Stochastic variance reduced gradient method for linear ill-posed inverse problems

Abstract

In this paper we apply the stochastic variance reduced gradient (SVRG) method, which is a popular variance reduction method in optimization for accelerating the stochastic gradient method, to solve large scale linear ill-posed systems in Hilbert spaces. Under {\it a priori} choices of stopping indices, we derive a convergence rate result when the sought solution satisfies a benchmark source condition and establish a convergence result without using any source condition. To terminate the method in an {\it a posteriori} manner, we consider the discrepancy principle and show that it terminates the method in finite many iteration steps almost surely. Various numerical results are reported to test the performance of the method.