A Double Sparse Regularization Approach For Ill-posed Inverse Problems

An operator equation from given noisy data and the operator is not known exactly are considered in infinite-dimensional sequence spaces (?)p with p ?(0,1).Firstly, dual regularized total least-squares is combined with l0-sparse penalty to establish the double parameters of non-convex sparse optimization problem:In particular, we consider operator equations where the operator can be characterized by a function, i.e. the operators A0 and A? can be characterized by functions k0,k?, B is a bilinear operator while p= 0, L is a bounded linear and continuously invertible operator, and ||x||0 refers to the number of nonzero elements.Next, the non-convex Tikhonov type functional was transform by means of a superposition operator into a convex formulation, existence and the stability for the method have been proved. Necessary optimality conditions in the format of complement system are obtained, A monotonically convergent scheme is used to prove that J ?,??,? is to be strictly monotone decreasing and to weakly converge to exact solution for the case p?(0,1).Bitter end, necessary optimality conditions for the case p= 0 in the format of a complementary system are obtained. A primal-dual active set strategy based on the Lagrange multiplier rule is proposed and analyzed to obtain convergence of solution to the regularization problem.