Chebyshev pseudospectral method in the reconstruction of orthotropic conductivity

• Published in: Inverse problems in science and engineering Vol. 29; no. 5; pp. 681 - 711
• Main Authors: Boos, Everton, Luchesi, Vanda M., Bazán, Fermín S. V.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 04.05.2021
• Subjects: Chebyshev pseudospectral method; inverse problem; Levenberg-Marquardt method; Morozov's discrepancy principle; Orthotropic conductivity reconstruction
• ISSN: 1741-5977; 1741-5985
• DOI: 10.1080/17415977.2020.1801675

In this paper, we present a method to reconstruct the spatially varying conductivity tensor in isotropic and orthotropic materials, involved in a two-dimensional transient anisotropic model with Robin boundary conditions. For the reconstruction, the partial differential equation is solved by a semi-discrete method that combines a pseudospectral collocation method for spatial variables and Crank-Nicolson for time. The conductivity tensor is reconstructed through a non-linear least-squares problem solved by Levenberg-Marquardt method (LMM), along with Morozov's discrepancy principle as stopping rule to cope with noise in the data. Unlike classic LMM implementations that mitigate poor conditioning in calculating iterates using nonsingular diagonal scaling matrices, in this paper, singular regularization matrices are used. The impact of such a modification is illustrated with numerical experiments using discrete differential operators as scaling matrices. Numerical results show that accurate conductivity values can be obtained using a fairly small number of discretization points at a very low computational cost.