Ill-posed problems

Introduction to the theory, numerical methods and applications of ill-posed
problems
Short description
Anatoly Yagola
Professor, Dr. Sc. (Phys.-Math)
Department of Mathematics, Faculty of Physics,
Lomonosov Moscow State University,
Moscow, Russia
yagola@physics.msu.ru
The first part of this course gives a basic introduction to the theory and numerical methods for
solving ill-posed problems. The second part deals with applications to inverse problems in
electronic microscopy, acoustics, astrophysics, geophysics.
.
Location and dates
Department of Mathematical Sciences, fall semester 2013.
November 4 – November 12, 2013.
Introductory lecture:
Should be fixed later.
Aim of the course
After a successful completion of the course the students will be able to apply mathematical
methods for solving inverse and ill-posed problems.
Target group
Graduate students in mathematics and physics.

Entry requirements
Basic undergraduate mathematics courses in linear algebra, calculus, integral equations.
Course organizers
Anatoly Yagola (yagola@physics.msu.ru ) and Larisa Beilina (larisa.beilina@chalmers.se )
Teachers
Anatoly Yagola (yagola@physics.msu.ru )
Course program
1)Elements of the theory of ill-posed problems . Ill-posed problems in physical sciences.
Definitions. Functional spaces and linear operators. Regularizing algorithms. Fundamental
properties of ill-posed problems. Ill-posed problems on compact sets. Sourcewise representation
and a posteriori error estimation. Tikhonov’s variational approach for constructing regularizing
algorithms. Choice of a regularization parameter.
2)Numerical methods for solving ill-posed problems with different constraints. Ill-posed
problems on compact sets of a special structure. Methods for minimization of
Tikhonov’s functional and the discrepancy. Conjugate gradients method and others.
3)Applications to inverse problems of astrophysics, electronic microscopy, acoustics,
astrophysics, geophysics.
Lectures
7 double hours.
Exam
Project work with a written report.

Registration
Please contact the course organizers for information.
Literature
1. A.N.Tikhonov, A.V.Goncharsky, V.V.Stepanov, A.G.Yagola. Numerical methods for the
solution of ill-posed problems. - Kluwer Academic Publishers, Dordrecht, 1995.
2. A.N.Tikhonov, A.S.Leonov, A.G.Yagola. Nonlinear ill-posed problems. V.1, 2. -
Chapman and Hall, London, 1998.