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Science in Siberia (scientific newspaper of Siberian Branch of Russian Academy of Sciences) N 40 (2725) October 8, 2009 INVERSE AND ILL-POSED PROBLEMS With the support of the Russian Foundation for Fundamental Research, Novosibirsk State University, several research institutes of the Siberian Branch of the Russian Academy of Sciences (SB RAS), Yugra Research Institute of Information Technology (URIIT), Baker Hughes Inc., Intel Corp., and Schlumberger Ltd., the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences held the international Conference and Workshop for Young Scientists called “Theory and Computational Methods for Inverse and Ill-posed Problems” in August 10-20, 2009. The conference was hosted by the Sobolev Institute of Mathematics and included participants from China, France, Kazakhstan, Kyrgyzstan, Turkey, Ukraine, USA, Uzbekistan, and 16 cities of the Russian Federation. Sergey I. Kabanikhin, Chairman of the Organizing Committee, - Chief Research Scientist, Wave Processes Laboratory, Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences; Chief of Laboratory of Mathematical Problems of Geophysics, Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of Russian Academy of Sciences;Professor, Chair of Theory of Functions, Novosibirsk State University. The scientific agenda of the conference and workshop included 42 plenary 40-minute lectures and over forty 20-minute presentations. The plenary lectures were given by academicians A. N. Konovalov and M. I. Epov, corresponding members of the Russian Academy of Sciences B. D. Annin, V. V. Vasin (Yekaterinburg), V. G. Romanov, I. A. Taimanov, and A. M. Fedotov (with co-author, academician Yu. I. Shokin), 36 Doctors of Science and 12 Doctors of Philosophy. Scientific reports were presented by 25 young scientists from Novosibirsk and 22 from other cities. The conference and workshop included special sections on high-performance and parallel computational algorithms (organized by Intel), problems and algorithms of tomography (organized by the Institute of Theoretical and Applied Mechanics, SB RAS), problems of seismology, geoelectrics, and induction logging (organized by Institute of Oil and Gas Geology and Geophysics, Baker Hughes, and Schlumberger). The attendees included participants from 69 educational, scientific, and research and production organizations. The scientific agenda of the conference and workshop included the theory of inverse and ill- posed problems, regularization methods, iterative and direct methods for solving inverse problems, numerical methods for solving direct and inverse problems of acoustics, tomography, geoelectrics, seismology, gravimetry, transport theory, etc. Special prizes for the most interesting lectures were awarded to Professor V. S. Belonosov, D.Sc. (Sobolev Institute of Mathematics, SB RAS), and Professor V. V. Pikalov, D.Sc. (Institute of Theoretical and Applied Mechanics, SB RAS). The Best Paper by a Young Scientist Award was shared by M. A. Shishlenin, Ph.D. (Sobolev Institute of Mathematics, SB RAS) and P. A. Chistyakov (Institute of Mathematics and Mechanics, Urals Branch of the Russian Academy of Sciences, Yekaterinburg). What Are Inverse and Ill-Posed Problems? First publications on inverse and ill-posed problems date back to the first half of the 20th century. Their subjects were related to physics (inverse problems of quantum scattering theory, electrodynamics, and acoustics), geophysics (inverse problems of electrical prospecting, seismology, and potential theory), astronomy, and other areas of science. Since the advent of powerful computers, the area of application for the theory of inverse and ill-posed problems has extended to almost all fields of science that use mathematical methods. The main areas of application are geophysics, astronomy, data visualization, medical and industrial tomography, non-destructive testing, remote sensing, and many others. In direct problems of mathematical physics, researchers try to find exact or approximate functions that describe various physical phenomena such as the propagation of sound, heat, seismic waves, electromagnetic waves, etc. In these problems, the media properties (expressed by the equation coefficients) and the initial state of the process under study (in the nonstationary case) or its properties on the boundary (in the case of a bounded domain and/or in the stationary case) are assumed to be known. However, it is precisely the media properties that are often unknown in real life. This leads us to inverse problems, in which it is required to determine the equation coefficients, or the unknown initial or boundary conditions, or the position, boundaries, and other properties of the domain where the process under study is taking place. Most of these problems are ill-posed (i.e., at least one of the three conditions of well-posedness does not hold: the condition of existence, or uniqueness, or stability of solutions with respect to small variations in the data). The unknown equation coefficients usually represent important media properties such as density, electrical conductivity, heat conductivity, etc. Also, in inverse problems it is often required to determine the location, shape, and structure of intrusions, defects, sources (of heat, waves, potential difference, pollution), and so on. Given such a wide variety of applications, it is no surprise that the theory of inverse and ill-posed problems has become one of the most rapidly developing areas of modern science. The foundations of the theory of inverse and ill-posed problems were laid in the Soviet Union in the middle of the 20th century. In the recent decades, however, Russian science has retreated from its leading role in the theory of inverse and ill-posed problems, for obvious reasons. Many talented specialists, including many young researchers, have left the country. Whereas there are over 11000 book titles containing the words “inverse problems” published outside of Russia, the number of such books published in Russia has been steadily declining. A hopeful sign, however, is the publication of the textbook “Inverse and Ill-posed Problems” by S. I. Kabanikhin (Sibirskoye Nauchnoye Izdatelstvo, 1998) officially recommended for university and college students majoring in Applied Mathematics and Information Technology, Applied Mathematics, Mechanics, and Applied Mechanics by the Science and Education Policy Board for Mathematics of the Ministry of Education and Science of the Russian Federation. In our everyday life we are constantly dealing with inverse and ill-posed problems and, given good mental and physical health, we are usually quick and effective in solving them. For example, consider our visual perception. It is known that our eyes are able to perceive visual information from only a limited number of points in the world around us at any given moment. Then why do we have an impression that we are able to see everything around? The reason is that our brain, like a powerful personal computer, completes the perceived image by interpolating and extrapolating the data received from the identified points. Clearly, the true image of a scene (generally, a three-dimensional color scene) can be adequately reconstructed from several points only if the image is more or less familiar to us, i.e., if we previously saw and sometimes even touched most of the objects in it. Thus, although the problem of reconstructing the image of an object and its surroundings from several points is ill-posed (i.e., there is no uniqueness or stability of solutions), our brain is capable of solving it rather quickly. This is due to the brain's ability to use its extensive previous experience (a priori information). In general, attempting to understand a substantially complex phenomenon or solve a problem where the probability of error is fairly high, we usually arrive at an unstable (ill-posed) problem. Ill-posed problems are ubiquitous in our daily lives (especially for those who are not looking for standard ways of solving problems). Indeed, everyone realizes how easy it is to make a mistake when reconstructing the events of the past from a number of facts of the present (for example, reconstructing the scene and motives of a crime based on available evidence, determining the cause and the development stages of a disease based on the results of a medical examination, and so on). The same is true for predicting the future (the course of someone’s life, the development of a country, or any sufficiently complex process) or reaching into inaccessible zones to explore their structure and internal processes (examining internal organs of a patient, exploring mineral deposits, exploring new regions of the Universe, etc.). Essentially every attempt to expand the limits of the direct sensory perception of the world (tactile, visual, aural, etc.) leads to ill-posed problems. It may seem that mathematicians turned to more complex, unstable (inverse and ill- posed) problems after they had mastered the methods for solving stable (well-posed) problems. However, historically this is not entirely true, since ill-posed problems have been around since ancient times and mathematicians just tried to solve them without introducing new terminology. An important common property of inverse and ill-posed problems is the instability of solutions with respect to small errors in the data. In the majority of cases that are of interest, inverse problems turn out to be ill-posed and, conversely, an ill-posed problem can usually be reduced to a problem that is inverse to some direct (well-posed) problem. Historically, however, inverse and ill-posed problems were often formulated and studied independently and in parallel, and therefore both terms are in use in the scientific literature today. To sum up, it can be said that specialists in inverse and ill-posed problems study the properties of and regularization methods for unstable problems. In other words, they develop and study stable methods for approximating unstable mappings. In terms of linear algebra, this means developing approximate methods of finding normal pseudo-solutions to systems of linear algebraic equations with rectangular, degenerate, or ill-conditioned matrices. In functional analysis, the main example of ill-posed problems is represented by an operator equation Aq = f, where A is a compact (completely continuous) operator. In some recent publications, certain problems of mathematical statistics are viewed as inverse problems of probability theory. From the point of view of information theory, the theory of inverse and ill-posed problems deals with the properties of maps from compact sets with high epsilon-entropy to tables with low epsilon- entropy. Historical Perspective It is well known that many mathematical concepts and problem formulations are products of studying physical phenomena. This is certainly true for the theory of inverse and ill-posed problems. Plato’s philosophical allegory about echo and shadows on the cave walls (i.e., the data of an inverse problem) being the only reality available to human cognition was a precursor to Aristotle’s solution to the problem of reconstructing the shape of the Earth from its shadow on the moon (projective geometry). The introduction of the physical concept of instantaneous speed led Isaac Newton to the discovery of the derivative, and the instability (ill-posedness) of the problem of numerical differentiation of an approximate function is still a subject of present-day research. Lord Rayleigh’s research in acoustics led him to the question of whether it is possible to determine the density of a non-uniform string from its sound (the inverse problem of acoustics), which brought about the development of seismic prospecting on one hand, and the theory of spectral inverse problems on the other hand. The study of the motions of celestial objects and the problem of estimating unknown parameters based on measurement results that contain random errors led Legendre and Gauss to overdetermined systems of algebraic equations and to the method of least squares. Cauchy proposed the steepest descent method for finding the minimum of a multivariate function. In 1948, L.V. Kantorovich generalized, developed, and applied these ideas to operator equations in Hilbert spaces. At present, the steepest descent method together with the conjugate gradient method are among the most popular methods for solving ill-posed problems. It should be noted that Kantorovich was the first to point out that if the problem is ill-posed, then the method he proposed converges only with respect to the objective functional. Thus, certain inverse and ill-posed problems have long been studied by researchers in various areas of science. Nevertheless, it was not until the early 20th century when the mathematical properties of ill-posed problems were described by Hadamard, and it became clear that a unified approach to solving such problems was necessary. The idea that there are no inadequate problems, but some problems are ill-posed discouraged some researchers while others aimed to find new approaches to solving these “bad” problems. R. Courant‘s opinion that unstable problems have no physical meaning did not prevent him from solving a strongly ill-posed problem of reconstructing a function from its spherical means. In 1953-55, S. L. Sobolev was a research advisor for V. K. Ivanov’s doctorate thesis “Studies of the inverse problem of the potential theory”, which became a theoretical basis for a series of inverse problems of gravimetric prospecting. From the classical Cauchy-Kovalevskaya theorem it follows that a wide range of inverse and ill-posed problems have a unique solution, but only in the class of analytic functions. L. V. Ovsyannikov proved that the requirement of analyticity with respect to variable which is orthogonal to a surface can be substantially relaxed. In further developing the method of scales of Banach spaces proposed by L. V. Ovsyannikov and L. Nirenberg, V. G. Romanov has shown that for a large class of inverse problems it is possible to get rid of the requirement of analyticity with respect to two variables, namely the spatial variable which is orthogonal to a surface and the time variable. The said research paved the road to studying multidimensional inverse problems of geophysics, where the base model is a horizontally-stratified medium. It is impossible to cover all aspects of the theory of inverse problems and its applications in a single article. We will mention just two areas whose emergence and development was largely due to substantial contribution of researchers from Akademgorodok, Novosibirsk, V. E. Zakharov and A. B. Shabat (the method of the inverse scattering problem), and A. S. Alekseev and S. V. Gol’din (inverse problems of geophysics). The method of the inverse scattering problem was applied in solving nonlinear equations of mathematical physics (the Korteweg-de Vries equation, the nonlinear Schrödinger equation, the Kadomtsev–Petviashvili equation, etc.) and stimulated new research in various areas of mathematics and physics (spectral theory of differential operators, classical algebraic geometry, relativistic strings, etc.). The method of the inverse scattering problem is considered one of the pearls of the mathematical physics of the 20th century. The results of A. S. Alekseev’s and S. V. Gol’din’s research in applying the spectral theory of inverse problems and integral geometry in geophysics became the theoretical basis of many geophysical methods (inverse kinematic and dynamic problems of seismology). It should be noted that the acknowledged success of the present generation of Siberian geophysicists was largely determined by high-quality instruction in mathematical at the Geology and Geophysics Department of Novosibirsk State University. The author had a privilege to work at the Geophysics Department when it was home to a creative team of geophysics teachers (S. V. Gol’din, L. A. Tabarovsky, M. I. Epov, Yu. A. Dashevsky, and others) and mathematics teachers (M. M. Lavrent’ev, A. S. Alekseev, V. G. Romanov, T. A. Godunova, and others). Discussions of what areas of mathematics need to be taught to geophysics students and what portion of the curriculum they should take were regular at the teachers’ meetings and often resembled discussions at scientific conferences. At present, some fellows and alumnae of this department are directors of research institutes (Institute of Oil and Gas Geology and Geophysics, Institute of Computational Mathematics and Mathematical Geophysics, Yugra Research Institute of Information Technology, etc.), while many others are employed by large companies such as Schlumberger, General Electric, Intel, Baker Hughes, etc. The list of examples can be continued, but we will conclude by pointing out that A. N. Tikhonov, V. K. Ivanov, and M. M. Lavrent’ev are recognized worldwide as the founders of the theory of ill-posed problems. Their works laid the foundation for the theory of inverse and ill-posed problems. One of their principal ideas was the necessity to restrict the class of possible solutions when studying ill-posed problems. The choice of the set where the approximate solution is sought (the well-posedness set) is of primary importance. In most cases this set is chosen to be compact, which ensures the convergence of regularizing algorithms, helps to choose the appropriate regularization parameter and estimate the deviation of the approximate solution from the exact solution to the ill-posed problem. Results of the related mathematical research were applied to solving a series of specific inverse problems of geophysics, radiolocation, astronomy, and medical tomography. Outstanding scientific achievements in these areas were honored with the Lenin Prize awarded to A. N. Tikhonov and V. K. Ivanov, and later with the USSR State Prize awarded to M. M. Lavrent'ev, Yu. E. Anikonov, V. R. Kireitov, V. G. Romanov, and S. P. Shishatskii. Since the late 20th century, the interest of mathematicians and other scientists in inverse and ill- posed problems has been growing at an unprecedented rate. Four prominent international journals devoted to this subject were founded within the span of a very short historical period, including the Journal of Inverse and Ill-Posed Problems edited by academician M. M. Lavrent'ev. International organizations such as Inverse Problems International Association and the International Society for Inverse Problems in Science and Engineering are active in promoting research. Dozens of large conferences on different aspects of the theory of inverse problems and its applications are held in the world each year. More details about the related journals, associations, and conferences can be found on the author’s page on the web site of the Sobolev Institute of Mathematics. Current Areas of Application for Inverse Problems Inverse problems have so many areas of application that a Google search for the phrase “inverse problems” currently yields about 2,070,000 hits, while the same search in Yahoo yields 8,810,000 hits. We will mention just a few examples of research results produced by the staff members, PhD students and research interns of the Wave Processes Laboratory of the Sobolev Institute of Mathematics, SB RAS. In the past 30 years, the laboratory has studied inverse problems of seismic prospecting, electrodynamics, medicine, chemistry, acoustics, and biology. Research collaborators included scientists from many institutes of the Siberian Branch of the Russian Academy of Sciences (Institute of Computational Mathematics and Mathematical Geophysics, Institute of Oil and Gas Geology and Geophysics, the Boreskov Institute of Catalysis, Institute of Cytology and Genetics, and Institute of Thermophysics), International Tomography Center (SB RAS), companies Schlumberger, Baker Hughes, and Intel, and other researchers from Austria, Brazil, Germany, Italy, Kazakhstan, China, USA, Turkey, Uzbekistan, Sweden, and Japan. One of the main purposes of this research was to study the uniqueness of solutions to inverse problems. It is especially important for inverse problems of geophysics, medicine, and nondestructive testing. The uniqueness theorem for solutions to such problems answers the question of how many and what kind of measurements are sufficient to be certain that they identify only one object (for example, a mineral deposit in geophysical problems, or a change in the structure of internal organs in problems of medicine, or an internal defect in problems of nondestructive testing). The second most important subject studied by the laboratory included the estimates of stability of solutions to inverse problems with respect to measurement errors (which inevitably occur in every experiment). At the next stage, the research focused on numerical methods for solving inverse problems. It was shown that conditional stability estimates for solutions of inverse and ill-posed problems make it possible to estimate the rate of convergence of numerical methods for solving inverse problems and establish new rules for choosing the regularization parameter in ill- posed problems. The full list of applied grant-related and other research projects carried out at the laboratory is too long for this article. We conclude with mentioning that the results presented in the book “Inverse Problems of Geoelectrics” by V. G. Romanov and S. K. Kabanikhin (“Nauka”, Moscow, 1991) were used as a basis for a new method of electromagnetic prospecting (patent No. 2062489). These are examples of the work of just one laboratory of one of the institutes. The list would be enormous if we mentioned research results of all laboratories of the institutes of the Siberian Branch of the Russian Academy of Sciences that deal with various kinds of inverse problems. The Effect of the Conference The overwhelming majority of the participants acknowledged that they learned a lot at the conference and workshop and agreed that it should become an annual event. The following are some of the conference recommendations: “In view of its high educational and scientific value, the conference and workshop should be held regularly. Since the theory and numerical methods for solving inverse and ill-posed problems are applied in almost all fields of science that use mathematical methods, we address the governing bodies of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk State University, and research institutes involved in physics and mathematics studies with the request to organize an educational and scientific advisory center called ‘The theory and numerical methods for solving inverse and ill-posed problems’. The mission of the center could include providing consultations and assistance for undergraduate and postgraduate students, young scientists, and all researchers who study the theory of inverse and ill-posed problems and apply it in practice; performing the analysis and classification of methods for the study and numerical solution of applied inverse problems; developing software packages for the numerical solution of inverse and ill-posed problems.” In conclusion, it is important to mention young scientists who actively participated in organizing the conference: M. A. Shishlenin, PhD, Vice-Chairman of the Organizing Committee; A. V. Penenko, Academic Secretary of the Conference, PhD student; D. A. Voronov, O. I. Krivorot’ko, N. S. Novikov, and V. A. Chembai, graduate students at Novosibirsk State University. Their example shows that there is a new generation of researchers capable of carrying on the scientific tradition and organizing conferences in the future. On behalf of all participants and organizers of the conference, I wish to thank the executive boards of the Sobolev Institute of Mathematics, the Russian Foundation for Fundamental Research, directors of the research institutes and organizations that supported the conference, and lecturers who shared their knowledge and experience with young scientists. Sergey I. Kabanikhin, Chairman of the Organizing Committee Original Russian version of the article at http://www.sbras.ru/HBC/hbc.phtml?10+519+1

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