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

作者：Bakushinsky Anatoly^*; Smirnova Alexandra; Liu Hui

来源：Journal of Inverse and Ill-Posed Problems, 2015, 23(3): 195-210.

DOI：10.1515/jiip-2013-0086

• TSVD
• irregular operator equation
• stopping rule
• image restoration
• inverse scattering problem
• iterative regularization

A problem of solving a (non)linear operator equation, F(u) = y, F : x -> y, on a pair of Hilbert spaces is addressed. A generalized Gauss-Newton scheme
u(n+1) = xi(n) - theta(F'*(u(n))F'(u(n)), tau(n))F'*(u(n)){F(u(n)) - y(delta) - F'(u(n))(u(n) - xi(n))}, u(0), xi(n) is an element of 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 coefficient is proven. A novel a posteriori stopping rule, designed to accommodate noise in both the data and the source condition, is justified. Practical aspects are illustrated with numerical simulations for a large-scale linear image de-blurring system as well as a nonlinear inverse scattering model.

• 出版日期2015-6