????₁-????₂ Sparse Regularization For Nonlinear Inverse Problems

In recent years,sparse regularization has attracted more and more attention due to its wide application background.The classical l1 sparse regularization methods usually can't give the most sparse solution,so the theory of non-convex sparse regularization has become a hot topic of many scholars.At present,for the non-convex sparse regularization method of linear ill posed inverse problems,there have been some researches on the regularization theory and numerical methods.However,there are few related conclusions in the study of non-convex sparse regularization of nonlinear ill posed inverse problems,so there are still many problems to be solved in the theoretical analysis and numerical methods of regularization.In this paper,we propose a non-convex sparse regularization method with ?(?)1-?(?)2 penalty terms,for the nonlinear ill posed problem,and use this sparse regularization method to solve the nonlinear compressed sensing problem.we analyze the coercivity,weak lower semi-continuity and Radon-Riesz property of ?(?)1-?(?)2 penalty terms,and discuss the well posedness of non-convex sparse regularized solutions,that is,the existence,stability and convergence of regularized solutions.The convergence rate of the regularized solution of O(?1/2)order is given under the appropriate source condition and nonlinear condition.Because the operator equation is nonlinear and the penalty term is non-convex and non-smooth,the general gradient method can't be directly applied to the ?(?)1-?(?)2 regularization function.Therefore,in this paper,the generalized conditional gradient algorithm is used to transform the original nonconvex sparse regularization functional into the structure of type F(x)+?(x),the convexity and smoothness of its simulation F(x)and penalty terms ?(x)are analyzed,a two-step soft threshold algorithm for non-convex sparse regularization is constructed,and the proof of convergence of the algorithm is given.The structure of the algorithm is simple and easy to implement,and its form is similar to the classical soft threshold algorithm.Finally,a numerical example of the algorithm in nonlinear compression sensing is given.The new algorithm proposed in this paper is compared with the classical l1 sparse regularization method to verify the effectiveness of the proposed algorithm.