In celebration of the 60th birthday of Professor Alemdar Hasanoğlu (Hasanov)

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Inverse and Ill-Posed Problems and Their Applications

According to Hadamard, a problem defined by the operator equation Az = u(1) (where z and u are elements of metric spaces Z and U, respectively) is correctly (or well) posed problem if the following three conditions are satisfied: (a) Eq. (1) is solvable for any u; (b) the solution of Eq.(1) is unique; (c) the solution of Eq.(1) is stable with respect to perturbations in the right-hand side u; i.e., that inverse operator exists, is defined throughout the space U, and is continuous. If one of the conditions (a)-(c) does not hold, the problem is called ill-posed. A lot of mathematical problems are ill-posed. Among them there are the following very well known examples: the Fredholm integral equation of the first kind; an operator equation (1) with a completely continuous operator in infinite-dimensional spaces, etc. A.N. Tikhonov proposed a special approach for solving ill-posed problems: for searching for stable solutions of unstable ill-posed problems it is necessary to use special regularizing operators (algorithms), if they exist, which give stable approximations to exact solution of unstable problems. Tikhonov has proposed also concrete regularizing operators for linear ill-posed operator equations in the Hilbert spaces, for minimization of functionals, for unstable problems of linear algebra, etc. At present, the theory of ill-posed problems is developed and is widely used to solve inverse problems in optics, spectroscopy, electrodynamics, plasma diagnostics, geophysics, astrophysics, image processing, etc. Regularizing algorithms, when being applied to process experimental data, significantly improve the accuracy with which the parameters of the physical objects are determined. The resolution of an experimental device can thus be greatly increased simply by using computer data processing without any expensive modification of the device itself. There is no doubt that the most effective systems with software packages for experimental work include programs based on regularizing algorithms. The methods for solving ill-posed problems now available can be successfully used in various branches of natural sciences. In my lecture I would like to introduce some fundamental results of the theory of linear and nonlinear ill-posed problems and its applications