scholarly journals Optimisation in the regularisation ill-posed problems

1986 ◽  
Vol 28 (1) ◽  
pp. 114-133 ◽  
Author(s):  
A. R. Davies ◽  
R. S. Anderssen
Keyword(s):  
Maximum Likelihood ◽  
Tikhonov Regularization ◽  
Cross Validation ◽  
Discrepancy Principle ◽  
Asymptotic Estimates ◽  
Deconvolution Problem ◽  
Ill Posed ◽  
The Relationship ◽  
Asymptotic Analyses ◽  
Selection Of

We survey the role played by optimization in the choice of parameters for Tikhonov regularization of first-kind integral equations. Asymptotic analyses are presented for a selection of practical optimizing methods applied to a model deconvolution problem. These methods include the discrepancy principle, cross-validation and maximum likelihood. The relationship between optimality and regularity is emphasized. New bounds on the constants appearing in asymptotic estimates are presented.

scholarly journals On tensor GMRES and Golub-Kahan methods via the T-product for color image processing

2021 ◽  
Vol 37 ◽  
pp. 524-543
Author(s):  
Mohamed El Guide ◽  
Alaa El Ichi ◽  
Khalide Jbilou ◽  
Rachid Sadaka
Keyword(s):  
Tikhonov Regularization ◽  
Cross Validation ◽  
Krylov Subspace ◽  
Color Image ◽  
Regularization Parameter ◽  
Color Image Processing ◽  
Regularization Technique ◽  
Image Sequence Processing ◽  
Ill Posed ◽  
Selection Of

The present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear tensor equations. We use the T-product for two tensors to define tensor tubal global Arnoldi and tensor tubal global Golub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based global approaches can be exploited to solve ill-posed problems arising from recovering blurry multichannel (color) images and videos, using the so-called Tikhonov regularization technique, to provide computable approximate regularized solutions. We also review a generalized cross-validation and discrepancy principle type of criterion for the selection of the regularization parameter in the Tikhonov regularization. Applications to image sequence processing are given to demonstrate the efficiency of the algorithms.

scholarly journals Morozov’s Discrepancy Principle For The Tikhonov Regularization Of Exponentially Ill-Posed Problems

2008 ◽  
Vol 8 (1) ◽  
pp. 86-98 ◽  
Author(s):  
S.G. SOLODKY ◽  
A. MOSENTSOVA
Keyword(s):  
Tikhonov Regularization ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
Operator Equations ◽  
Order Of Accuracy ◽  
Finite Dimensional ◽  
Dimensional Version ◽  
Ill Posed ◽  
Linear Operator Equations ◽  
Morozov’S Discrepancy Principle

Abstract The problem of approximate solution of severely ill-posed problems given in the form of linear operator equations of the first kind with approximately known right-hand sides was considered. We have studied a strategy for solving this type of problems, which consists in combinating of Morozov’s discrepancy principle and a finite-dimensional version of the Tikhonov regularization. It is shown that this combination provides an optimal order of accuracy on source sets

scholarly journals DISCREPANCY SETS FOR COMBINED LEAST SQUARES PROJECTION AND TIKHONOV REGULARIZATION

2017 ◽  
Vol 22 (2) ◽  
pp. 202-212
Author(s):  
Teresa Reginska
Keyword(s):  
Least Squares ◽  
Tikhonov Regularization ◽  
Discrepancy Principle ◽  
Order Of Accuracy ◽  
Finite Dimensional ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Data Error ◽  
Two Parameter ◽  
Regularized Solution

To solve a linear ill-posed problem, a combination of the finite dimensional least squares projection method and the Tikhonov regularization is considered. The dimension of the projection is treated as the second parameter of regularization. A two-parameter discrepancy principle defines a discrepancy set for any data error bound. The aim of the paper is to describe this set and to indicate its subset such that for regularization parameters from this subset the related regularized solution has the same order of accuracy as the Tikhonov regularization with the standard discrepancy principle but without any discretization.

Learning Susceptibility of a Pathogen to Antibiotics Using Data from Similar Pathogens

2009 ◽  
Vol 48 (03) ◽  
pp. 242-247 ◽  
Author(s):  
S. Andreassen ◽  
L. Leibovici ◽  
M. Paul ◽  
A. Zalounina
Keyword(s):  
Maximum Likelihood ◽  
Antibiotic Therapy ◽  
Cross Validation ◽  
Optimal Size ◽  
Empirical Antibiotic Therapy ◽  
Training Set ◽  
Validation Set ◽  
Using Data ◽  
Selection Of

Summary Objectives: Selection of empirical antibiotic therapy relies on knowledge of the in vitro susceptibilities of potential pathogens to antibiotics. In this paper the limitations of this knowledge are outlined and a method that can reduce some of the problems is developed. Methods: We propose hierarchical Dirichlet learning for estimation of pathogen susceptibilities to antibiotics, using data from a group of similar pathogens in a bacteremia database. Results: A threefold cross-validation showed that maximum likelihood (ML) estimates of susceptibilities based on individual pathogens gave a distance between estimates obtained from the training set and observed frequencies in the validation set of 16.3%. Estimates based on the initial grouping of pathogens gave a distance of 16.7%. Dirichlet learning gave a distance of 15.6%. Inspection of the pathogen groups led to subdivision of three groups, Citrobacter, Other Gram Negatives and Acinetobacter, out of 26 groups. Estimates based on the subdivided groups gave a distance of 15.4% and Dirichlet learning further reduced this to 15.0%. The optimal size of the imaginary sample inherited from the group was 3. Conclusion: Dirichlet learning improved estimates of susceptibilities relative to ML estimators based on individual pathogens and to classical grouped estimators. The initial pathogen grouping was well founded and improvement by subdivision of the groups was only obtained in three groups. Dirichlet learning was robust to these revisions of the grouping, giving improved estimates in both cases, while the group-based estimates only gave improved estimates after the revision of the groups.

scholarly journals ON MOROZOV’S DISCREPANCY PRINCIPLE FOR NONLINEAR ILL-POSED EQUATIONS

2009 ◽  
Vol 79 (2) ◽  
pp. 337-342 ◽  
Author(s):  
M. T. NAIR
Keyword(s):  
Tikhonov Regularization ◽  
Nonlinear Problems ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Operator Equations ◽  
Ill Posed ◽  
Morozov’S Discrepancy Principle ◽  
General Source

AbstractMorozov’s discrepancy principle is one of the simplest and most widely used parameter choice strategies in the context of regularization of ill-posed operator equations. Although many authors have considered this principle under general source conditions for linear ill-posed problems, such study for nonlinear problems is restricted to only a few papers. The aim of this paper is to apply Morozov’s discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems under general source conditions.

IMPROVEMENT OF SPRING DURUM WHEAT SELECTION METHODOLOGIES UNDER CLIMATE CHANGE CONDITIONS

1970 ◽  
pp. 33-36
Author(s):  
А. I. Grabovets ◽  
V. P. Kadushkina ◽  
S. А. Kovalenko
Keyword(s):  
Grain Yield ◽  
Durum Wheat ◽  
Correlation Coefficient ◽  
Harvest Index ◽  
Unit Area ◽  
Correlation Coefficients ◽  
Chemical Mutagenesis ◽  
Main Method ◽  
The Relationship ◽  
Selection Of

With the growing aridity of the climate on the Don, it became necessary to improve the methodology for conducting the  breeding of spring durum wheat. The main method of obtaining the source material remains intraspecific step hybridization. Crossings were performed between genetically distant forms, differing in origin and required traits and properties. The use of chemical mutagenesis was a productive way to change the heredity of genotypes in terms of drought tolerance. When breeding for productivity, both in dry years of research and in favorable years, the most objective markers were identified — the size of the aerial mass, the mass of grain per plant, spike, and harvest index. The magnitude of the correlation coefficients between the yield per unit area and the elements of its structure is established. It was most closely associated with them in dry years, while in wet years it decreased. Power the correlation of the characteristics of the pair - the grain yield per square meter - the aboveground biomass averaged r = 0.73, and in dry years it was higher (0.91) than in favorable ones (0.61 - 0.70) , between the harvest and the harvest index - r = 0.81 (on average). In dry years, the correlation coefficient increased to 0.92. Research data confirms the greatest importance of the mass of grain from one ear and the plant in the formation of grain yield per unit area in both dry and wet years. In dry years, the correlation coefficient between yield and grain mass per plant was on average r = 0.80; in favorable years, r = 0.69. The relationship between yield and grain mass from the ear was greater — r = 0.84 and r = 0.82, respectively. Consequently, the breeding significance of the aboveground mass and the productivity of the ear, as a criterion for the selection of the crop, especially increases in the dry years. They were basic in the selection.

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