**Nikiforov**

Vladimir O.

D.Sc., Prof.

Vladimir O.

D.Sc., Prof.

doi: 10.17586/2226-1494-2015-15-6-1147-1154

doi: 10.17586/2226-1494-2015-15-6-1147-1154

# METHOD OF TRAINING EXAMPLES IN SOLVING INVERSE ILL-POSED PROBLEMS OF SPECTROSCOPY

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**Article in**English

**For citation:**Sizikov V.S., Stepanov A.V. Method of training examples in solving inverse ill-posed problems of spectroscopy.

*Scientific and Technical Journal of Information Technologies, Mechanics and Optics*, 2015, vol. 15, no. 6, pp. 1147–1154.

**Abstract**

**Subject of Study.**The paper deals with further development of the method of computational experiments for solving ill-posed problems, e.g., the inverse spectroscopy problem. This method produces an effective (nonoverstated) estimate for solution error of the first-kind equation. **Method of Research.** An equation is solved by the Tikhonov regularization method. We have obtained nonoverstated estimate for solution error and a new principle for choosing the regularization parameter on the basis of the truncating singular number spectrum of an operator. It is proposed to estimate the truncation magnitude by results of solving model (training, learning) examples close to an initial example (problem). This method takes into account an additional information about the solution. **Main Results.** We have derived a new, more accurate estimate for regularized solution error using the truncation parameter *g*. Ways for determining *g* according to the results of solving model examples are proposed. The method of modeling or training is applied to solving the inverse spectroscopy problem (restoration of a fine spectrum structure by solving integral equation on the basis of an experimental spectrum and the spread function of a spectral device). The method makes it possible to resolve close lines and select weak lines. **Practical Relevance.** The proposed method can be used to restore smoothed and noisy spectra, in other words, to enhance the resolution of spectral devices by mathematical and computer processing of experimental spectra.

**Keywords:**ill-posed problems, Tikhonov regularization, solution error, method of training examples, inverse problem of spectroscopy, integral equation, spread function of spectral device, measured spectrum, training spectra, restored spectrum.

**Acknowledgements.**This work was supported by the Russian Foundation for Basic Research (RFBR), grant №13-08-00442.

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