SIXTH EU FRAMEWORK PROGRAMME TRANSFER OF KNOWLEDGE TODEQ

OPERATOR THEORY METHODS FOR DIFFERENTIAL EQUATIONS



Regularization methods for ill-posed problems
of analysis and statistics



Lectures by
Sergei V. Pereverzev
(Johann Radon Institute for Computational and Applied Mathematics (RICAM) )


14 May –25 May

Sergei V. Pereverzev will read minicourse at the Stefan Banach International Mathematical Center in Warsaw, ul ¦niadeckich 8, room 405. Six lectures, 1,5 hours each, are planned, and some tutorials/exercises depending on a need. The main part of lectures will be given on May 16-18: Wednesday, 16:00, Thursday, 10:15. Friday, 10:15, and Wednesday, May 23, 16:00.

Moreover, on Friday, May 25, 15:00 Sergei V. Pereverzev will have a lecture on the seminar of the Center of Science and Technology.

The course is addressed mainly to mathematicians interested in modern applied mathemetics. There is no registration fee. We offer to cover accommodation and living expenses for some number of participants (limited, available according to the order of applications). The accommodation places will be in the Banach Center guest rooms and, if there is a need, a cheap accommodation elsewhere will be available. PhD/MSc students are asked to provide a recommendation letter from their supervisor.

Please send the application containing Name, E-mail, University, Status ( PhD student, MSc student, postdoc, adiunkt, professor) before the application deadline of 30th April 2007 to the e-mail address.


Poster

Description:

Ill-posed equations arise frequently in the context of inverse problems, where it is the aim to determine unknown characteristics of the underlying process from the data corrupted by measurement noise. When such noise is assumed to be random, this is a problem of statistical estimation. When it is assumed to be chosen, not at random, but by an antagonistic opponent, this is a problem of optimal recovery. While the two problems are superficially different, there is a number of underlying similarities, and the results obtained for one problem may be exploited for the other. It is the aim of the course to discuss a general approach to regularization of ill-posed problems for deterministic as well as for stochastic noise models.

Plan:

  1. Examples for motivation: Numerical differentiation, Cauchy problem for Elliptic Equations, Natural linearization for parameter identification.
  2. Essentially ill-posed linear operator equations. A general view of the problem of regularization.
  3. The best possible accuracy for stochastic and deterministic noise models.
  4. Balancing principle for an adaptive regularization and its extension.
  5. Applications and Discussions.

Bibliography:

  1. Peter Mathé and Sergei V. Pereverzev, Regularization of some linear ill-posed problems with discretized random noisy data, Mathematics of Computations, v.75, no 256, 2006, pp. 1913--1929.
  2. S. Pereverzev, E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Analysis , v.43, no 5, 2005, pp. 2060--2076.
  3. Alexander Goldenshluger and Sergei V. Pereverzev, On adaptive inverse estimation of linear functionals in Hilbert scales, Bernoulli, v.9 , no. 5, 2003, pp. 783--807.
  4. Peter Mathé and Sergei V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Problems, v. 19, 2003, pp. 789--803.
  5. H. W. Engl, P. Fusek, S.V. Pereverzev, Natural linearization for the identification of nonlinear heat transfer laws, Journal of Inverse and Ill-posed Problems, v.13, no 6, 2005, pp. 567 ­ 582.


Contact:

Teresa Regińska
Institute of Mathematics
Polish Academy of Sciences
ul. Śniadeckich 8,
P.O. Box 21, 00-956 Warszawa, POLAND
tel.: +48 22 52 28 100
fax: +48 22 62 25 750
e-mail