An adaptive heavy ball method for ill-posed inverse problems

In this paper we consider ill-posed inverse problems, both linear and nonlinear, by a heavy ball method in which a strongly convex regularization function is incorporated to detect the feature of the sought solution. We develop ideas on how to adaptively choose the step-sizes and the momentum coefficients to achieve acceleration over the Landweber-type method. We then analyze the method and establish its regularization property when it is terminated by the discrepancy principle. Various numerical results are reported which demonstrate the superior performance of our method over the Landweber-type method by reducing substantially the required number of iterations and the computational time.