Abstract
We consider linear ill‐posed problems in Hilbert spaces with noisy right hand side and given noise level. For approximation of the solution the Tikhonov method or the iterated variant of this method may be used. In self‐adjoint problems the Lavrentiev method or its iterated variant are used. For a posteriori choice of the regularization parameter often quasioptimal rules are used which require computing of additionally iterated approximations. In this paper we propose for parameter choice alternative numerical schemes, using instead of additional iterations linear combinations of approximations with different parameters.
Highlights
We consider the operator equationAu = f, f ∈ R(A), (1.1)where A ∈ L(H, F ) is a linear continuous operator, and H, F are Hilbert spaces with corresponding inner products (., .) and norms
In this paper we propose for parameter choice alternative numerical schemes, using instead of additional iterations linear combinations of approximations with different parameters
This paper is devoted to several quasioptimal rules for choice of α = α(δ)
Summary
Where A ∈ L(H, F ) is a linear continuous operator, and H, F are Hilbert spaces with corresponding inner products (., .) and norms. Standard regularization method for solving problem (1.1) is the Tikhonov method uα = (αI + A∗A)−1A∗fδ. The accuracy of this approximation may be increased by iteration. With generating function gα(λ) = (α + λ)−1 in case of Tikhonov and Lavrentiev m method and gα(λ). This paper is devoted to several quasioptimal rules for choice of α = α(δ). These rules require computing of iterated approximations. In this paper we propose for parameter choice alternative numerical schemes, where instead of additional iterations linear combinations of approximations with different parameters are used.
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