REGULARIZATION THEORY OF INVERSE PROBLEMS A BRIEF REVIEW

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: Edge-detection is ill-posed! • Image restoration: blurred image response function band-limited: invisible objects exist Existence if and only if • Interpolation Given such that find unknown object

WHAT ABOUT LEARNING? Learning from examples can be regarded as the problem of approximating a multivariate function from sparse data Data: the training set obtained by sampling according to some probability distribution Unknown: an estimator such that with high probability Is this an ill-posed problem? Is this an inverse problem? Next talk! predicts

MATHEMATICAL FRAMEWORK Hilbert spaces; find linear continuous: given a noisy Ill-posedness: such that or • Pseudosolutions: • Generalized solution or is unbounded or the minimum norm pseudosolution • Generalized inverse Finding is ill-posed bounded Remark: well-posedness does not imply stability closed

REGULARIZATION ALGORITHMS A regularization algorithm for the ill-posed problem is a one-parameter family • • of operators such that: is linear and continuous • Semiconvergence: given such that noisy version of there exists Tikhonov method Two major points: 1) how to compute the minimum 2) how to fix the regularization parameter

COMPUTATION Two ‘easy’ cases: • • is a convolution operator with kernel is a compact operator Singular system:

THE REGULARIZATION PARAMETER Basic definition: Let be a measure of the amount of noise affecting the datum Then a choice is optimal if Example: Discrepancy principle: solve • Generalized to the case of noisy models • Often oversmoothing Other methods: GCV, L-curve…

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: convex subset of the source space Open problem: is this a regularization method?

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?