Lipschitz Stability for Backward Heat Equation with Application to Fluorescence Microscopy

In this work we study a Lipschitz stability result in the reconstruction of a compactly supported initial temperature for the heat equation in Rⁿ, from measurements along a positive time interval and over an open set containing its support. We employ a nonconstructive method which ensures the existence of the stability constant, but it is not explicit in terms of the parameters of the problem. The main ingredients in our method are the compactness of support of the initial condition and the explicit dependency of solutions to the heat equation with respect to it. By means of Carleman estimates we obtain an analogous result for the case when the observation is made along an exterior region ω × (τ, T), such that the unobserved part Rⁿ\ω is bounded. In the latter setting, the method of Carleman estimates gives a general conditional logarithmic stability result when initial temperatures belong to a certain admissible set, without the assumption of compactness of support and allowing an explicit stability constant. Furthermore, we apply these results to deduce similar stability inequalities for the heat equation in R and with measurements available on a curve contained in R ×[0, ∞), leading to the derivation of stability estimates for an inverse problem arising in 2D fluorescence microscopy. In order to further understand this Lipschitz stability, in particular, the magnitude of its stability constant with respect to the parameters of the problem, a numerical reconstruction is presented based on the construction of a linear system for the inverse problem in fluorescence microscopy. We investigate the stability constant by analyzing the condition number of the corresponding matrix.