Projected Newton method for a system of Tikhonov-Morozov equations

TL;DR

This paper introduces a Newton-type method for solving combined Tikhonov normal equations and Morozov's discrepancy principle, ensuring convergence with step size bounds and reducing computational costs via Krylov subspace projection.

Contribution

It presents a novel Newton-based approach for the combined Tikhonov-Morozov system with proven convergence and efficiency improvements through Krylov subspace projection.

Findings

Method guarantees convergence with step size bounds.

Projection onto Krylov subspace reduces computational cost.

Applicable to non-linear Tikhonov-Morozov systems.

Abstract

In this paper we derive a Newton type method to solve the non-linear system formed by combining the Tikhonov normal equations and Morozov's discrepancy principle. We prove that by placing a bound on the step size of the Newton iterations the method will always converge to the solution. By projecting the problem onto a low dimensional Krylov subspace and using the method to solve the projected non-linear system we show that we can reduce the computational cost of the method.