Ill-Posed Problems

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Investigations are carried on in the following main directions:

• ill-posed problems, including methods of regularization, operator equations, problems of integral geometry;
• inverse problems of mathematical physics, including problems of finding coefficients of equations basing on various additional information about their solutions; theorems of existence and uniqueness of solution for inverse problems, estimates of conditional stability.

The staff of the Department take active part in training of scientific personnel, teach in NSU (Novosibirsk State University) and other higher educational institutions of Novosibirsk. Under their guidance 15 doctor theses and over 100 Ph. D. theses were prepared and successfully defended.

In the Laboratory of Ill-Posed Problems headed by Academician M. M. Lavrent’ev, the theory of inverse problems connected with seismoprospecting, photometry, and astrophysics is studied. Integral geometry problems and the theory of operator equations are considered. M. M. Lavrent’ev is the editor-in-chief of the Journal of Inverse and Ill-Posed Problems.

In the Laboratory of Wave Phenomena headed by Professor V. G. Romanov, a wide range of inverse problems of the theory of wave propagation arising in electrodynamics, seismoprospecting, and acoustics are investigated. Inverse problems of the theory of transfer, electroelasticity, and tomography are also considered.
A number of new theorems of existence, uniqueness, and stability of solution were proved in the Laboratory, numerical methods of solving inverse problems of electromagnetism, acoustics, elasticity have been developed.
Members of the Laboratory V. G. Romanov and S. I. Kabanikhin are members of the editorial board of the Journal of Inverse and Ill-Posed Problems.
A scientific seminar which deals with the theory and numerical methods of inverse problems is permanently held in the Laboratory.

In the Laboratory of Inverse Problems of Mathematical Physics headed by Professor Yu. E. Anikonov, investigations on inverse problems for general evolutionary equations and kinetic equations are carried on. Various problems of recovering body shapes by functionals of their projections on planes are studied. Questions of existence, uniqueness, and stability of solution to linear and nonlinear problems of integral geometry are considered. Applications of the theory of multidimensional inverse problems to tomography, geophysics, sociology, in particular, for modeling of ethnic processes, are also studied.
Yu. E. Anikonov is a member of the editorial board of the Journal of Inverse and Ill-Posed Problems and heads the scientific seminar “Inverse Problems of Mathematical Physics”.

In the Laboratory of Numerical Methods of Solution of Inverse Problems headed by Professor A. L. Bukhgeim, the problems of finding memory and/or coefficients of hyperbolic or parabolic (integro-differential) equations are solved. Questions of existence, uniqueness, and stability of solutions to such problems are investigated. In these problems, the method of weight a priori estimates of Carleman type is successfully applied. For a number of differential equations the problems of continuation of solutions from given sets (including discrete ones) are solved, for which formulas of Carleman type are obtained. The interconnection of some inverse problems for the transfer equation (tomography problem) with the theory of A-analytical functions is also investigated in the Laboratory. Inversion formulas for the problem of emission tomography and their numerical algorithms were obtained.