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

Roman Chapko, B. Tomas Johansson

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)2010-2021
JournalZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
Volume98
Issue number11
Early online date17 Sep 2018
DOIs
Publication statusPublished - 1 Nov 2018

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Biharmonic Equation
Iteration
Discrepancy Principle
Stopping Rule
Incomplete Data
Noisy Data
Ill-posed Problem
Reformulation
Inverse Problem
Uniqueness
Moment
Derivative
Formulation
Operator

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.

Cite this

Chapko, R., & Johansson, B. T. (2018). An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 98(11), 2010-2021. https://doi.org/10.1002/zamm.201800102
Chapko, Roman ; Johansson, B. Tomas. / An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation. In: ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. 2018 ; Vol. 98, No. 11. pp. 2010-2021.
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Chapko, R & Johansson, BT 2018, 'An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation', ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, vol. 98, no. 11, pp. 2010-2021. https://doi.org/10.1002/zamm.201800102

An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation. / Chapko, Roman; Johansson, B. Tomas.

In: ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 98, No. 11, 01.11.2018, p. 2010-2021.

Research output: Contribution to journalArticle

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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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Chapko R, Johansson BT. An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. 2018 Nov 1;98(11):2010-2021. https://doi.org/10.1002/zamm.201800102