Journal ArticleDOI
Definitions and examples of inverse and ill-posed problems
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In this article, the authors consider definitions and classification of inverse and ill-posed problems and describe some approaches which have been proposed by outstanding Russian mathematicians A. N. Tikhonov, V. K. Ivanov and M. M. Lavrentiev.Abstract:
Abstract The terms “inverse problems” and “ill-posed problems” have been steadily and surely gaining popularity in modern science since the middle of the 20th century. A little more than fifty years of studying problems of this kind have shown that a great number of problems from various branches of classical mathematics (computational algebra, differential and integral equations, partial differential equations, functional analysis) can be classified as inverse or ill-posed, and they are among the most complicated ones (since they are unstable and usually nonlinear). At the same time, inverse and ill-posed problems began to be studied and applied systematically in physics, geophysics, medicine, astronomy, and all other areas of knowledge where mathematical methods are used. The reason is that solutions to inverse problems describe important properties of media under study, such as density and velocity of wave propagation, elasticity parameters, conductivity, dielectric permittivity and magnetic permeability, and properties and location of inhomogeneities in inaccessible areas, etc. In this paper we consider definitions and classification of inverse and ill-posed problems and describe some approaches which have been proposed by outstanding Russian mathematicians A. N. Tikhonov, V. K. Ivanov and M. M. Lavrentiev.read more
Citations
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References
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Book
Inverse Problems for Maxwell's Equations
V. G. Romanov,Sergey Kabanikhin +1 more
TL;DR: In this paper, the authors considered the Cauchy problem for one-dimensional medium in case of the current located at a point and the problem of determining the medium permittivity.
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Dynamical Inverse Problems of Distributed Systems
TL;DR: In this article, a monograph dealing with problems of dynamical reconstruction of unknown variable characteristics (distributed or boundary disturbances, coefficients of operator etc.) for various classes of systems with distributed parameters (parabolic and hyperbolic equations, evolutionary variational inequalities etc.).
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Integral Equations of First Kind
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Inverse problems of wave processes
TL;DR: One-dimensional inverse problems for string equation peculiarities of solution -formulation of the direct problem the first method of solution of inverse problem method of linear integral equations the case of discontinuous s(y) the Gel'fand-Levitan equation for second-order hyperbolic equations some cases of explicit solution on connection of inverse problems with nonlinear ordinary differential equations.