TITLE:

Fast Krylov-secant solvers for systems of nonlinear
partial differential equations

ABSTRACT:

We introduce two inexact algorithms for the solution of large systems
of algebraic equations, both of which with the potential to achieve
higher-order convergence and based on Krylov-Broyden updates, i.e.,
Broyden updates restricted to the Krylov subspace of the inexact linear
solver.  The Krylov-Eirola-Nevanlinna algorithm is
an inexact extension of the nonlinear Eirola-Nevanlinna iteration, which is, in
turn, based on the Broyden iterative solver.  The higher-order Krylov-Newton
algorithm is an efficient inexact composite Newton method.  We study the
application of these algorithms to a set of test problems including a
nonlinear integral equation, as well as nonlinear systems arising from
the finite-difference discretization of a steady-state and a time-dependent
partial differential equations.  We show the higher robustness of the newly
proposed methods by contrasting them to the standard Newton and Broyden's
algorithms, as well as their higher computational efficiency based on runtime
operation counts.  We discuss the use of preconditioners
in the context of Krylov-Broyden updates and give theoretical interpretation
of the proposed methods within the framework of standard secant-method
theory.