Recovery of a coefficient in a diffusion equation from large time data^*

Abstract This paper considers the determination of a spatially varying coefficient in a parabolic equation from time trace data. There are many uniqueness theorems known for such problems but the reconstruction step is severally ill-posed: essentially the problem comes down to trying to reconstruct an analytic function from values on a strip. However, we look at an even more restricted data where the measurements are not made on the whole time axis but only for large values adding further to the ill-conditioning situation. In addition, we do not assume the initial state is known. Uniqueness is restored by making changes to the boundary condition, in particular, to the impedance parameter, for each of a series of measurements. We show that an undefined implementation of the above paradigm leads to both uniqueness and an effective reconstruction algorithm. Extension is also made to the case of fractional model and to replacing the parabolic equation with a damped wave equation.