Speaker
Description
A discretized inverse problem of deconvolution with single- or multi-dimensional Gaussian function as the apparatus function [1] is found in a number of optical and spectroscopic applications. Because of its instability [1, p. 12; 2, p. 32-37], it is necessary to reduce it to some regularized analogue [1; 2, p. 47], after which it can be solved by common methods for solving linear systems of equations or optimization.
The so-obtained result has a meaning of approximate solution, signal noise influence filtered by certain superimposition of special restrictions, either on the solution or on parameters of its search. The value of the key parameter of regularization method applied [2, p. 47], in turn, should be selected according to known principles. Proper selection [1, p. 55-61; 2, p. 66] of the key parameter reduces the signal noise contribution to the solution to the lowest possible level with maximum possible preservation of signal information and compliance of its a priori set properties, if any.
Despite the uniqueness of result obtained by the chosen regularization method at specific implementations of signal noise and values of its key parameter, there remains the problem of competition of various regularized solutions. It consists in the fact that selection of different regularization methods, their auxiliary parameters, and implementation of signal noise can yield solutions that formally have comparable explanatory power, but differ in their features. An effective way to overcome this is the use of global optimization methods and genetic algorithms. Application of these methods radically increases the time required for calculations, and hence methods for further acceleration of solving the deconvolution problem, if possible, are in demand.
We have developed a number of techniques that may enable acceleration of solving discretized problems of deconvolution without significant loss of accuracy regardless of the regularization method. We considered specific examples to show that application of alternative schemes for improving discrete approximation of deconvolution problem and reducing its dimensionality (involving non-standard methods of interpolation and taking edge effects into account [1, p. 35], as well as breaking the separation problem into many equivalent subproblems) makes it possible to accelerate computing processes, at least when obtaining solutions of a certain class.
The work was carried out in the framework of the state assignment for Budker INP SB RAS and RFBR project no. 19-05-50046. The work was done at the shared research center SSTRC on the basis of the Novosibirsk FEL/VEPP-4-VEPP-2000 complex at BINP SB RAS, using equipment supported by project RFMEFI62119X0022.
[1] Hansen P. C., Nagy J. G., O'Leary D. P. Deblurring Images, Matrices, Spectra, and Filtering. SIAM, Philadelphia, 2006. 130 p.
[2] Leonov A.S. Solving ill-posed inverse problems. Outline of the theory, practical algorithms and demonstrations in Matlab (in Russian). 2nd ed. M.: "LIBROKOM" Book House, 2012. 336 p.
