Abstract
Most problems in mechanics and physics have the form ‘Find the effect of this cause.’ There are numerous examples: Find how this structure is deformed when these forces are applied to it. Find how heat diffuses through a body when a heat source is applied to a boundary. Find how waves are bent, or absorbed, as they pass through a nonhomogeneous medium.
As an orthodox mathematician, he believes his formula more than his eyes and common sense, and doesn’t see the incongruity in it.
Academician A.N. Krylov, on a formula in an article by Levi-Civita.
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References
The first book devoted to ill-posed problems was
A.N. Tikhonov and V.Y. Arsenin, Solution of Ill-Posed Problems, John Wiley, New York, 1977. This is invaluable as a guide to the early literature. It uses the methods of functional analysis, and has many instructive examples from the theory of Fredholm integral equations. The reader who has studied the present book will have more than sufficient background knowledge in functional analysis to understand it.
Classical treatment of the abstract theory of ill-posed problems is to be found in the rather difficult
V.A. Morozov, Methods of Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984.
Perhaps the best introduction to the theory of the inverse problems we have studied in this chapter is
C.W. Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg, Braunschweig, 1993. This motivates the study of inverse problems by many examples taken from different areas of mathematics, physics and engineering. It provides a very brief summary of functional analysis and then applies it to the inverse problem stated as a Fredholm integral equation of the first kind, or more generally as the equation Ax = y. The principal aim of Chapter 8 has been to expand on Groetsch’s treatment, trying to fill in some of the steps which he left to the reader. The book has a valuable guide to the literature.
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© 1996 Kluwer Academic Publishers
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Lebedev, L.P., Vorovich, I.I., Gladwell, G.M.L. (1996). Applications to Inverse Problems. In: Functional Analysis. Solid Mechanics and Its Applications, vol 41. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0169-8_8
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DOI: https://doi.org/10.1007/978-94-009-0169-8_8
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