Biography of Michael V. Klibanov Professor, Ph.D. and Doctor of Science in Physics and Mathematics and an outstanding expert in inverse problems

Michael (Mikhail) Victor Klibanov was born in 1950 in Samara, Russia, a city located on the banks of the river Volga in the European part of the country. His mother, Ada, was a chemistry instructor in a local college and his father, Victor, was an aviation engineer. His parents are still alive and currently reside in New York City. Michael graduated from high school with honours in June 1967. That summer he became a student at the Mathematics Faculty of Novosibirsk State University (NSU; Novosibirsk, Siberia). The NSU is one of the top three Russian universities, along with Moscow State and Saint Petersburg State Universities. The NSU is located in the Novosibirsk Scientific Center, a micro-city named ‘Akademgorodok’, which is about 25 km outside of Novosibirsk City. Akademgorodok has many research institutions, which form the Siberian Branch of the Russian Academy of Science. This provides a very creative research atmosphere in that area.

Professor Klibanov earned his Master of Science degree in Mathematics, with honours, in 1972. In 1973, he became a doctoral level graduate student at the Computing Center of Siberian Branch of the Russian Academy of Sciences. This Computing Center is one of the research institutions in Akademgorodok. Michael's thesis advisor was Professor Mikhail Mikhailovich Lavrent’ev, a member of the Russian Academy of Sciences and the Founder of the theory of multidimensional coefficient inverse problems (MCIPs) for partial differential equations (PDEs). In the beginning of the 1960s, Lavrent’ev also published some foundational works on the Theory of Ill-Posed Problems (along with A.N. Tikhonov and V.K. Ivanov). Professor Klibanov defended his Candidate of Science in Physics and Mathematics (an equivalent to a Ph.D.) in 1977 and then went back to Samara, the city of his childhood.

In Samara, he received the position of Associate Professor of Mathematics at the Samara State University. He worked there until 1989, when he emigrated to the US. For about two years after 1977, Michael was looking for a new direction of research. His dream was to do something quite unusual and important. In 1979, he focused on a very challenging and important problem, which (at that time) was a serious hindrance to progress for the entire field of MCIPs. This was the issue of proof of the so-called global uniqueness results for MCIPs. Michael's interest was in MCIPs with a single measurement event only, as it had been a traditional focus of the Lavrent’ev group. Single measurement means the minimal amount of information, which is both economical and good for some applications. It was at that time, the so-called local uniqueness theorems were proven for these problems. In other words, the common assumption was that the unknown coefficient is either sufficiently small (in a certain norm), or piecewise analytic, or something similar. The true challenge was to prove such a uniqueness theorem, in which the main assumption would be only that the unknown coefficient belongs to one of the main Banach spaces used in PDE theory: C^k , H^k , etc.

Professor Klibanov worked very hard for two years (1979–1980) on this challenge. As one of the self-tormenting episodes of that period, he often recalls a vacation with his wife Vera, on the beach of the Black Sea in August of 1980. While everyone was swimming and enjoying life, he was working hard on this problem at the beach. Still, success was nowhere in sight. However, while boarding the plane on his flight back to Samara, he, all of a sudden, got a vague idea to take a more careful look at Chapter 4 of the newly published book of M.M. Lavrent’ev, V.G. Romanov and S.P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis (Moscow, Nauka, 1980; English translation: AMS, Providence, RI, 1986). The chapter is full of long formulae with very little explanation. At the first glance, those formulae seemed very intimidating to him. Nevertheless, by the end of August of 1980, after some very careful reading, he saw that this chapter actually was very informative. Indeed, it contained a painstaking derivation of pointwise Carleman estimates for a general parabolic operator with variable coefficients, as well as for the ultra-hyperbolic operator with , , in its principal part (a particular case is the wave operator ). Next, these estimates are applied there for proofs of Holder stability and uniqueness results for ill-posed Cauchy problems for those operators, including the Cauchy problem for a general elliptic operator as a particular case of the parabolic operator.

So, Professor Klibanov raised the following question in his mind: The main difference between the coefficient inverse problem I study and those Cauchy problems is that I have initial data, in addition to those Cauchy data. However, I need these initial data because the coefficient is unknown in my case. Is it possible to somehow combine these two things? This question was the starting point of his truly revolutionary discovery, which has quickly catapulted him to the very top echelon among experts in inverse problems.

In September and October of 1980, after a long and self-tormenting effort with many sleepless nights, he developed a breakthrough scheme, which has made it possible to apply Carleman estimates to the above challenging problem. While initially he had only a modest goal to prove the global uniqueness for at least one MCIP, to his fascination and disbelief, he saw that his method was working for basically all MCIPs, as long as the Carleman estimate was valid for the underlying PDE operator. However, since Carleman estimates are proven for the three main types of PDEs of the second order: hyperbolic, parabolic and elliptic ones, the resulting method embraces a broad class of MCIPs. Furthermore, it became clear that Holder stability estimates can also be obtained the same way. It turned out that Dr A.L. Buhgeim (another pupil of M.M. Lavrent’ev) independently and simultaneously developed a similar concept. Since Soviet Mathematics Doklady (currently Doklady Mathematics) is a journal featuring rapid short publications (usually without proofs), it was decided to publish a joint paper in this journal1. The first complete proofs are published by these two authors separately in Proceedings2 (the work of A.L. Buhgeim ‘Carleman estimates for Volterra operators and uniqueness of inverse problems’ was published on pages 56–64 of the same Proceedings). Also, see 3–6 for some subsequent publications of Michael on this subject. Since then, this technique became very popular under the name ‘The Buhgeim–Klibanov method’ and has been featured in numerous publications of many authors. This technique still remains the only one which enables us to prove global uniqueness theorems for MCIPs with single measurement events. Actually, this method is the true core of Klibanov's research career.

In 1980s, Professor Klibanov also worked on the so-called phase problem in optics (PPO). PPO deals with the recovery of a compactly supported complex valued function from the absolute value of its Fourier transform. This absolute value is the amplitude of the optical signal, which is actually measured in optics. Michael has proved a number of pioneering global uniqueness results for PPO 7–10, which have not been surpassed to this day.

In 1986, he defended his Doctor of Science degree in Physics and Mathematics in the Computing Center of the Siberian Branch of the Russian Academy of Sciences (Novosibirsk). His dissertation was based on two discoveries: the Buhgeim–Klibanov method and global uniqueness theorems for PPO. He cherished this degree as one of his highest achievements. In the former Soviet Union (so as in the current Russian) system, the Doctor of Science degree is much higher than the Candidate of Science. The Doctor of Science opens the door to the Full Professor position in Russia, which is impossible for a Candidate of Science to achieve.

After relocating to the United States in 1989, Professor Klibanov focused his research efforts on applications of Carleman estimates to the numerical treatment of inverse problems. He realized that the so-called Quasi-Reversibility Method of French mathematicians R. Lattes and J.-L. Lions is well-suited for these problems in the linearized case. His first interest was in a linearized MCIP for a parabolic PDE11. Next, he focused on the hyperbolic case. The first required goal was to prove the Lipschitz stability estimate for the Cauchy problem for a hyperbolic PDE with Dirichlet and Neumann data at the lateral boundary of the time cylinder. Michael (with co-authors) was the first one who figured out how to do this using Carleman estimates12,13. At that time, he did not know much about Control Theory. So, he was truly surprised to find out that his result was actually a long-standing unsolved problem in Control Theory, where it is called the observability inequality. The observability inequality is equivalent to the exact controllability, which is of primary interest in the control community. Thus, the methodology for proving the observability inequality by Carleman estimate has a big impact. More precisely, prior to12,13, such an estimate was proven by Lop Fat Ho only for the wave equation without lower order terms. Lop Fat Ho has developed the so-called method of multipliers. But since any Carleman estimate is independent of low-order terms of the corresponding PDE, Michael's method has handled a quite more general case in which low-order terms were included. So, since works12,13, Carleman estimates became one of the important tools in Control Theory, see also14. Michael (with co-authors) also published some numerical works, where the theory of12,13 was applied to numerical studies using the Quasi-Reversibility Method 15–17. It was shown in these publications that the Quasi-Reversibility Method provides a very stable solution for the above-mentioned Cauchy problem.

Concerning MCIPs, Professor Klibanov has always felt dissatisfaction with the numerical methods for these problems. Indeed, since the field of inverse problems is an applied one, he felt that just proving theorems was insufficient. As soon as theorems are established, one should still figure out how to compute those problems. On the other hand, regardless of all the nice theorems, numerical methods for MCIPs were relying only on the so-called small perturbation approach. In this approach, one should assume that the first guess for the solution is known a priori with a very good accuracy. The latter, however, is a rare case in applications. In recent years, some numerical methods have been published which do not use that approach. Michael has always valued these works quite highly (see, e.g. citations in 18–20). Still, these works deal with the data resulting from many measurements. On the other hand, Michael's interests have always been with MCIPs with single measurement data. Initially, Dr A. Timonov and he developed the so-called convexification algorithm6,21, see also22. However, since the convexification uses layer stripping with respect to one of the spatial variables, it cannot image the medium deeply. This is because the derivative with respect to this variable is involved in the original PDE operator. So, recently Michael and Dr L. Beilina came up with a globally convergent technique, which is based on a different idea: the layer stripping is conducted with respect to the positive parameter s of the Laplace transform18,19. Since the derivative with respect to s is not involved in the original PDE operator, this new method can image rather large domains. Just as Klibanov's first result on Carleman estimates, the technique of18,19 is applicable to quite a broad class of MCIPs both for hyperbolic and parabolic PDEs. This is his current focus of research. The idea of ‘eliminating’ the unknown coefficient from the PDE via the differentiation still comes from his earlier works on Carleman estimates.

Most recently, Professor Klibanov decided to work on another challenging problem of testing the technique of18,19 on real-life experimental data. He has organized an interdisciplinary team of mathematicians and engineers. First, engineers measured these data. Then, mathematicians applied the technique of18,19 for the imaging of dielectric abnormalities20. Experimental data were collected using a state-of-the-art technology in the picosecond regime (1 ps = 10⁻¹² s = 10⁻³ ns). The goal was to image the locations and refractive indices of those dielectric inclusions. No a priori assumption about the structure of the scattering background medium was made. It just so happened that the team did not know the values of the refractive indices of dielectrics they were using in their experiments. In other words, they have worked with blind data from the start, which is the most difficult case. So, these indices were measured only a posteriori, i.e. after images were computed by the mathematical sub-team. To the great joy of the entire team, comparison of measured and computational results consistently demonstrated an excellent accuracy of the computed ones. Hence, the result of20 clearly speaks very highly about the globally convergent numerical method of18,19.

A remarkable feature of Michael's mathematics is that his description is lucid. It gives adequate information for interested readers and actually suggests further directions of research. Throughout his career, he managed to invent elegant solutions for very important and quite challenging problems and, while doing this, he never followed the footsteps of others. On the contrary, he demonstrated many times his ability to have a fresh look and surprising approaches to those problems. His method using Carleman estimate gained a truly great popularity, not only because this method is very effective, but also because his exposition inspires others to innovate. Although Michael realizes that his style might draw some criticism from people who pursue theoretical sophistication in the field, he keeps proudly saying that he pursues the same style which his mentor Academician Lavrent’ev actually did.

To summarize, the following are the main pioneering results of Michael V. Klibanov, which made him a distinguished member of the international inverse problems community:

1.	Introduction of the powerful tool of Carleman estimates in the field of MCIPs 1–4,5,6.
2.	Introduction of Carleman estimates in the Control Theory for proofs of observability estimates6,12–14.
3.	Development of globally convergent numerical methods for MCIPs with the single measurement data6,17–22.
4.	Global uniqueness theorems for the PPO 7–10.

Main publications of Professor Michael V. Klibanov, Ph.D. and Doctor of Science in Physics and Mathematics are compiled in the References.

Michael is at present at the peak of his research career. He is very energetic, full of plans and ideas. We wish him great health, great creativity, many more excellent research results and many, many happy years of life.

George S. Dulikravich, Robert P. Gilbert, Alemdar Hasanov, Bernd Hofmann, Sergey I. Kabanikhin, Mikhail M. Lavrent’ev, Vladimir G. Romanov, Paul E. Sacks and Masahiro Yamamoto.