An Analysis of Equivalent Operator Preconditioning for Equation-Free Newton–Krylov Methods

Abstract

We consider the computation of a fixed point of a time-stepper using NewtonKrylov methods, and propose and analyze equivalent operator preconditioning for the resulting linear systems. For a linear, scalar advection-reaction-diffusion equation, we investigate in detail how the convergence rate depends on the choice of preconditioner parameters and on the time discretization. The results are especially valuable when computing fixed points of a coarse time-stepper in the equation-free multiscale framework, in which one simulates an unavailable coarse-scale model by wrapping a set of computational routines around appropriately initialized fine-scale simulations. Both analytical results and numerical experiments are presented, showing that one can speed up the convergence of iterative methods significantly for a wide range of parameter values in the preconditioner.

Numerical Bifurcation Analysis of PDEs From Lattice Boltzmann Model Simulations: a Parsimonious Machine Learning Approach

“…the construction of a black-box map on the macroscopic scale. By doing so, one can perform multiscale numerical analysis, even for microscopically large-scale systems tasks by exploiting the algorithms (toolkit) of matrix-free methods in the Krylov subspace [19,[23][24][25][26][27], thus bypassing the need to construct explicitly models in the form of PDEs. In the case when the macroscopic variables are not known a-priori, one can resort to nonlinear manifold learning algorithms such as Diffusion maps [28][29][30][31] to identify the intrinsic dimension of the slow manifold where the emergent dynamics evolve.…”

A Multilevel Algorithm to Compute Steady States of Lattice Boltzmann Models

“…the extraction of the coarse state at time t * + ∆t. The resulting coarse time-∆t map can then be used in conjunction with projective integration methods [7] to accelerate time integration, or with a matrix-free algorithm to directly compute coarse steady states [17][18][19]. We refer to [12] for a recent review and references to related methods.…”