january 2006 - the mathworks a multilevel method for ill-posed problems malena i. español tufts...

January 2006 - The Mathworks

A Multilevel Method for Ill-Posed Problems

Malena I. EspañolTufts University


Outline

Problem Regularization Multilevel method Implementation Future work


Discrete Ill-Posed problem

holds condition Picard Discrete

noise (white)unknown is

gap without aluessingular v Decaying

:Properties

e

matrix dconditione-ill large, a is where

,

model theand ,given , Find

nm

true

true

RA

ebbAx

bAx

0 10 20 30 4010

-15

10-10

10-5

100

105

i

i|uiTbtrue|


Need for regularization

n

ii

eui

bu

n

i

Tiii

T

i

Ti

i

trueTix

uVUA

1

1

bygiven issolution exact The

.SVD thebe Let

0 10 20 30 4010

-15

10-10

10-5

100

105

i| uiTbtrue|

| uiTe|


Regularization methods

n

i i

T

i

iTik

k

i i

T

TSVD

bux

bux

i

i

1i22

2

1i

:tionRegulariza Tikhonov

: (TSVD) SVD Truncated


LSQR: Regularization method

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

),( :solution LSQR bAAAKx TTk

iterations

rela

tive e

rror


Multilevel: Introduction

hhh bxA

hhh bxA 222

hhh bxA 444


Downsampling (“going down”)

)2(4

112212

2 hj

hj

hj

hj vvvv

h

h

hD

v

v

v

v

v

v

v

vv

v

vP

23

2

1

7

6

5

4

3

2

1

2

1210 000

0012 100

0000 121

4

1


Interpolation (“going up”)

h

h

hI

v

v

v

v

vv

v

v

v

v

vP

7

6

5

4

3

2

1

23

2

1

2

100

200

1 10

020

01 1

002

00 1

2

1

)(2

1 21

212

22

hj

hj

hj

hj

hj

vvv

vv


Basic multilevel algorithm

hII xPPx 4hhh bxA

hDDhIIhDD bPPxPPAPP 4

Solve small system

downsampling

A and b

interpolating

x


Implementation

xPPx II~bAx

bxPAPPP IIDD

~~ Solve small system

downsampling

b

interpolating

function x=nm(A,b,iter)n=size(A,1);while length(b)>31 b=fwei(b);endx=lsqr(@pap,b,1e-8,iter,[],[],[],A);while length(x)<n x=interpol(x);end

x~


Implementation

bAx

downsampling

b

3

2

1

7

6

5

4

3

2

1

1210 000

0012 100

0000 121

2

1

b

b

b

b

b

b

b

bb

b

bPD

function z=fwei(y)n=length(y);m=floor(n/2);z=zeros(m,1);z(1:m)=.5*y(2:2:n-1) + .25*(y(1:2:n-2)+y(3:2:n));


Implementation

)))))~((((( xPPAPP IIDD

bAx

bxPAPPP IIDD

~~

downsampling

b

Solve small system using LSQR

x=lsqr(@pap,b,1e-8,iter,[],[],[],A);

xAxA T~~ and ~~

needs LSQR

)))))~((((( xPPAPP TTTTD

TD II

function y = pap(x,A,transp_flag)n=size(A,1);m=length(x);if (nargin > 2) & strcmp(transp_flag,'transp') while length(x)>n x=interpol(x); end y=A'*x; while length(y)>m y=fwei(y); endelse while length(x)>n x=interpol(x); end y=A*x; while length(y)<m y=fwei(y); endend


Implementation

bAx

bxPAPPP IIDD

~~

downsampling

b

Solve small system using LSQR

xPPx II~

interpolating

x~

function z=interpol(x)n=length(x);m=2*n+1;z=zeros(m,1);z(1)=x(1)/2;z(m)=x(n)/2;z(2:2:m-1)=x(1:n);z(3:2:m-2)=.5*(x(1:n-1) +x(2:n));


Results

0 5 10 15 20 25 30 3510

-3

10-2

10-1

100

pmrnsd

lsqrmultilevel


Future Work

Fine tune parameters Find where it can be used as a

preconditioner Extend to 2D, 3D

january 2006 - the mathworks a multilevel method for ill-posed problems malena i. español tufts...

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