The Levenberg–Marquardt regularization for the backward heat equation with fractional derivative

Abstract

. The backward heat problem with time-fractional derivative in Caputo’s sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg–Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a Hölder-type source condition. Numerical examples for one and two dimensions are provided.