Ill-Posedness of Inverse Diffusion Problems by Jacobi's Theta Transform (Journal of Mathematical Study, 2018)
The subject is the ill-posedness degree of some inverse problems for thetransient heat conduction. We focus on three of them: the completion of missingboundary data, the identification of the trajectory of a pointwise source and the recoveryof the initial state. In all of these problems, the observations provide over-specifiedboundary data, commonly called Cauchy boundary conditions. Notice that the thirdproblem 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 differentmathematical tools to provide detailed and complete proofs for these results.Jacobi Theta functions, complemented with the Jacobi Imaginary Transform, turn outto 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 aboutthe observation of the initial data problem. It is actually exponentially ill-posed.
Keywords: Integral operators; regular kernels; Jacobi transform; separated variables approximation.