Saturday, November 13, 2010

What are inverse or ill-posed problems? Part 1.

First publications on inverse and ill-posed problems appeared in first half of 20th century. They were related to scientific research in physics (inverse problem of quantum theory of scattering, electrodynamics and acoustics), geophysics (inverse problems of electric sounding, seismology, theory of potential), astronomy and other areas of natural sciences. With the emergence of computers, inverse problems penetrated all scientific disciplines that use mathematical methods. Major application are geophysics, astronomy, data visualization, medical and industrial tomography, defect detection and distance sounding and many others.

In the direct problem of mathematical physics, researchers tend to find (explicitly or approximately) the functions describing various physical phenomena, such as the propagation of sound, heat and seismic waves, electromagnetic waves, and so on. The properties of the medium (the equation coefficients), as well as the initial state of the process (nonstationary case) or its properties on the boundary (in the case of a limited area and / or in the stationary case) are assumed to be known. However, it is the properties of the medium, in practice, are often unknown. This means that you need to formulate and solve the inverse problem, which is required to determine any coefficients of the equations, or unknown initial or boundary conditions or the location, boundaries and other properties of the region in which the process under study. These objectives are in most cases ill-posed (i.e., at least one of the three properties of well-posedness - a condition of existence, uniqueness and stability of the solution with respect to small variations in the data of the problem, is violated). And the unknown coefficients of the equations are as a rule, the density, electrical conductivity, thermal conductivity and other important properties of the medium. Also, very often in inverse problems is required to find the location, shape and structure of the inclusions, defects, sources (heat, vibration, stress, pollution) and so on. It is no wonder that with such a broad set of applications, the theory of inverse and ill-posed problems since its inception has been one of the most rapidly developing areas of modern science.

Everyone solves inverse and ill-posed problem every minute. Humans solve them, as a rule, quickly and efficiently (if they are in good health and a clear mind). Let's consider, for example, visual perception. It is known that our vision fixes only a finite number of points around us. Then how are we seeing everything? Brain (in this situation - a powerful personal computer) uses seen points (interpolates and extrapolates) to recover all that the eye did not percept. It is clear that to recover true picture (in the general case - the volume and color) using only several points can only be the case when it is already more or less familiar with (most of the subjects and images we have seen, and sometimes touched by hands). That is, despite the strong ill-posedness (non-uniqueness and instability of the solution) of the problem (to restore the observed object and all that surrounds it by several points), the brain solves this problem pretty quickly. Why? It uses a wealth of experience (a priori information). And anyway, if we want to understand something complex enough to solve the problem, the probability of error which is large enough, we tend to arrive at an unstable (ill-posed) problem.

We can say that people (especially inclined to seek unconventional solutions) are constantly facing inverse and ill-posed problems. In fact, everyone understands how easy to make a mistake when trying to restore the past, based on some facts of the present (trace the motives and details of the crime on the available evidence, understand the reasons for initiation and development stages of the disease on the survey results and so on). Or look into the future (anticipated development of the child, the direction of the country and another complicated processes). Or penetrate into the zone of inaccessibility and to understand what is happening there (to investigate the internal human organs, to detect mineral deposits, to learn something new about the universe and so on). In fact, any attempt to expand the boundaries of direct (sensory, visual, auditory, etc.) perception of the world leads to an ill-posed problems. It would seem that we can say that, having learned to solve the stable (well posed) problem, mathematicians have moved on to more complex unstable (inverse and ill-posed) problems. But historically this is not true, since all the ages man has been surrounded by ill-posed problems, and mathematicians tried to solve such problems, dispensing with the relevant terms.

Inverse and ill-posed problems are united by one important feature - the instability of solutions to small errors in measurement data. In most interesting cases, inverse problems are ill-posed and such problems, as a rule, can be formulated as inverse with respect to certain direct (well posed) problems. But since inverse and ill-posed problems are historically formulated and studied quite often independently and in parallel, nowadays both terms are used in the scientific literature.

Summarizing, we can say that experts in the inverse and ill-posed problems
are involved in the study of properties and methods for the regularization of unstable problems. In other words, mathematicians are trying to create and explore sustainable methods of approximation of unstable maps. From the viewpoint of linear algebra this is a search of approximate methods for finding the normal pseudosolutions of systems of linear algebraic equations with rectangular, singular or poorly conditioned matrices. In functional analysis, a prime example of ill-posed problem is an operator equation Aq = f, where A is compact (completely continuous). Recently published studies interpret some problems of mathematical statistics as the inverse problem of probability theory. From the viewpoint of information theory, experts in the inverse and ill-posed problems investigate the properties of maps of compacts with a high epsilon-entropy in the table with a small epsilon-entropy.

In this introductory article we have learned about definitions of inverse and ill-posed problems, examples of their practical application and possible areas of development.
Next part will cover historical background of inverse and ill-posed problems.

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