Discrepancy principle for generalized GN iterations combined with the reverse connection control

作者：Bakushinsky Anatoly^*; Smirnova Alexandra

来源：Journal of Inverse and Ill-Posed Problems, 2010, 18(4): 421-431.

DOI：10.1515/JIIP.2010.019

• Stopping rule
• discrepancy principle
• ill-posed problem
• iterative regularization
• Frechet derivative
• Gauss-Newton's method

In this paper we investigate the generalized Gauss-Newton method in the following form
x(n+1) = xi(n) - theta(F'*(x(n)) F'(x(n)), tau(n)) F'*(x(n)){(F(x(n)) - f delta) - F'(x(n))(x(n) - xi(n))}. x(0,) xi(n) is an element of D subset of H-1
The modified source condition
(x) over cap = xi(n) - is an element of(F'*(x(n)) F'(x(n)))S-p(n), p >= 1/2.
which depends on the current iteration point xn, is used. We call this inclusion the undetermined reverse connection. The new source condition leads to a much larger set of admissible control elements xi(n) as compared to the previously studied versions, where xi(n) = xi. The process is combined with a novel a posteriori stopping rule, where N = N (delta, f delta) is the number of the first transition of parallel to F(x(n)) - f delta parallel to through the given level delta(omega) 0 < omega < 1, i.e.,
parallel to F(x(N(delta,f delta))) - f delta parallel to <= delta(omega) < parallel to F(x(n)) - f delta parallel to, 0 <= n < N (delta, f delta).
The convergence analysis of the proposed algorithm is given.

• 出版日期2010-10