An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

Roman Chapko; B. Tomas Johansson

doi:10.1002/zamm.201800102

An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

Roman Chapko, B. Tomas Johansson

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown.

Original language	English
Pages (from-to)	2010-2021
Journal	ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
Volume	98
Issue number	11
Early online date	17 Sept 2018
DOIs	https://doi.org/10.1002/zamm.201800102
Publication status	Published - 1 Nov 2018

Bibliographical note

This is the peer reviewed version of the following article: Chapko R, Johansson BT. An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation. Z Angew Math Mech. 2018;98:2010–2021, which has been published in final form at https://doi.org/10.1002/zamm.201800102. This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.

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10.1002/zamm.201800102

An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation
This is the peer reviewed version of the following article: Chapko R, Johansson BT. An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation. Z Angew Math Mech. 2018;98:2010–2021, which has been published in final form at https://doi.org/10.1002/zamm.201800102. This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.
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abstract = "An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown.",

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AB - An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown.

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An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

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