S e r g e i L' v o v i c h S o b o l e v

Key words: computational mathematics and cybernetics,
difference analogues of embedding theorems

Work during and after the Great Patriotic War

During the Great Patriotic War and in the post-war period the importance of computational mathematics and computer engineering grew dramatically. Sergei L'vovich actively engaged himself into the work connected with development of those fields in this country. Working for many years at the post of chief deputy director of the Institute of Atomic Energy which was headed by I.V.Kurchatov, Sergei L'vovich took active and direct part in solution of important applied problems which were meaningful for defense. Exactly at that time, in December 1951, Sergei L'vovich Sobolev gained the highest State decoration: he was awarded the title of a Hero of Socialist Labour for exceptional services for his State. Sergei L'vovich was twice awarded the State Prize of First Degree for his works made at the Kurchatov Institute of Atomic Energy.

Sergei L'vovich Sobolev is one of the first mathematicians who understood the special importance and necessity of the rapid development of computational mathematics and cybernetics in this country. Since 1952 till 1960 S.L.Sobolev headed the Chair of Computational Mathematics at the Moscow State University which was the first such chair in this country and played an inestimable role in the development of this important field of modern mathematics. Sergei L'vovich succeeded in creating a solidary collective of enthusiasts at this chair having set high level of research work and ensured graduation of the first qualified specialists in computational mathematics. In solution of the problems of computational mathematics Sergei L'vovich widely applied the apparatus of functional analysis, partial differential equations and the theory of functions; not seldom the problems of computational mathematics themselves are set in his works as problems of functional analysis. Sergei L'vovich wrote in this connection that "to concieve the theory of computations without Banach spaces is impossible just as trying to concieve it without computers. "
As early as in the pre-war years Sergei L'vovich published a number of works on estimation the sums of values of functions which are given on a grid: those were the first works on difference analog of embedding theorems. This direction of research gained substantial development and now is a necessary tool for studing of convergence derivation of estimates for grid solutions and error. Qualitative study of solutions to difference equations and investigation of stability for many classes of grid problems is reduced to the study of behavior of the Green's function of grid problems. Sergei L'vovich obtained exact estimates of the asymptotic behavior of the difference Green's function for the Laplace equation.
Studying the convergence and stability of algorithms for solution of problems of mathematical physics, S.L.Sobolev introduced concepts of uniform completely continuity and correct approximation by a sequence of operators for the theory of approximation, and also the notion of regular and nonregular closure of a computational algorithm (some continuous analog obtained by passing to the limit from relations comprising a computational algorithm). If the closure of an algorithm is regular, then we may expect that it will be stable under various perturbations. These works drew attention to questions of the general theory of computational algorithms and to the techniques and methods for solving large systems of equations.

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