GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Deflation of conjugate gradients with applications to boundary value problems
SIAM Journal on Numerical Analysis
Stabilization of unstable procedures: the recursive projection method
SIAM Journal on Numerical Analysis
A flexible inner-outer preconditioned GMRES algorithm
SIAM Journal on Scientific Computing
Applied numerical linear algebra
Applied numerical linear algebra
Multigrid
SIAM Journal on Matrix Analysis and Applications
Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum
SIAM Journal on Scientific Computing
Telescopic projective methods for parabolic differential equations
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing
SIAM Journal on Scientific Computing
Inexact Krylov Subspace Methods for Linear Systems
SIAM Journal on Matrix Analysis and Applications
Inexact Matrix-Vector Products in Krylov Methods for Solving Linear Systems: A Relaxation Strategy
SIAM Journal on Matrix Analysis and Applications
Constraint-Defined Manifolds: a Legacy Code Approach to Low-Dimensional Computation
Journal of Scientific Computing
Numerical stability analysis of an acceleration scheme for step size constrained time integrators
Journal of Computational and Applied Mathematics
Second-order accurate projective integrators for multiscale problems
Journal of Computational and Applied Mathematics
Recycling Krylov Subspaces for Sequences of Linear Systems
SIAM Journal on Scientific Computing
Accuracy analysis of acceleration schemes for stiff multiscale problems
Journal of Computational and Applied Mathematics
Smooth initialization of lattice Boltzmann schemes
Computers & Mathematics with Applications
Smooth initialization of lattice Boltzmann schemes
Computers & Mathematics with Applications
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The long-term dynamic behavior of many dynamical systems evolves on a low-dimensional, attracting, invariant slow manifold, which can be parameterized by only a few variables ("observables"). The explicit derivation of such a slow manifold (and thus, the reduction of the long-term system dynamics) is often extremely difficult or practically impossible. For this class of problems, the equation-free framework has been developed to enable performing coarse-grained computations, based on short full model simulations. Each full model simulation should be initialized so that the full model state is consistent with the values of the observables and close to the slow manifold. To compute such an initial full model state, a class of constrained runs functional iterations was proposed (Gear and Kevrekidis, J. Sci. Comput. 25(1), 17---28, 2005; Gear et al., SIAM J. Appl. Dyn. Syst. 4(3), 711---732, 2005). The schemes in this class only use the full model simulator and converge, under certain conditions, to an approximation of the desired state on the slow manifold. In this article, we develop an implementation of the constrained runs scheme that is based on a (preconditioned) Newton-Krylov method rather than on a simple functional iteration. The functional iteration and the Newton-Krylov method are compared in detail using a lattice Boltzmann model for one-dimensional reaction-diffusion as the full model simulator. Depending on the parameters of the lattice Boltzmann model, the functional iteration may converge slowly or even diverge. We show that both issues are largely resolved by using the Newton-Krylov method, especially when a coarse grid correction preconditioner is incorporated.