Inverse problems of vibrational spectroscopy

Many problems of physical chemistry belong to the class of inverse problems, in which from known experimental data of the object we need to determine some of its properties based on a certain model connecting these properties with measured characteristics. Inverse problems typically lead to mathematical models that are not well-posed in the sense of Hadamard, i.e. to the ill-posed problems. This means that they may not have a solution in the strict sense, solutions may not be unique and/or may not depend continuously on the input data. Mathematical problems possessing such properties are called ill-posed problems, mostly due to the nonstability of solutions with respect to data perturbations. The theory of ill-posed problems (advanced by A. N. Tikhonov and his scientific school in 1960s) investigates and develops the effective stable numerical (regularization) methods for the solution of the ill-posed problems. In the background of this theory lies an understanding of the underdetermined character of the illposed problems and a concept of the regularizing operator (algorithm).