A nonstandard approximation of pseudoinverse and a new stopping criterion for iterative regularization

Abstract

A problem of solving a (non)linear operator equation, ( ) = , : X → Y, on a pair of Hilbert spaces is addressed. A generalized Gauss-Newton scheme0 , ∈ X, is investigated. Three basic groups of generating functions are considered. A nonstandard approximation of pseudoinverse through a "gentle" iterative truncation is presented. Its optimality on the class of generating functions with the same correctness coe cient is proven. A novel a posteriori stopping rule, designed to accommodate noise in both the data and the source condition, is justi ed. Practical aspects are illustrated with numerical simulations for a large-scale linear image de-blurring system as well as a nonlinear inverse scattering model. Mathematical preliminaries and optimality property . IntroductionConsider the inverse problem ( ) = , :where a (non)linear operator is mapping between two Hilbert spaces X, Y, and the exact data is contaminated by noise ‖ − ‖ ≤ .(1.2) Assume, for now, that is Fréchet di erentiable with a Lipschitz continuous derivative, i.e., there exist constants , ≥ 0 such thatfor any , ∈ X. One of the best known numerical algorithms for solving minimization problem (1.2) is the Gauss-Newton scheme:(1.4) It can be viewed as a simpli ed version of the full Newton method applied to the normal equation, which allows to avoid evaluation of the second derivative operator and to get a linear equation with a self-adjoint operator at every step of the iterative process. Suppose that̂ is a solution to ( ) = , maybe nonunique. In case when ὔ * (̂ ) ὔ (̂ ) is not boundedly invertible, A. Bakushinsky [3] suggested regularizing (1.4) iteratively +1 = − [ ὔ * ( ) ὔ ( ), ] ὔ * ( ){ ( ) − − ὔ ( )( − )}, 0 , ∈ X, > 0, lim →∞ = 0, (1.5)