Prediction-correction matrix splitting iteration algorithm for a class of large and sparse linear systems

Abstract

For the large and sparse linear systems, we utilize the efficient splittings of the system matrix and introduce an intermediate variable. The main contribution of this paper is that a prediction-correction matrix splitting iteration algorithm is constructed from the view of numerical optimization to solve the derived equation instead, which is inspired by the idea of adaptive parameter update. The novel algorithm adopts the prediction and correction two-step iteration, which uses information with delay to define the iterations. The global convergence results are established and the algorithm enjoys at least a Q-linear convergence rate under some suitable conditions. Further, a preconditioned version is also presented. Compared with some well-known algorithms, numerical experiments show the efficiency and effectiveness of the new proposal with application to the three-dimensional convection-diffusion equation and the image restoration problems.