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.

scholarly journals Existence of approximate solution to abstract nonlocal Cauchy problem

1992 ◽  
Vol 5 (4) ◽  
pp. 363-373 ◽  
Author(s):  
L. Byszewski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Approximate Solution ◽  
Closed Subset ◽  
Nonlocal Condition ◽  
Right Hand ◽  
The Right ◽  
Nonlocal Cauchy Problem

The aim of the paper is to prove a theorem about the existence of an approximate solution to an abstract nonlinear nonlocal Cauchy problem in a Banach space. The right-hand side of the nonlocal condition belongs to a locally closed subset of a Banach space. The paper is a continuation of papers [1], [2] and generalizes some results from [3].

scholarly journals Approximate Solution of Perturbed Volterra-Fredholm Integrodifferential Equations by Chebyshev-Galerkin Method

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
K. Issa ◽  
F. Salehi
Keyword(s):  
Approximate Solution ◽  
Galerkin Method ◽  
Integrodifferential Equation ◽  
Numerical Results ◽  
Integrodifferential Equations ◽  
Right Hand ◽  
The Right

In this work, we obtain the approximate solution for the integrodifferential equations by adding perturbation terms to the right hand side of integrodifferential equation and then solve the resulting equation using Chebyshev-Galerkin method. Details of the method are presented and some numerical results along with absolute errors are given to clarify the method. Where necessary, we made comparison with the results obtained previously in the literature. The results obtained reveal the accuracy of the method presented in this study.

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 Simplification method for nonlinear equations of monotone type in Banach space

2021 ◽  
Vol 23 (2) ◽  
pp. 185-192
Author(s):  
Irina P. Ryazantseva
Keyword(s):  
Banach Space ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Original Equation ◽  
Ill Posed ◽  
Simple Operator ◽  
The Right ◽  
Dual Mapping ◽  
The Given ◽  
Monotone Type

Abstract. In a Banach space, we study an operator equation with a monotone operator T. The operator is an operator from a Banach space to its conjugate, and T=AC, where A and C are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator T of the original equation, but a more simple operator A, which is B-monotone, B=C−1. The existence of the operator B is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

A Variant of Projection-Regularization Method for Ill-Posed Linear Operator Equations

2020 ◽  
pp. 2150008
Author(s):  
Bechouat Tahar ◽  
Boussetila Nadjib ◽  
Rebbani Faouzia
Keyword(s):  
Approximate Solution ◽  
Linear Operator ◽  
Tikhonov Regularization ◽  
Hilbert Spaces ◽  
Regularization Method ◽  
Tikhonov Regularization Method ◽  
Operator Equations ◽  
Ill Posed ◽  
Rank Approximation ◽  
Linear Operator Equations

In this paper, we report on a strategy for computing the numerical approximate solution for a class of ill-posed operator equations in Hilbert spaces: [Formula: see text]. This approach is a combination of Tikhonov regularization method and the finite rank approximation of [Formula: see text]. Finally, numerical results are given to show the effectiveness of this method.

The Canonical TSVD Method for L-Generalized Solution of Linear Ill-Posed Equations

2011 ◽  
Vol 211-212 ◽  
pp. 177-181
Author(s):  
Zhi Liu
Keyword(s):  
Approximate Solution ◽  
Regularization Method ◽  
Generalized Solution ◽  
Discrete Method ◽  
Numerical Examples ◽  
Ill Posed

The Canonical TSVD method is a kind of robust regularization method. In this paper, we first applied the Canonical TSVD method to finding L-generalized solution. We discussed a discrete method and how to compute the approximate solution of the Canonical TSVD method, then we gave some numerical examples. Finally, we drew some conclusions.

Dual Regularized Total Least Squares And Multi-Parameter Regularization

2008 ◽  
Vol 8 (3) ◽  
pp. 253-262 ◽  
Author(s):  
S. LU ◽  
S.V. PEREVERZEV ◽  
U. TAUTENHAHN
Keyword(s):  
Least Squares ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Differential Operators ◽  
Total Least Squares ◽  
Right Hand ◽  
Regularization Parameters ◽  
Ill Posed ◽  
Regularized Total Least Squares ◽  
Parameter Regularization

AbstractIn this paper we continue our study of solving ill-posed problems with a noisy right-hand side and a noisy operator. Regularized approximations are obtained by Tikhonov regularization with differential operators and by dual regularized total least squares (dual RTLS) which can be characterized as a special multi-parameter regularization method where one of the two regularization parameters is negative. We report on order optimality results for both regularized approximations, discuss compu-tational aspects, provide special algorithms and show by experiments that dual RTLS is competitive to Tikhonov regularization with differential operators.

scholarly journals On the stoppage rule in solution for monotone ill-posed problems

2016 ◽  
Vol 13 (4) ◽  
pp. 63-67
Author(s):  
Nguyễn Bường
Keyword(s):  
Banach Space ◽  
Noisy Data ◽  
Initial Problem ◽  
Right Hand ◽  
Ill Posed ◽  
The Right

The purpose of this note is to present an  iteractive method for solving a regularized equation for nonlinear monotone ill-posed problems in Banach space and  to  study its stoppage rule so that iteractive sequence converges to  a solution of initial problem,  as the noisy data of the right-hand side converges to its exact.

scholarly journals EXTRAPOLATION OF TIKHONOV REGULARIZATION METHOD

2010 ◽  
Vol 15 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Uno Hämarik ◽  
Reimo Palm ◽  
Toomas Raus
Keyword(s):  
Approximate Solution ◽  
Tikhonov Regularization ◽  
Noise Level ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Noisy Data ◽  
Tikhonov Regularization Method ◽  
Guaranteed Accuracy ◽  
Numerical Examples ◽  
Ill Posed

We consider regularization of linear ill‐posed problem Au = f with noisy data fδ, ¦fδ - f¦≤ δ . The approximate solution is computed as the extrapolated Tikhonov approximation, which is a linear combination of n ≥ 2 Tikhonov approximations with different parameters. If the solution u* belongs to R((A*A) n ), then the maximal guaranteed accuracy of Tikhonov approximation is O(δ 2/3) versus accuracy O(δ 2n/(2n+1)) of corresponding extrapolated approximation. We propose several rules for choice of the regularization parameter, some of these are also good in case of moderate over‐ and underestimation of the noise level. Numerical examples are given.

Editorial Note

1946 ◽  
Vol 11 (1) ◽  
pp. 2-2
Keyword(s):  
Editorial Note ◽  
Speech Sounds ◽  
Left Hand ◽  
Hand Column ◽  
Right Hand ◽  
Infant Speech ◽  
The Right

In the article “Infant Speech Sounds and Intelligence” by Orvis C. Irwin and Han Piao Chen, in the December 1945 issue of the Journal, the paragraph which begins at the bottom of the left hand column on page 295 should have been placed immediately below the first paragraph at the top of the right hand column on page 296. To the authors we express our sincere apologies.

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