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 imageunknown object Interpolation response function band-limited: invisible objects existExistence if and only if Givenfind such that

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

MATHEMATICAL FRAMEWORK findgiven a noisysuch that the minimum norm pseudosolution Generalized solution Generalized inverse Remark: well-posedness does not imply stability Pseudosolutions:or Findingis ill-posedboundedclosed Hilbert spaces;linear continuous: Ill-posedness:or is unbounded

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

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

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

ITERATIVE METHODS 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: Open problem:is this a regularization method? convex subset of the source space Iterative methods can be used: to solve the Tikhonov minimum problem as regularization algorithms

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?