On the second-order asymptotical regularization of linear ill-posed inverse problems

In this paper, we establish an initial theory regarding the second-order asymptotical regularization (SOAR) method for the stable approximate solution of ill-posed linear operator equations in Hilbert spaces, which are models for linear inverse problems with applications in the natural sciences, ima...

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Bibliographic Details
Published in:Applicable analysis Vol. 99; no. 6; pp. 1000 - 1025
Main Authors: Zhang, Y., Hofmann, B.
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 25.04.2020
Taylor & Francis Ltd
Subjects:
Algorithms
Asymptotic properties
asymptotical regularization
Convergence
convergence rate
discrepancy principle
Hilbert space
Ill posed problems
index function
Inverse problems
Iterative methods
Linear ill-posed problems
Linear operators
Michael Klibanov
qualification
Regularization
second-order method
source condition
ISSN:0003-6811, 1563-504X, 1563-504X
Online Access:Get full text
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Summary:In this paper, we establish an initial theory regarding the second-order asymptotical regularization (SOAR) method for the stable approximate solution of ill-posed linear operator equations in Hilbert spaces, which are models for linear inverse problems with applications in the natural sciences, imaging and engineering. We show the regularizing properties of the new method, as well as the corresponding convergence rates. We prove that, under the appropriate source conditions and by using Morozov's conventional discrepancy principle, SOAR exhibits the same power-type convergence rate as the classical version of asymptotical regularization (Showalter's method). Moreover, we propose a new total energy discrepancy principle for choosing the terminating time of the dynamical solution from SOAR, which corresponds to the unique root of a monotonically non-increasing function and allows us to also show an order optimal convergence rate for SOAR. A damped symplectic iterative regularizing algorithm is developed for the realization of SOAR. Several numerical examples are given to show the accuracy and the acceleration effect of the proposed method. A comparison with other state-of-the-art methods are provided as well.
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ISSN:0003-6811
1563-504X
1563-504X
DOI:10.1080/00036811.2018.1517412