Identifying the Conductivity of a One-Dimensional Medium from Time Averaged and Undersampled Potential Data

The identification of coefficients appearing in differential equations has been extensively considered before. The direct problem we refer to is a two point initial BVP for a linear parabolic PDE in one spatial dimension. Our inverse problem (IP) consists of identifying the position dependent conductivity from potential data. Given the existence of a solution to the IP, we provide the weak counterpart of Kitamura and Nakagiri's [1977] uniqueness conditions and of the IP solution method due to Ponzini and Crosta [1988]. We first average the original PDE over time, then replace said average by the arithmetic mean of potential values in order to model data undersampling. In both cases we provide uniqueness conditions and stability estimates of the Gronwall-Bellmann type, which we apply to analytical and numerical examples. Kitamura S., Nakagiri S., 1977, SICON, 15, pp 785 - 802. Ponzini G., Crosta G., 1988, Transport in Porous Media, 3, pp 415-436.