An Inertial Splitting Algorithm For Monotone Inclusions

In this thesis,we mainly study to solve monotone inclusion problems and common solution problems in Hilbert space.The monotone inclusion problems are to find a zero point of the sum of a set-valued monotone operator and a single-valued monotone operator;the common solution problems are to find a point,which is both a solution of the monotone inclusion problems and a fixed point of nonexpansive mappings.For the common solution problems,we introduce an inertial method into the splitting algorithms,and propose a new inertial splitting algorithm.We show that if the single-valued operator is monotone and Lipschitz continuous,and the set-valued operator is maximal monotone,then the iterates generated by the algorithm weakly converge to a solution.Numerical experimental results show that the introduction of inertial methods improves the performance of the algorithm.For the monotone inclusion problems,we introduce a new inertial splitting algorithm,combining viscosity approximations and inertial methods.Then,we prove that the iterates generated by the inertial splitting method strongly converge to the solution of monotone inclusions,if the single-valued operator is monotone and Lipschitz continuous,and the set-valued operator is maximal monotone.Numerical experiments show that our algorithm has fewer iteration steps and faster convergence,compared with existing splitting algorithms.