Method of training examples in solving inverse ill-posed problems of spectroscopy

TL;DR

This paper introduces a new approach to training in solving ill-posed inverse spectroscopy problems, utilizing spectral truncation and model examples to improve regularization and error estimation.

Contribution

It proposes a novel method for estimating solution errors and selecting regularization parameters based on spectral truncation and model problem solutions.

Findings

Effective error estimates for first-kind equations

New principle for choosing regularization parameters

Numerical example demonstrating improved solution accuracy

Abstract

Further development of the method of computational experiments for solving ill-posed problems is given. The effective (unoverstated) estimate for solution error of the first-kind equation is obtained using the truncating singular numbers spectrum of an operator. It is proposed to estimate the magnitude of the truncation by results of solving model (training, learning) examples close to the initial example (problem). This method takes into account an additional information about the solution and gives a new principle for choosing the regularization parameter and error estimate for equation solution by the Tikhonov regularization method. The method is illustrated by a numerical example from the inverse problem of spectroscopy.