A Class of Active Set Randomized Iterative Methods for Solving Constrained Discrete Ill-Posed Problems

Abstract:

Linear discrete ill-posed problems arise from many scientific computation and engineering application areas. Considering the solution to large scale ill-posed problems with box constraints， A novel class of randomized internal and external iterative methods are proposed based on the active set strategy， which contain a two-level iteration that， for the external iteration， updates the active set and free variable set， and orthogonally project the iterate onto the feasible boundary， while for the internal iteration， solves the unconstrained linear system with the Krylov subspace methods. The proposed novel active set algorithm is to utilize the efficient randomized method for the internal iteration and the step size is chosen by the Armijo criterion， so that the objective function value can be monotonically decreased with the increase of the number of iterations. Numerical experiments on the image restoration show the efficiency of the proposed algorithm. Under the condition of the discrepancy principle， the computational complexity， iteration steps， and CPU time of the novel active set randomized iteration methods are less than those of previous methods.

Key words:linear discrete ill-posed problems;box constraints;randomized iteration;active method;internal and external iteration