Inverse Problems

Example

Direct problem

given polynomial

find zeros

Inverse problem

given zeros

find polynomial

Well-posedness

A problem is well posed if

Existence

- there exists a solution of the problem

Uniqueness

- there is at most one solution of the problem

Stability

- the solution depends continuously on the data

Example (ill-posed problem)Operator

}0)0(:]1,0[{]1,0[: yCyYCXK

t

dssxtKx0

)()(

Norm

problem is not stable

.

Perturb by y )/sin( 2 t

error in data

error in solution

1

Inverse problem

Given , compute such that , ie., x yKx 'yx Yy

The worst-case errorYXK :

linear

bounded

YX , Banach

Y

X

.

.

XX 1

stronger norm

11:. xcx

X

Inverse problem

Given , compute such that x yKx Yy

In general, we do not have the data …y

… but the perturbed data …y

},,:sup{:).,,(1111ExKxXxxExx

YX F

The worst-case errorYXK :

linear

bounded

YX , Banach

Y

X

.

.

XX 1

stronger norm

11:. xcx

X

Worst case error:

},,:sup{:).,,(111ExKxXxxE

YX F

Assume

-

-

- extra information for solutions and

Y

yy

)(XKy

21

Ex

21

Ex

The worst-case error (example)

YXK : ]1,0[2LYX

t

dssxtKx0

)()(

}0)0(',0)1(:)1,0({: 21 xxHxX

stronger norm 2'':1 L

xx

2.L

3/13/2

1).,,( EE F

It can then be shown:

Regularisation Theory

- compact operator

- one to one

-

YXK :

For , we would want to solve)(XKy

yKx

We actually know ... Yy

yy

yKx

Xdim

Problem!

???)(XKy

Find an approximation for x x

Aim

XXKK )(:1

Idea: Construct a suitable bounded approximation

of

XYR :

- small error (hopefully not much worse than the worst case error!)

- depends continuously on x y

Approximation

Ryx

)(XKy

Regularisation Strategy

XXKK )(:1

Idea: Construct a suitable bounded approximation

of

XYR :

Definition: A regularisation strategy is a family of linear and bounded operators

such that

0,: XYR

XxxKxR

,lim0

Theorem: (due to being compact)

1- is not uniformly bounded

2- Convergence is not uniform, but point wise

R

K

Error

yKx

End problem... Perturbed problem...

)(XKy

Yy

yRx :,

xx ,

xKxRR

xKxRyyR

xyRyRyR

XYR : XXKK )(:1approximations

of

Error

yxK )(

End problem...

)(XKy

xKxRRxx ,

When 0

R

0 xKxR

0

Perturbed problem...

Yy

XYR : XXKK )(:1approximations

of

MinimizationxKxRRxx

,

xKxRR min

Regularisation Strategy

XXKK )(:1

Idea: Construct a suitable bounded approximation

of

XYR :

Definition: A regularisation strategy is a family of linear and bounded operators

such that

0,: XYR

XxxKxR

,lim0

The worst-case error (example)

YXK : ]1,0[2LYX

t

dssxtKx0

)()(

}0)0(',0)1(:)1,0({: 21 xxHxX

stronger norm 2'':1 L

xx

2.L

3/13/2

1).,,( EE F

It can then be shown:

]1,0[2LY

}0)0(|)1,0({ 2 xLxX

Example of a regularisation strategy

YXK : ]1,0[2LYX

t

dssxtKx0

)()(

2.L

':1 yyK

Regularisation strategy:

)2/()2/(

)(

tyty

tyR

Example of a regularisation strategy

It can be shown, for a priori information

)2/()2/(

)(

tyty

tyR

221

,2

Eccxx

L

ExL

2''

Choose3

3 /)( Ec

3/13/2

1).,,( EE F

Then…

3/13/2),(2

EcxxL

asymptotically optimal

FilteringYXK :

compact

),( jjj yx singular system for K

jjj j

xyyx ),(1

1

is the solution of yKx

It can be shown

jj yx , orthonormal systems such that

...21 singular values of K

jjj yKx and jjj xyK *

Filtering

jjj j

xyyx ),(1

1

is the solution of yKx

Regularisation strategy (Filtering):

jjj j

j xyyq

yR ),(),(

:1

regularizing filter :q

1),( q

1),( q 0 when

)()(),( cRcq

Tykhonov Regularisation

YXK : compact

),( jjj yx singular system for K

jjj j

j xyyq

yR ),(),(

:1

)(),( 2

2

q

Rewrite :

Landweber Iteration

yKx

yaKxKaKIx ** )(

Iterative process

;00 x yaKxKaKIx mm *1* )(

Then

,yRx mm

1

0

** )(m

k

km KKaKIaRwhere

Landweber Iteration

1

0

** )(m

k

km KKaKIaR

YXK : compact and 210K

a

XYRm :

defines a regularization strategy

It can be shown…

Choices for m

accuracy of : large

stability of : small

an optimal choice can be made…

mR

mR

m

m

Conclusion

-Worst case error

- Regularisation strategies

- Filtering

- Tykhonov Regularisation

- Landweber Iteration