , we also have

f

k

ckk

f

k=j

ckk

+ cjj.

This leads to data compression.

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Example

We denote by VJ the space of piecewise constant basis func-tions on [0,1] with width h = 2J and dimension N = 2J.

The space V0 has one basis function: the constant function1 = 1.For V1 the usual basis functions are depicted here:

The coefficients ck behave like

ck = h1

2 a+h/2

ah/2

f(x)dx h1

2f(a).

Can we chose a more appropriate basis?

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First approach: construct basis for VJ by expanding the

basis of VJ1.

Figure 1: An alternative basis for V1.

Figure 2: Together with the functions above an alternative basis for V2.

But: the new basis is no longer orthogonal.For the test function f(x) = x, the drop in the coefficientsseems to be like h3/2 (23/2 3).

generation |ck|

0 0.3541 0.1252 0.0443 0.016

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A better alternative basis for VJ

Construct an orthogonal basis {i} for VJ. This leads to

Figure 3: A better alternative basis for V1.

Figure 4: Together with the two functions above a better alternative basisfor V2.

Note that these are the Haar wavelets!

(x) = 0,0(x) = 2(x), 0,1(x) = 3(x), 1,1(x) = 4(x).

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Now:

ck = h1

2

aah/2 f(x)dx

a+h/2a f(x)dx

= h

1

2

aah/2(f(a) + (x a)f

(a) + . . .)dx

a+h/2

a (f(a) + (x a)f(a) + . . .)dx

1

4f(a)h3/2.

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Comparison of this basis and homogeneous one:

Figure 5: Approximation of f (top left) with 32, 10, 9 basis functions,respectively.

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Figure 6: Approximation of f with 22, 10 homogeneous basis functions,respectively.

If wavelets used as basis functions have several momentsequal to zero, then reduction will be better.

Because, if

v =lZZ

cl,Il,I +J1k=I

lZZ

dl,kl,k,

then

|dk,J| Chd+3/2,

where d is the number of moments equal to zero.

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Example

M L2-error for spline d = 1 L2-error for spline d = 3

256 1.149 103 2.829 104

128 1.149 103 2.829 104

64 1.356 103 2.829 104

32 2.623 103 1.143 103

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3. Denoising

Suppose we have a signal with some noise. We can make awavelet decomposition up to a certain depth.If the coefficients of the wavelets remain relatively large(say |dj, k| > ), then we have some localized noise.So cancel these contributions by forcing dj,k = 0 at L levelsdeep.

Then, we can reconstruct the filtered signal.

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The role of L can be seen as follows:

Figure 7: Filtering of f with L = 1, 3, 5 levels ( = 0.1).

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The role of is shown here:

Figure 8: Filtering of f (top) with L = 1, 2 levels ( = 0.01).

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But: if the original signal is not periodic, we encounter

problems at the boundaries.

Figure 9: Filtering of non-periodic f with 3 levels ( = 0.01).

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4. Preconditioning

Consider

D(u) = f

on a domain , with the differential operator D elliptic. IfD is linear, this would lead to a linear system

Dc = r,

with

Dj,k = (j, Dk), rj = (j, f).

The matrix D is called the stiffness matrix. If we use aniterative method to solve this system, the speed of con-vergence strongly depends on the condition number of thematrix D, (D) = DD1, with

D = supc=1

cTDc, D11 = infc=1

cTDc.

For symmetric matrices, we can express the condition num-ber in terms of the eigenvalues:

(D) =max

min.

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Example

We consider

D(u) = d2u

dx2+ u = f, x (0, 1),

with periodical boundary conditions.We assume that the numerical solution v can be writtenas a linear combination of certain scaling functions whichspan VJ.

Galerkins method and integration by parts gives us

Dc = r,

with

Di,j =

di,J

dx,

dj,J

dx

+ (i,J, j,J),

and, as before

rj = (j, f).

For linear B-splines:di,J

dx,

dj,J

dx

=

2J/2

d

dx(2Jxi)2J/2

d

dx(2Jxj)dx.

The derivatives are piecewise constant, and therefore wederive

Di,j =

2N2 + 23

, i = j,

N2 + 16 , i = (j 1) mod N,0 else.

This is a circulant matrix, that is Di,j = d(i j).

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We define the symbol of a circulant matrix as

D(z) =

j

Di,jzij.

For D now follows

max = maxzN=1

|D(z)|, min = minzN=1

|D(z)|.

For the differential equation we have

D(z) =

j

di,Jdx ,

dj,J

dx

+ (i,J, j,J)

zij.

We write

D(z) = D1(z) + D2(z),

with

D1(z) = j

di,J

dx,

dj,J

dx

zij, D2(z) =

j

(i,J, j,J)zij.

The second term can be recognized as D2(z) = R(z). Inthe same manner we can derive D1(z) = 2

2JR(z).Using this and the 2-scale relations, we can derive

D(z) = N2(2 z z1) +1

6(4 + z+ z1).

Calculating the biggest and smallest eigenvalue, we see that

1 = 4N2 + 13

.

If N , 1 .

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We now use wavelets

v =k,j

dk,jk,j.

The stiffness matrix D then looks like

Dl,m,j,k =

dm,l

dx,

dk,j

dx

+ (m,l, k,j).

After some algebra using Riesz functions and 2-scale rela-tions, we can show that

2(D) C, for all J.

Comparison

2J 1 216 1024.3 45.432 4096.3 49.7

64 16384.3 52.9128 65536.3 55.4256 262144.3 57.3

Again: this strongly depends on the periodicity of the bound-ary conditions.

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5. Adaptive grids

We consider a hyperbolic PDE

u(x, t)

t= F(u,

u

x, . . .),

together with initial condition and periodic boundary con-ditions.

The approximation of the initial condition u(x, 0) is done

by

v(x, 0) =N1i=0

ci,I(0)i,I(x) +J1j=I

iIj

di,j(0)i,j(x).

Here N = 2J is the amount of intervals on the coarsestgrid I, the set Ij is a subset of all possible wavelets on thegrids j = I , . . . , J 1. These sets Ij are found by making awavelet decomposition and leave out all wavelet coefficients

below a certain threshold .

But: this would mean that we have to approximate thefunction on the finest grid first!

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We ignore all contributions below a wavelet for which

|dk,l| . (filled circles mean |dk,l| > , open circles mean:|dk,l| )

Adaptivity means that wavelets left out in previous timesteps can occur again.

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Example: Burgers equation

Figure 10: Approximation on t = 0, 112

, 212

, 312

, 412

, 512

, for Burgers equation.

The number of basis functions with coefficient above threshold is 32, 56,122, 114, 114 and 114, respectively.

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Example: wave equation

Figure 11: Approximation on t = 0, 0.3 and 0.5, respectively, for the wave

equation. The number of basis functions with coefficient above threshold isapproximately 60.

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6. Integral Equations

We consider

u(x) =

K(x; t)u(t)dt + f(x).

We take

v(x) =

j

cjj(x).

Galerkins method would leave us with

Ac = r,

with

Aj,k = (j, k)

j(x)K(x; t)k(t)dxdt, rj = (j, f).

Often this A is well conditioned, but full.

Using wavelets reduces the number of nonzero elements.

We represent

v(x) =lZZ

cl,Il,I +J1k=I

lZZ

dl,kl,k.

Look at the second term of A

Kl,m,j,k =

m,l(x)K(x; t)k,j(t)dxdt.

We make the following assumption on K(x; t) d

xdK(x; t)

+

d

tdK(x; t)

Cd 1|x t|d+1 ,for a certain d > 0.

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Making use of the Taylor series of K(x; t) and taking a

wavelet with d zero moments, we can derive

|Kl,m,j,k| C1

|x0 t0|d+1.

Using this we can bring down the number of nonzero ele-ments (or better: the number of elements with value abovea certain threshold ).

Example

N = 106 = 109 = 1012

24 74% 92% 92%48 19% 85% 96%96 5.1% 54% 78%

192 1.1% 16% 50%384 0.34% 3.5% 25%

Table 1: The number of elements above threshold , with Daubechieswavelets with K = 2.

N = 106 = 109 = 1012

24 66% 92% 92%48 12% 93% 96%96 3.1% 47% 90%

192 0.85% 12% 56%384 0.32% 2.4% 21%

Table 2: The number of elements above threshold , with Daubechieswavelets with K = 5.

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