LA TEORIA DELLA REGOLARIZZAZIONE NELLA COMPUTER …
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Transcript LA TEORIA DELLA REGOLARIZZAZIONE NELLA COMPUTER …
REGULARIZATION THEORY OF INVERSE PROBLEMS
- A BRIEF REVIEW -
Michele Piana, Dipartimento di Matematica, Università di Genova
PLAN
• Ill-posedness
• Applications
• Regularization theory
• Algorithms
SOME DATES
1902 (Hadamard) - A problem is ill-posed when its solution is not
unique, or it does not exist or it does not depend
continuously on the data
Early sixties - ‘…The crux of the difficulty was that numerical
inversions were producing results which were
physically unacceptable but were mathematically
acceptable…’ (Twomey, 1977)
1963 (Tikhonov) - One may obtain stability by exploiting additional
information on the solution
1979 (Cormack and Hounsfield) – The Nobel prize for Medicine and
Physiology is assigned ‘for the
developement of computed assisted
tomography’
EXAMPLES
• Differentiation:
f1 ( x ) f ( x ) f 2 ( x ) f ( x ) sin( x )
Edge-detection is ill-posed!
• Image restoration:
g : blurred image
k : response function
| f1 ( x) f 2 ( x) |
| f1( x ) f 2( x ) |
f : unknown object
g( x) (k f )( x)
k band-limited: invisible objects exist
Existence if and only if
ˆ ( ) |2 d
ˆ
|
g
(
)
/
k
• Interpolation
Given
l
z ( xi , yi ) X Y i 1 find
such that f ( xi ) yi
i 1,...l
f : X Y
WHAT ABOUT LEARNING?
Learning from examples can be regarded as the problem
of approximating a multivariate function from sparse data
Data: the training set
z ( xi , yi ) , xi X yi Y
obtained by sampling
l
i 1
X Y
according to some
probability distribution
Unknown: an estimator f : X Y such that f (x ) predicts
y with high probability
Is this an ill-posed problem?
Is this an inverse problem?
Next talk!
MATHEMATICAL FRAMEWORK
H, K Hilbert spaces; A : H K linear continuous:
find f given a noisy g such that g Af
Ill-posedness: Ker ( A) 0 or R( A) K or
A
1
is unbounded
• Pseudosolutions: u , Au g min or u , A Au A g
• Generalized solution u : the minimum norm pseudosolution
• Generalized inverse
Finding
u
A : A g u
is ill-posed
( A
bounded
Remark: well-posedness does not imply stability
R(A) closed
u
u
A A
g
g
)
REGULARIZATION ALGORITHMS
A regularization algorithm for the ill-posed problem
is a one-parameter family
R
g Af
of operators such that:
• R : K H is linear and continuous
• lim 0 R Af A Af f X
• Semiconvergence: given g noisy version of Af
R
g
A
Af
R
g
A
Af
such
that
opt
opt
there exists
Tikhonov method
Af g
2
K
f
Two major points:
2
H
min
R ( A A I )1 A
1) how to compute the minimum
2) how to fix the regularization parameter
COMPUTATION
Two ‘easy’ cases:
•
A is a convolution operator with kernel k
kˆ ( ) gˆ ( )
ˆ
( R g )( )
2
ˆ
k ( )
•
A
is a compact operator
Singular system: Au j j v j
j
j 1
R g
2
j
( g, v j )u j
Av j j u j
THE REGULARIZATION PARAMETER
Basic definition:
Let be a measure of the amount of noise affecting the datum
Then a choice ( ) is optimal if
lim 0 R g u
lim 0 ( ) 0
Example: Discrepancy principle: solve
AR g g
• Generalized to the case of noisy models
• Often oversmoothing
Other methods: GCV, L-curve…
g:
ITERATIVE METHODS
Iterative methods can be used:
• to solve the Tikhonov minimum problem
• as regularization algorithms
In iterative regularization schemes:
• the role of the regularization parameter is played by the
iteration number
• The computational effort is affordable for non-sparse matrix
New, tighter prior constraints can be introduced
Example:
f n1
PC ( f n A ( g Af n ) )
C : convex subset of the source space
Open problem: is this a regularization method?
f0 0
CONCLUSIONS
• There are plenty of ill-posed problems in the applied sciences
• Regularization theory is THE framework for solving linear
ill-posed problems
• What’s up for non-linear ill-posed problems?