Cubic convergence order yielding iterative regularization methods for ill-posed Hammerstein type operator equations

Argyros, Ioannis K and George, Santhosh and Shobha, Monnanda Erappa Cubic convergence order yielding iterative regularization methods for ill-posed Hammerstein type operator equations. Rendiconti del Circolo Matematico di Palermo. ISSN 0009-725X

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Abstract

For the solution of nonlinear ill-posed problems, a Two Step Newton-Tikhonov methodology is proposed. Two implementations are discussed and applied to nonlinear illposed Hammerstein type operator equations KF(x) = y, where K defines the integral operator and F the function of the solution x on which K operates. In the first case, the Fr´echet derivative of F is invertible in a neighbourhood which includes the initial guess x0 and the solution ˆ x. In the second case, F is monotone. For both cases, local cubic convergence is established and order optimal error bounds are obtained by choosing the regularization parameter according to the the balancing principle of Pereverzev and Schock (2005).We also present the results of computational experiments giving the evidence of the reliability of our approach

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