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Accelerated iterative algorithms for the Cauchy problem in steady-state anisotropic heat conduction

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Abstract

In the framework of stationary anisotropic heat conduction (the Laplace-Beltrami equation) without heat sources, we investigate both theoretically and numerically the acceleration of the two iterative algorithms of Kozlov et al. (U.S.S.R. Computational Mathematics and Mathematical Physics 31:45–52, 1991) for the accurate, convergent and stable reconstruction of the missing temperature and normal heat flux on an inaccessible boundary of the domain occupied by a solid from the knowledge of Cauchy data on the remaining and accessible boundary. For each of the two algorithms with relaxation studied, this inverse Cauchy problem for the Laplace-Beltrami equation with exact data is transformed into an equivalent fixed point problem for an associated operator that is defined on and takes values in a suitable function space, and also accounts for the relaxation parameter. Consequently, the convergence of each relaxation algorithm reduces to analysing the properties of the corresponding operator and this enables us to determine the admissible range for the relaxation parameter, as well as a criterion for selecting its optimal value at each iteration, for each iterative procedure and exact Cauchy data. In case of perturbed Cauchy data, regularisation is achieved by terminating the iteration according to the discrepancy principle. The numerical implementation is realised for two-dimensional homogeneous anisotropic solids using the finite element method and confirms a significant reduction in the number of iterations and hence CPU time required for the two relaxation algorithms proposed to achieve convergence, for both exact and perturbed Cauchy data, provided that the dynamical selection of the optimal value for the relaxation parameter is employed.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei, 010014, Bucharest, Romania

    Mihai Bucataru & Liviu Marin

  2. Gheorghe Mihoc – Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 13 Calea 13 Septembrie, 050711, Bucharest, Romania

    Mihai Bucataru & Liviu Marin

Authors
  1. Mihai Bucataru
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  2. Liviu Marin
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M.B. and L.M. wrote the main manuscript text, whilst M.B. prepared all figures and tables. Both authors reviewed the manuscript.

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Correspondence to Liviu Marin.

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Bucataru, M., Marin, L. Accelerated iterative algorithms for the Cauchy problem in steady-state anisotropic heat conduction. Numer Algor 95, 605–636 (2024). https://doi.org/10.1007/s11075-023-01583-0

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