We investigate the numerical reconstruction of the missing thermal boundary conditions on an inaccessible part of the boundary in the case of steady-state heat conduction in anisotropic solids from the knowledge of over-prescribed noisy data on the remaining accessible boundary. This inverse problem is tackled by employing a variational formulation that transforms it into an equivalent control problem; four such approaches are discussed thoroughly. The numerical implementation is realised for the 2D case via the boundary element method for perturbed Cauchy data, whilst the numerical solution is stabilised/regularised by stopping the iterative procedure according to Morozov’s discrepancy principle (Morozov, VA. On the solution of functional equations by the method of regularization. Doklady Mathematics 1966; 7: 414–417).

中文翻译：

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稳态各向异性热传导中柯西问题的 Landweber-Fridman 算法

在各向异性固体中稳态热传导的情况下，我们根据剩余可访问边界上的超规定噪声数据的知识，研究了边界不可访问部分缺失热边界条件的数值重建。这个逆问题是通过使用变分公式来解决的，该公式将其转换为等效的控制问题；对四种这样的方法进行了深入讨论。对于 2D 情况，通过扰动 Cauchy 数据的边界元方法实现数值实现，同时根据 Morozov 差异原理（Morozov, VA. On the solution of function equations by the正则化方法。Doklady 数学 1966；7：414–417）。