Inverse Problems
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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
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