Convolution regularization method for backward problems of linear parabolic equations

In this paper, we consider a class of severely ill-posed backward problems for linear parabolic equations. We use a convolution regularization method to obtain a stable approximate initial data from the noisy final data. The convergence rates are obtained under an a priori and an a posteriori regularization parameter choice rule in which the a posteriori parameter choice is a new generalized discrepancy principle based on a modified version of Morozov's discrepancy principle. The log-type convergence order under the a priori   regularization parameter choice rule and log⁡loglog⁡log-type order under the a posteriori regularization parameter choice rule are obtained. Two numerical examples are tested to support our theoretical results.