A comparison of the Landweber method and the Gauss--Newton method for an inverse parabolic boundary value problem
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.
scientific article; zbMATH DE number 2091434