Many problems of science, technology and engineering are posed in the form of an operator equation of the first kind with an operator and a right part approximately known. Often such problems turn out to be ill-posed. The theory of solving linear and nonlinear ill-posed problems is advanced greatly today (see for example [1, 2]). A general scheme for constructing regularizing algorithms on the base of Tikhonov variational approach is considered in [2]. It is very well known that ill-posed problems have unpleasant properties even in the cases when there exist stable methods (regularizing algorithms) of their solution. So it is recommended to study all a priori information, to find all physical constraints, which may make it possible to construct a well-posed mathematical model of the physical phenomena.
Computational programs for linear ill-posed problems with different a priori information (monotonicity, convexity, known number of extremes, sourcewise representation of an unknown solution, etc.) could be found in [1] and other author’s publications, and could be generalized for nonlinear problems also. In these cases, if we know error levels of experimental data, it is possible also to calculate error estimates of unknown solutions or so called a posteriori error estimates. If the constraints are not sufficient to formulate a well-posed problem then it is necessary to use all these constraints but for constructing regularizing algorithms we must know error levels of experimental data. As examples of successful applications of these regularizing algorithms to practical problems we consider inverse problems of vibrational spectroscopy and electron microscopy.
REFERENCES
1. Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., and Yagola, A.G., Numerical Methods for the Solution of Ill-Posed Problems. Kluwer Academic Publ., Dordrecht, (1995).
2. Tikhonov, A.N., Leonov, A.S., and Yagola, A.G., Nonlinear Ill-Posed Problems. V. 1, 2. Chapman and Hall, London, (1995).
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