Increasing the Accuracy of Solving Discrete Ill-Posed Problems by the Random Projection Method

2018 ◽  
Vol 54 (5) ◽  
pp. 842-852 ◽  
Author(s):  
E. G. Revunova
Keyword(s):  
Projection Method ◽  
Random Projection ◽  
Ill Posed

scholarly journals On the Generalization of the Random Projection Method for Problems of the Recovery of Object Signal Described by Models of Convolution Type

2021 ◽  
pp. 25-34
Author(s):  
O.G. Revunova ◽  
◽  
A.V. Tyshcuk ◽  
О.О. Desiateryk ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Projection Method ◽  
Random Projection ◽  
Convolution Type ◽  
Kernel Matrix ◽  
Input Output ◽  
Technical Problems ◽  
Discrete Representation ◽  
Ill Posed ◽  
Output Transformation

Introduction. In technical systems, there is a common situation when transformation input-output is described by the integral equation of convolution type. This situation accurses if the object signal is recovered by the results of remote measurements. For example, in spectrometric tasks, for an image deblurring, etc. Matrices of the discrete representation for the output signal and the kernel of convolution are known. We need to find a matrix of the discrete representation of a signal of the object. The well known approach for solving this problem includes the next steps. First, the kernel matrix has to be represented as the Kroneker product. Second, the input-output transformation has to be presented with the usage of Kroneker product matrices. Third, the matrix of the discrete representation of the object has to be found. The object signal matrix estimation obtained with the help of pseudo inverting of Kroneker decomposition matrices is unstable. The instability of the object signal estimation in the case of usage of Kroneker decomposition matrices is caused by their discrete ill posed matrix properties (condition number is big and the series of the singular numbers smoothly decrease to zero). To find solutions of discrete ill-posed problems we developed methods based on the random projection and the random projection with an averaging by the random matrices. These methods provide a stable solutions with a small computational complexity. We consider the problem of object signals recovering in the systems where an input-output transformation is described by the integral equation of a convolution. To find a solution for these problems we need to build a generalization for two-dimensional signals case of the random projection method. Purpose. To develop a stable method of the recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. Results and conclusions. We developed the method of a stable recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. The stable estimation of the object signal is provided by Kroneker decomposition of the kernel matrix of convolution, computation of random projections for Kroneker factorization matrices, and a selection of the optimal dimension of a projector matrix. The method is illustrated by its application in technical problems.

scholarly journals The random projection method in goodness of fit for functional data

2007 ◽  
Vol 51 (10) ◽  
pp. 4814-4831 ◽  
Author(s):  
J.A. Cuesta-Albertos ◽  
E. del Barrio ◽  
R. Fraiman ◽  
C. Matrán
Keyword(s):  
Projection Method ◽  
Functional Data ◽  
Goodness Of Fit ◽  
Random Projection

scholarly journals Expanding the applicability of a Two Step Newton-type projection method for ill-posed problems

2014 ◽  
Vol 51 (1) ◽  
pp. 141-166
Author(s):  
Ioannis K. Argyros ◽  
Monnanda E. Shobha ◽  
Santhosh George
Keyword(s):  
Projection Method ◽  
Ill Posed

Random projection method for ultra wideband signal acquisition

2011 ◽  
Vol 18 (6) ◽  
pp. 14-21 ◽  
Author(s):  
Jie LIN ◽  
Guang-ming SHI ◽  
Xu-yang CHEN ◽  
Li ZHANG
Keyword(s):  
Projection Method ◽  
Random Projection ◽  
Ultra Wideband ◽  
Signal Acquisition ◽  
Wideband Signal

On the semilocal convergence of a two-step Newton-like projection method for ill-posed equations

2013 ◽  
Vol 40 (3) ◽  
pp. 367-382
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Projection Method ◽  
Semilocal Convergence ◽  
Ill Posed
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