Conditional Stability and Numerical Reconstruction of Initial Temperature

In this paper, we address an inverse problem of reconstruction ofthe initial temperature in a heat conductive system when somemeasurement data of temperatureare available, which may be observed in a subregioninside or on the boundary of the physical domain, along a timeperiod which may start at some point, possibly far away from theinitial time. A conditional stability estimate is first achieved bythe Carleman estimate for such reconstruction. Numericalreconstruction algorithm is proposed based on the outputleast-squares formulation with the Tikhonov regularization using thefinite element discretization, and the existence and convergence ofthe finite element solution are presented. Numerical experiments arecarried out to demonstrate the applicability and effectiveness ofthe proposed method.