An optimal regularization method for convolution equations on the sourcewise represented set

2015 ◽  
Vol 23 (5) ◽  
Author(s):  
Ye Zhang ◽  
Dmitry V. Lukyanenko ◽  
Anatoly G. Yagola
Keyword(s):  
Integral Equation ◽  
Regularization Method ◽  
A Priori ◽  
Noisy Data ◽  
Optimal Recovery ◽  
Multidimensional Case ◽  
A Priori Information ◽  
Ill Posed ◽  
Priori Information ◽  
Convolution Equations

AbstractIn this article, we consider an inverse problem for the integral equation of the convolution type in a multidimensional case. This problem is severely ill-posed. To deal with this problem, using a priori information (sourcewise representation) based on optimal recovery theory we propose a new method. The regularization and optimization properties of this method are proved. An optimal minimal a priori error of the problem is found. Moreover, a so-called optimal regularized approximate solution and its corresponding error estimation are considered. Efficiency and applicability of this method are demonstrated in a numerical example of the image deblurring problem with noisy data.

scholarly journals Method of Constructing a Nonlinear Approximating Scheme of a Complex Signal: Application Pattern Recognition

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 737
Author(s):  
Oksana Mandrikova ◽  
Bogdana Mandrikova ◽  
Anastasia Rodomanskay
Keyword(s):  
Noise Suppression ◽  
A Priori ◽  
Noisy Data ◽  
Adaptive Threshold ◽  
Complex Signal ◽  
A Priori Information ◽  
Threshold Functions ◽  
Useful Signal ◽  
Natural Data ◽  
Priori Information

A method for identification of structures of a complex signal and noise suppression based on nonlinear approximating schemes is proposed. When we do not know the probability distribution of a signal, the problem of identifying its structures can be solved by constructing adaptive approximating schemes in an orthonormal basis. The mapping is constructed by applying threshold functions, the parameters of which for noisy data are estimated to minimize the risk. In the absence of a priori information about the useful signal and the presence of a high noise level, the use of the optimal threshold is ineffective. The paper introduces an adaptive threshold, which is assessed on the basis of the posterior risk. Application of the method to natural data has confirmed its effectiveness.

scholarly journals Elimination of hidden a priori information from remotely sensed profile data

2006 ◽  
Vol 6 (4) ◽  
pp. 6723-6751 ◽  
Author(s):  
T. von Clarmann ◽  
U. Grabowski
Keyword(s):  
Degrees Of Freedom ◽  
A Priori ◽  
Michelson Interferometer ◽  
A Priori Information ◽  
Loss Of Information ◽  
Use Of Data ◽  
Remote Measurements ◽  
Ill Posed ◽  
Priori Information ◽  
Additional Loss

Abstract. Profiles of atmospheric state parameters retrieved from remote measurements often contain a priori information which causes complication in the use of data for validation, comparison with models, or data assimilation. For such applications it often is desirable to remove the a priori information from the data product. If the retrieval involves an ill-posed inversion problem, formal removal of the a priori information requires resampling of the data on a coarser grid, which, however, is a prior constraint in itself. The fact that the trace of the averaging kernel matrix of a retrieval is equivalent to the number of degrees of freedom of the retrieval is used to define an appropriate information-centered representation of the data where each data point represents one degree of freedom. Since regridding implies further degradation of the data and thus causes additional loss of information, a re-regularization scheme has been developed which allows resampling without additional loss of information. For a typical ClONO2 profile retrieved from spectra as measured by the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS), the constrained retrieval has 9.7 degrees of freedom. After application of the proposed transformation to a coarser information-centered altitude grid, there are exactly 9 degrees of freedom left, and the averaging kernel on the coarse grid is unity. Pure resampling on the information-centered grid without re-regularization would reduce the degrees of freedom to 7.1.

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