Backward step control for Hilbert space problems

TL;DR

This paper develops a backward step control method for solving nonlinear equations in Banach and Hilbert spaces, ensuring global convergence and efficient residual reduction, with applications to nonlinear elliptic boundary value problems.

Contribution

It introduces a backward step control approach for Hilbert space problems, providing theoretical convergence guarantees and practical numerical methods for nonlinear equations.

Findings

Guaranteed bounds on residual norm decrease

Efficient Krylov-Newton and multilevel methods demonstrated

Numerical results for Carrier and minimum surface equations

Abstract

We analyze backward step control globalization for finding zeros of G\^ateaux-differentiable functions that map from a Banach space to a Hilbert space. The results include global convergence to a distinctive solution characterized by propagating the initial guess by a generalized Newton flow with guaranteed bounds on the discrete nonlinear residual norm decrease and an (also numerically) easily controllable asymptotic linear residual convergence rate. The convergence theory can be exploited to construct efficient numerical methods, which we demonstrate for the case of a Krylov-Newton method and an approximation-by-discretization multilevel framework. Both approaches optimize the asymptotic linear residual convergence rate, either over the Krylov subspace or through adaptive discretization, which in turn yields practical and efficient stopping criteria and refinement strategies that balance the nonlinear residuals with the relative residuals of the linear systems. We apply these methods to the class of nonlinear elliptic boundary value problems and present numerical results for the Carrier equation and the minimum surface equation.