A comparison of the Landweber method and the Gauss--Newton method for an inverse parabolic boundary value problem
type: Item
instance of: scholarly article
title: A comparison of the Landweber method and the Gauss--Newton method for an inverse parabolic boundary value problem
zbMATH Open document ID: 1056.65093
author: Philipp Kügler
author: R. S. Chapko
published in: Journal of Computational and Applied Mathematics
publication date: 2004-08-16T00:00:00Z
review text: Thermal imaging is one of the nondestructive material testing techniques designed to generate information -- \textit{e.g.} existence of inclusion or cracks -- on the interior of a physical body. The corresponding mathematical formulation consists of an ill-posed inverse problem which gives rise to high instabilities, so that regularization techniques are required. The paper considers, associated to the linear (direct) problem \[ \left .\begin{matrix}{\partial u}/{\partial t} = \Delta u & \text{ in } & D\times (0,T) \\ u(\cdot , 0)= \;\;0 & \text{ in } & D \\ u = \;\;0 & \text{ on } &\Gamma_1\times [0,T] \\ u = \;\;f & \text{ on } &\Gamma_2\times [0,T] \\ \end{matrix}\; \right], \] an inverse problem, namely: determine \(\Gamma_1\;\) from knowledge of the heat flux \[ \frac{\partial u}{\partial \nu} = g\; \text{ on }\; \Gamma_2\times [0,T]. \] Here \(\;T>0\;\), \(\;D:=D_2\setminus \bar D_1\;\), being \(\;D_1, D_2\subset \mathbb{R}^2\;\) bounded and simply connected domains, \(\;\bar D_1\subset D_2,\;\) with boundaries \(\; \Gamma_i:=\partial D_i\;\) of class \(\;C^2\;\). Numerical experiments are carried out in order to evaluate and compare the actual performance of the Landweber and the regularized Gauss-Newton methods when applied to this inverse problem. The obtained results allow the authors to conclude that the Landweber algorithm is slow but stable in the presence of data perturbation -- independently of the initial guess. As regards to the compared performances: for convex inclusions, the Landweber method shows up to be more accurate, while the Newton technique fares better in the presence of nonconvex inclusions.
Mathematics Subject Classification ID: 65M32
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 80A23
zbMATH DE Number: 2091434
zbMATH Keywords: thermal imaging
zbMATH Keywords: boundary identification
zbMATH Keywords: Landweber method
zbMATH Keywords: heat equation
zbMATH Keywords: regularized Gauss-Newton method
zbMATH Keywords: nondestructive material testing
zbMATH Keywords: ill-posed inverse problem
zbMATH Keywords: regularization
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
full work available at URL: j.cam.2003.12.019
OpenAlex ID: W2020844590
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DOI: 10.1016/J.CAM.2003.12.019
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description: scientific article; zbMATH DE number 2091434