Ill-Posedness of Inverse Diffusion Problems by Jacobi's Theta Transform

Abstract

The subject is the ill-posedness degree of some inverse problems for the transient heat conduction. We focus on three of them: the completion of missing boundary data, the identification of the trajectory of a pointwise source and the recovery of the initial state. In all of these problems, the observations provide over-specified boundary data, commonly called Cauchy boundary conditions. Notice that the third problem is central for the controllability by a boundary control of the temperature. Presumably, they are all severely ill-posed, a relevant indicator on their instabilities, as formalized by G. Wahba. We revisit these issues under a new light and with different mathematical tools to provide detailed and complete proofs for these results. Jacobi Theta functions, complemented with the Jacobi Imaginary Transform, turn out to be a powerful tool to realize our objectives. In particular, based on the Laptev work [Matematicheskie Zametki 16, 741-750 (1974)], we provide new information about the observation of the initial data problem. It is actually exponentially ill-posed.