Stability and complexity of global iterative solvers for the Kadanoff-Baym equations

TL;DR

This paper analyzes the stability and computational complexity of various global iterative solvers for the Kadanoff-Baym equations, highlighting their potential advantages and challenges compared to traditional time-stepping methods.

Contribution

It provides a comparative study of multiple global-in-time iterative methods, including fixed point, Jacobian-free, and Newton-Krylov approaches, for solving the Kadanoff-Baym equations.

Findings

Several iterative methods achieve stable convergence at large times

Number of iterations scales linearly with the number of time steps

Fixed point iteration often fails to converge at large times

Abstract

Although the Kadanoff-Baym equations are typically solved using time-stepping methods, iterative global-in-time solvers offer potential algorithmic advantages, particularly when combined with compressed representations of two-time objects. We examine the computational complexity and stability of several global-in-time iterative methods, including multiple variants of fixed point iteration, Jacobian-free methods, and a Newton-Krylov method using automatic differentiation. We consider the ramped and periodically-driven Falicov-Kimball and Hubbard models within time-dependent dynamical mean-field theory. Although we observe that several iterative methods yield stable convergence at large propagation times, a standard forward fixed point iteration does not. We find that the number of iterations required to converge to a given accuracy with a fixed time step size scales roughly linearly with the number of time steps. This scaling is associated with the formation of a propagating front in the residual error, whose velocity is method-dependent. We identify key challenges which must be addressed in order to make global solvers competitive with time-stepping methods.