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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
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References
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Book
Inverse Problem Theory and Methods for Model Parameter Estimation
TL;DR: This chapter discusses Monte Carol methods, the least-absolute values criterion and the minimax criterion, and their applications to functional inverse problems.
Book
Inverse Acoustic and Electromagnetic Scattering Theory
David Colton,Rainer Kress +1 more
TL;DR: Inverse Medium Problem (IMP) as discussed by the authors is a generalization of the Helmholtz Equation for direct acoustical obstacle scattering in an Inhomogeneous Medium (IMM).
Book
Regularization of Inverse Problems
TL;DR: Inverse problems have been studied in this article, where Tikhonov regularization of nonlinear problems has been applied to weighted polynomial minimization problems, and the Conjugate Gradient Method has been used for numerical realization.
Book
The Mathematics of Computerized Tomography
TL;DR: In this paper, the Radon transform and related transforms have been studied for stability, sampling, resolution, and accuracy, and quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.
Book
Introduction to Inverse Problems in Imaging
Mario Bertero,Patrizia Boccacci +1 more
TL;DR: Part 2 Linear inverse problems: examples of linear inverse problems singular value decomposition (SVD) inversion methods revisited Fourier based methods for specific problems comments and concluding remarks.