The backward problem of parabolic equations with the measurements on a discrete set.

*(English)*Zbl 1431.65152Summary: The backward problems of parabolic equations are of interest in the study of both mathematics and engineering. In this paper, we consider a backward problem for the one-dimensional heat conduction equation with the measurements on a discrete set. The uniqueness for recovering the initial value is proved by the analytic continuation method. We discretize this inverse problem by a finite element method to deduce a severely ill-conditioned linear system of algebra equations. In order to overcome the ill-posedness, we apply the discrete Tikhonov regularization with the generalized cross validation rule to obtain a stable numerical approximation to the initial value. Numerical results for three examples are provided to show the effect of the measurement data.

##### MSC:

65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |

35R25 | Ill-posed problems for PDEs |

65F22 | Ill-posedness and regularization problems in numerical linear algebra |

35R30 | Inverse problems for PDEs |

##### Software:

Regularization tools
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\textit{J. Cheng} et al., J. Inverse Ill-Posed Probl. 28, No. 1, 137--144 (2020; Zbl 1431.65152)

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