Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold
Antonios Zagaris Christophe Vandekerckhove C. William Gear Tasso J. Kaper Ioannis G. Kevrekidis
In [C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes, SIAM J. Appl. Dyn. Syst. 4 (2005), 711--732], we developed the family of constrained runs algorithms to find points on low-dimensional, attracting, slow manifolds in systems of nonlinear differential equations with multiple time scales. For user-specified values of a subset of the system variables parametrizing the slow manifold (which we term observables and denote collectively by $u$), these iterative algorithms return values of the remaining system variables $v$ so that the point $(u,v)$ approximates a point on a slow manifold. In particular, the $m-$th constrained runs algorithm ($m = 0, 1, \ldots$) approximates a point $(u,v_m)$ that is the appropriate zero of the $(m+1)-$st time derivative of $v$. % The accuracy with which $(u,v_m)$ approximates the corresponding point on the slow manifold with the same value of the observables has been established in [A. Zagaris, C. W. Gear, T. J. Kaper and I. G. Kevrekidis, Analysis of the accuracy and convergence of equation-free projection to a slow manifold, ESAIM: M2AN 43(4) (2009) 757--784] for systems for which the observables $u$ evolve exclusively on the slow time scale. There, we also determined explicit conditions under which the $m-$th constrained runs scheme converges to the fixed point $(u,v_m)$ and identified conditions under which it fails to converge. Here, we consider the questions of stability and stabilization of these iterative algorithms for the case in which the observables $u$ are also allowed to evolve on a fast time scale. The stability question in this case is more complicated, since it involves a generalized eigenvalue problem for a pair of matrices encoding geometric and dynamical characteristics of the system of differential equations. We determine the conditions under which these schemes converge or diverge in a series of cases in which this problem is explicitly solvable. We illustrate our main stability and stabilization results for the constrained runs schemes on certain planar systems with multiple time scales, and also on a more-realistic sixth order system with multiple time scales that models a network of coupled enzymatic reactions. Finally, we consider the issue of stabilization of the $m-$th constrained runs algorithm when the functional iteration scheme is divergent or converges slowly. In that case, we demonstrate on concrete examples how Newton's method and Broyden's method may replace functional iteration to yield stable iterative schemes.
keywords: DAEs inertial manifolds. singular perturbations legacy codes Iterative initialization
An equation-free approach to coarse-graining the dynamics of networks
Katherine A. Bold Karthikeyan Rajendran Balázs Ráth Ioannis G. Kevrekidis
We propose and illustrate an approach to coarse-graining the dynamics of evolving networks, i.e., networks whose connectivity changes dynamically. The approach is based on the equation-free framework: short bursts of detailed network evolution simulations are coupled with lifting and restriction operators that translate between actual network realizations and their appropriately chosen coarse observables. This framework is used here to accelerate temporal simulations through coarse projective integration, and to implement coarse-grained fixed point algorithms through matrix-free Newton-Krylov. The approach is illustrated through a very simple network evolution example, for which analytical approximations to the coarse-grained dynamics can be independently obtained, so as to validate the computational results. The scope and applicability of the approach, as well as the issue of selection of good coarse observables are discussed.
keywords: Complex networks coarsegraining network evolution equation-free approach graph limits.

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