Sparse Regularization Of Ill-posed Problem

We prove the existence, the stability and the convergence of the regularizationmethod with two regularization parameters in topological spaces. And we derive theexistence, the stability and the convergence of the sparse regularization in the Hilbertspace. In addition, we study sufficient conditions for the penalty term that guaranteethe well-posedness of the method in the Hilbert space, and investigate to whichconditions are also necessary. Moreover, under conditions of the uniqueness ofminimizing sparse solution and the finite basis injectivity property of F, we haveshown that the minimisers ofT, y converge linearly to x as||y y||0.Finally, we give a simple summary of this article and its conclusion.