# Andrey Nikolayevich Tikhonov

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### Mathematician — Academician AS USSR

Alexander Nitusov

**Andrey Nikolayevich Tikhonov ** (also **Andrei Nikolaevich**) — famous Soviet mathematician and geo-physicist, who set up theory of differential equations with minor parameter at the major derivative as special domain in asymptotic analysis. He is also famous for his numerous solution algorithms of various applied problems.

He was an academician AS USSR, double bearer of honourable title “Hero of the Socialist Labour” and was awarded with umber of other highest state awards.

## Family and learning

**Andrey Nikolayevich
Tikhonov**

A.N. Tikhonov was born on the 30 th of October 1906 in town of Gagarin (previously Gzhatsk), in Smolensk region (West Russia). His father – Nikolay Vasilyevich Tikhonov was a tradesman, the mother – Maria Nikolaevna took care of the two sons (Andrey and the elder Nikolay). In 1910 the Tikhonovs moved to Moscow where the boys visited a city's gymnasium (secondary school). With outbreak of the Civil War (1918—1921) they first moved into Ukraine but in 1919 returned back to Moscow.

Exhausting World War I, which resulted in the (Russian) October Socialist Revolution and then continued as the Civil War brought national economy to total collapse. Life was extremely difficult at those times, so Andrey had to start working as an office clerk at the Agronomists Service of the Byelorussian–Baltic Railways, at the age of 13 (in 1919).

However, in 1922 he successfully passed exams for the curriculum of so-called “workers' faculty”1 — the evening courses of general education. He also intensively prepared himself to entering exams at the Moscow State University (MSU) and in September of the same year began studying there at the physical and mathematical faculty.

After graduation, in 1927, he stayed as a postgraduate at the Scientific-Research Institute of Mathematics, which was a part of the MSU. After that he remained at the university as a lecturer on mathematics. His first scientific researches he performed still being a student. They were concerned with *topology * and *functional analysis *. Thus in 1926 he introduced a notion of topological space – so-called “Tikhonov's product” (of topological spaces). He also proved theorems of *bi-compactness **of bi-compact spaces product * (in other words, the product of any set of compact topological spaces is compact) and of existence of a static point at continuous mappings in topological spaces.

## Scientific work in the 1930-s

In 1933 MSU underwent serious reorganisation, which -besides others- resulted in establishing of special Physical Faculty. Tikhonov received an appointment at its new chair of mathematics. At the same time he was also appointed as scientific specialist at the Moscow Geophysical Institute. In 1936 he successfully submitted and defended dissertation (habilitation) thesis, “On Functional Equations of Volterra, Type and their Implementation in Equations of Mathematical Physics”.

His principal mathematic interests lay in the field of solution methods of the equations of mathematical physics, especially problems of stability, in inverse problems and dependence of differential equations solutions from a minor parameter by derivatives. He also made essential contribution to *theoretical geophysics * and to *modelling of physical and chemical processes*. He proved the *unicity theorem * (uniqueness theorem) for *equation of heat conductivity*.

In 1936 he was awarded doctorate and was appointed as a *professor * (at the MSU); now he held the chair of mathematics at the Physical Faculty. In 1937 he also started his work at a new Institute of Theoretical Geophysics AS USSR, which had been established by famous mathematician O.Y. Schmidt2.

In 1939 he was elected as a corresponding member of the AS USSR at the department of natural mathematical and natural sciences on speciality “Geophysics and mathematical physics”.

In 1948 Tikhonov received the governmental directive, which entrusted him with urgent organisation of the “Computational Laboratory” for modelling and calculating of atomic explosion processes. In 1950 he also performed fundamental research on theory and methods of implementation of electromagnetic fields in researches of the Earth crust.

## Tikhonov's Mathematical Theories

In 1951 he and A.A. Samarskiy (Samarskiy Alexander Andreevich) jointly published a monograph “Equations of Mathematical Physics”. Their most important common works in 1956-1963 included researches on various variants of *differential schemas for * solving *differential equations with partial derivatives*, their stability and precession on non-homogenous nets. Their work also resulted in *theory of homogenous differential schemas*.

In 1953 academician M.V. Keldysh (Keldysh Mstialav Vsevolodovich) established new Institute of Applied Mathematics (IAM), within academy's structure. Tikhonov received position of its Deputy Director, which he held for many years.

In 1963-1966 he conducted researches on *exploration of natural resources*, which led him to most famous ** theory of ill-posed problems **and inverse problems. He developed method of *ill-posed problems solution * by implementation of ** regularisation method. ** Here the notion of

*regularisation*and a

*regularising operator*were introduced. He also researched ill-posed problems related to processing of nuclear physics experimental results, to problems of optimal control, optimal planning problems and some problems of linear algebra.

He reported on solution methods of the ill-posed problems at the International Congress of Mathematicians in 1966. Importance of the subject has been subsequently proved by the world recognition.

Generally speaking his regularisation method and solving of the ill-posed problems became the basic item of all his subsequent scientific work.

On the 1 st of July 1966 Tikhonov was elected an academician (real member) of the AS USSR, department of mathematics, specialisation “Mathematics”.

## Progress of Mathematics at the MSU

In the same 1966 he headed a chair of computational mathematics at the MSU Faculty of Mechanics and Mathematics. In 1970, chiefly due to personal efforts of M.V. Keldysh, new Faculty of Computational Mathematics and Cybernetic (CMC) was established at the MSU (also famous under Russ. abbreviation “VMK”) and Tikhonov was appointed as its dean.

Since then his name was closely connected with all following important events and achievements of mathematics at the MSU (and not only there). Thus, he headed work of several scientific teams, which performed the following researches:

- Automation of programming and mathematical support (algorithms and programming) for digital electronic computers. It integrated efforts of such leading mathematicians as M.R. Shura-Bura (Shura-Bura Mikhail Romanovich ), S.S. Lavrov (Lavrov Svyatoslav Sergeevich), E.A. Zhogolev (Zhogolev Evgeny Andreevich), L.N. Korolev (Korolev Lev Nikolaevich ), V.P. Ivannikov (Ivannikov Viktor Petrovich), N.P. Trifonov.
- Numerical solution methods of typical computational problems:
- Solution algorithms for
*linear algebra*problems (V.V. Voevodin); -
*Optimisation*of problem solution methods*on functions? classes*(N.S. Bakhvalov, N.M. Korobov, I.F. Sharygin, Ya.M. Zhileikin); - Solution methods for
*common differential equations*(N.S. Bakhvalov, S.S. Gaisaryan, A.D. Gorbunov, A.N. Tikhonov, A.A. Samarskiy (Samarskiy Alexander Andreevich)); - Solution methods for
*differential equations with discontinuous coefficients*(A.N. Tikhonov, A.A. Samarskiy); - Solution methods of
*stationary and nonstationary problems*(A.A. Samarskiy, E.G. Dyakonov, V.B. Andreev); - Solution methods of
*some ill-posed problems*(A.N. Tikhonov, A.B. Bakushinskiy, P.N. Zaikin, V.A. Morozov); - Methods and algorithms of
*optimisation problems, including games problems*(V.G. Karmanov, F.P. Vasilyev, Y.A. Greimer, B.T. Polyak); - Modelling methods for
*production processes as mass service systems*(L.A. Lyusternik, G.P. Klimov).

- Solution algorithms for
- Numerical solution methods for problems of hydro- and aerodynamics, including calculations of the gas flows in nozzles and diffuser sections, flow about blunt shaped bodies, boundary-layer in a viscous liquid, implementation of characteristics (graphs) method (G.I. Petrov, V.V. Rusanov, G.F. Telenin, L.A. Chudov, G.S. Roslyakov, V.M. Paskonov, N.N. Kuznetsov, B.M. Pavlov).
- Numerical solution methods for problems of electro-dynamics, including calculations of electromagnetic oscillations propagation in wave-guides, in layered medium, in conductive mediums/bodies (A.N. Tnkhonov, A.G. Sveshnikov, V.I. Dmitriev, I.P. Kotik).
- Structures deciphering analysis, automation of calculations in structural analysis of crystals (N.P. Zhidkov, B.M. Shchedrin).
- Tikhonov also supervised design and manufacturing of the -world?s only- ternary computer SETUN? (1956) and (serial) SETUN?-70 (1970) of the MSU. Dr. N.P. Brusentsov (Brusentsov Nikolay Petrovich) has been its inventor and chief-designer, E.A. Zhogolev (Zhogolev Evgeny Andreevich) leading programmer.

In 1978, soon after the death of academician M.V. Keldysh, Tikhonov was appointed as director of the M.V. Keldysh Institute of Applied Mathematics (AS USSR) and held that position until 1998.

He headed the MSU Faculty of Computational Mathematics and Cybernetic until 1990.

Andrey Nikolaevich Tikhonov died in the evening of 7.11.1993 г. in Moscow and has been buried at the city's central cemetery.

## State Awards

In 1966 Tikhonov was awarded the “Lenin Premium”, “for outstanding contribution to computational mathematics”.

He was also twice awarded the title “Hero of the Socialist Labour” and, in 1953, decorated with “Lenin Order”, “…for exceptional merits in executing special governmental directives”.

In 1953 he was awarded the “Stalin Premium” (later named *State Premium *) for successful work in atom project.

In 1963 he received “Lomonosov Premium” (the 1 st grade) for his research “On Solution of ill-posed Problems” and in 1981 the “Premium of the USSR Council of Ministers”.

He was six times decorated with the “Lenin Order” (1951, 1953, 1953, 1966, 1971, 1986) - the sixth one was given to his 80 years birthday, in 1986. He also was decorated with three orders “Red Banner of Labour” (1945, 1949, 1961), an order “October Revolution” (1975) (to the 250 anniversary of the Academy of Sciences ) and M.V. Keldysh Gold Medal (for series of works, “On Regularisation Methods in Broad Spectrum of ill-posed Problems of Mathematical Physics”), in 1990.

Tikhonov has authored (and co-authored) more than 500 publications (monographs, course-books, articles). Among them the most famous: “On Stability of inverse Problems” – 1943, “Methods of Mathematical Physics” – 1951, “Methods of ill-posed Problems Solution” – 1974, “Regularisation Algorithms and a priory Information” – 1983, “Non-linear ill-posed Problems” – 1985, “Numerical Solution Methods of ill-posed Problems” – 1990.

Notes

1. “Workers' faculties” – special educational courses successfully functioning in the 1920—1930-s. They were set-up according to the governmental directive of 17.09.1920 with the purpose of additional training, education upgrading and eventual preparing working young people for entering universities, institutes, colleges, etc. That should increase general level of the young people – older as 16, who were otherwise not able to systematically study at schools. Studying at the faculty was officially recognised as professional occupation the students received state scholarships. In the 1960-s they were transformed into “Preparation courses”.

2.**O.Y. Schmidt ** – http://en.wikipedia.org/

Sourses:

- More on A.N. Tikhonov – www-history.mcs.st-and.ac.uk
- ru.wikipedia.org
- Андрей Николаевич Тихонов
**.**Неофициальный сайт ВМиК (ВМК) МГУ: http://cmc-msu.ru/. - ИПМ им. М.В.Келдыша РАН. Страницы памяти http://keldysh.ru/.
- Биография.ру. Тихонов Андрей Николаевич www.biografija.ru/.