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

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

scholarly journals ON THE SELF-REGULARIZATION OF ILL-POSED PROBLEMS BY THE LEAST ERROR PROJECTION METHOD

2014 ◽  
Vol 19 (3) ◽  
pp. 299-308 ◽  
Author(s):  
Alina Ganina ◽  
Uno Hamarik ◽  
Urve Kangro
Keyword(s):  
Approximate Solution ◽  
Projection Method ◽  
Regularization Method ◽  
The Self ◽  
Noise Levels ◽  
Right Hand ◽  
Ill Posed ◽  
The Right

We consider linear ill-posed problems where both the operator and the right hand side are given approximately. For approximate solution of this equation we use the least error projection method. This method occurs to be a regularization method if the dimension of the projected equation is chosen properly depending on the noise levels of the operator and the right hand side. We formulate the monotone error rule for choice of the dimension of the projected equation and prove the regularization properties.

Wavelet projection methods for solving pseudodifferential inverse problems

2014 ◽  
Vol 12 (03) ◽  
pp. 1450025 ◽  
Author(s):  
E. P. Serrano ◽  
M. I. Troparevsky ◽  
M. A. Fabio
Keyword(s):  
Inverse Problem ◽  
Approximate Solution ◽  
Inverse Problems ◽  
Pseudodifferential Operator ◽  
Projection Method ◽  
Projection Methods ◽  
Wavelet Basis ◽  
Partial Knowledge ◽  
Orthonormal Wavelet ◽  
Ill Posed

We consider the Inverse Problem (IP) associated to an equation of the form Af = g, where A is a pseudodifferential operator with symbol[Formula: see text]. It consists in finding a solution f for given data g. When the operator A is not strongly invertible and the data is perturbed with noise, the IP may be ill-posed and the solution must be approximate carefully. For the present application we regard a particular orthonormal wavelet basis and perform a wavelet projection method to construct a solution to the Forward Problem (FP). The approximate solution to the IP is achieved based on the decomposition of the perturbed data calculating the elementary solutions that are nearly the preimages of the wavelets. Based on properties of both, the basis and the operator, and taking into account the energy of the data, we can handle the error that arises from the partial knowledge of the data and from the non-exact inversion of each element of the wavelet basis. We estimate the error of the approximation and discuss the advantages of the proposed scheme.

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.

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