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Main fields of research | ![]() |
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Ill-Posed Problems
The foundation of ill-posed problems, or incorrect problems, was laid
by Academician A. N. Tikhonov. Later, several scientific schools
on this theory formed, including the school of Moscow scientists headed
by A. N. Tikhonov, the school of Novosibirsk scientists headed
by M. M. Lavrent’ev, and the school of Ekaterinburg scientists
headed by V. K. Ivanov. The theory of ill-posed problems developed
also in France, Germany, Italy, and USA. At the Institute of Mathematics, investigations in the theory of ill-posed problems and inverse problems of mathematical physics are carried on in the department headed by Academician M. M. Lavrent’ev and in the laboratories headed by Corresponding Member of RAS V. G. Romanov and doctors of physical and mathematical sciences Yu. E. Anikonov and A. L. Bukhgeim. In all, 10 doctors work in the scientific collective: V. P. Golubyatnikov, S. I. Kabanikhin, A. I. Khisamutdinov, V. R. Kireitov, G. S. Lbov, V. A. Sharafutdinov, I. A. Taimanov, Yu. N. Valitskii, V. G. Yakhno, S. M. Zerkal’ and also a large group of candidates of sciences (Ph. D.). Investigations are carried on in the following main directions:
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. 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. 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. |
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© 2004, Sobolev
Institute of Mathematics of the SB RAS, Novosibirsk Omsk Branch of the Institute |
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