Linear Operator Equations in Hilbert Spaces

2009 ◽  
pp. 1-7
Author(s):  
Pavel B. Bochev ◽  
Max D. Gunzburger
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Operator Equations ◽  
Linear Operator Equations

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.

scholarly journals On ill-posedness concepts, stable solvability and saturation

2018 ◽  
Vol 26 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Bernd Hofmann ◽  
Robert Plato
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Solution Space ◽  
Nonlinear Problems ◽  
Special Role ◽  
Well Posedness ◽  
Operator Equations ◽  
Nonlinear Operators ◽  
Nonlinear Operator Equations ◽  
Linear Operator Equations

AbstractWe consider different concepts of well-posedness and ill-posedness and their relations for solving nonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard and Nashed are recalled which are appropriate for linear operator equations. For nonlinear operator equations, stable respective unstable solvability is considered, and the properties of local well-posedness and ill-posedness are investigated. Those two concepts consider stability in image space and solution space, respectively, and both seem to be appropriate concepts for nonlinear operators which are not onto and/or not, locally or globally, injective. Several example situations for nonlinear problems are considered, including the prominent autoconvolution problems and other quadratic equations in Hilbert spaces. It turns out that for linear operator equations, well-posedness and ill-posedness are global properties valid for all possible solutions, respectively. The special role of the nullspace is pointed out in this case. Finally, non-injectivity also causes differences in the saturation behavior of Tikhonov and Lavrentiev regularization of linear ill-posed equations. This is examined at the end of this study.

On nonnegative solvability of linear operator equations

1994 ◽  
Vol 18 (1) ◽  
pp. 88-108 ◽  
Author(s):  
Ruey-Jen Jang-Lewis ◽  
Harold Dean Victory
Keyword(s):  
Linear Operator ◽  
Operator Equations ◽  
Linear Operator Equations
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