1 / 11

REGULARIZATION THEORY OF INVERSE PROBLEMS

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

joanna
Download Presentation

REGULARIZATION THEORY OF INVERSE PROBLEMS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. REGULARIZATION THEORY OF INVERSE PROBLEMS - A BRIEF REVIEW - Michele Piana, Dipartimento di Matematica, Università di Genova

  2. PLAN • Ill-posedness • Applications • Regularization theory • Algorithms

  3. 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’

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

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

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

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

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

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

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

  11. 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?

More Related