A numerical solution of a two-dimensional IHCP. (English) Zbl 1296.65125
Authors’ abstract: We will first study the existence and uniqueness of the solution of a two-dimensional inverse heat conduction problem (IHCP) which is severely ill-posed, i.e., the solution does not depend continuously on the data. We propose a stable numerical approach based on the finite difference method and the least squares scheme to solve this problem in the presence of noisy data. We prove the convergence of the numerical solution, then, to regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. The stability and accuracy of the scheme presented is evaluated by comparison with the singular value decomposition method.
Reviewer: Christian Clason (Graz)
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 35K05 | Heat equation |
| 65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
inverse heat conduction problem; finite difference method; consistency; stability; convergence; least squares method; Tikhonov regularization method; ill-posed problem; comparison of methods; singular value decomposition methodReferences:
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