The computation of center manifolds pertains particularly to the design of controllers for stabilizing large-scale systems possessing uncontrollable modes [1-4]. Restricting the dynamics of large-scale continuous and discrete systems on the center manifold allows the stability analysis and the control of the full system based on a low-scale system whose order is determined by the dimension of the generalized eigenspace associated to the eigenvalues with zero real part. However, an important assumption for the computation of center manifolds is that explicit, reasonably accurate closed form dynamical models as appearing in the form of e.g. a system of ordinary differential equations are available. However, for many contemporary complex systems, macroscopic equations are often not available in a closed form. In these circumstances, conventional continuum algorithms cannot be used. Furthermore for large-scale black-box legacy codes the computation of center manifolds is far from trivial. The Equation-Free aproach [5,6] can be used to establish a link between traditional continuum numerical analysis and large-scale legacy codes and microscopic/ stochastic simulation of complex systems. This mathematics assisted computational methodology enables large-scale black-box and microscopic-level simulators to perform system-level analysis directly, without the need to pass through an intermediate, coarsegrained, macroscopic-level, "conventional" description of the system dynamics. The backbone of the method is the 'on-demand' identification of the quantities required for continuum numerics (coarse residuals, the action of coarse slow Jacobians, eigenvalues, Hessians, etc). These are obtained by repeated, appropriately initialized calls to an existing time-stepping routine, which is treated as a black box. The key assumption is that deterministic, macroscopic, coarse models exist and close for the expected behavior of a few macroscopic system observables, yet they are unavailable in closed form. In this work the Equation-free concept is exploited for the approximation of the coarse-grained center manifold on co-dimension one bifurcations of black-box legacy simulators or microscopic/stochastic simulators. The simulator is treated as a constantly evolving experiment and the approximation of a local center manifold is provided in a polynomial form whose coefficients are computed by wrapping around the available simulator an optimization algorithm. This is a three-step protocol including [7] (a) the detection of the (coarse-grained) nonhyperbolic equilibrium, (b) the stability analysis of the critical point(s), and (c) the approximation of a local center manifold by identifying the quantities required for the optimization algorithm (coarse residuals, Hessians, etc). The proposed method is demonstrated through kinetic Monte Carlo simulations- of simple reactions taking place on catalytic surfaces- whose dynamics exhibit coarse-grained turning points and Andronov-Hopf criticalities.
Equation-Free Computation of Center Manifolds for Black-Box/ LargeScale Simulators
2015
Abstract
The computation of center manifolds pertains particularly to the design of controllers for stabilizing large-scale systems possessing uncontrollable modes [1-4]. Restricting the dynamics of large-scale continuous and discrete systems on the center manifold allows the stability analysis and the control of the full system based on a low-scale system whose order is determined by the dimension of the generalized eigenspace associated to the eigenvalues with zero real part. However, an important assumption for the computation of center manifolds is that explicit, reasonably accurate closed form dynamical models as appearing in the form of e.g. a system of ordinary differential equations are available. However, for many contemporary complex systems, macroscopic equations are often not available in a closed form. In these circumstances, conventional continuum algorithms cannot be used. Furthermore for large-scale black-box legacy codes the computation of center manifolds is far from trivial. The Equation-Free aproach [5,6] can be used to establish a link between traditional continuum numerical analysis and large-scale legacy codes and microscopic/ stochastic simulation of complex systems. This mathematics assisted computational methodology enables large-scale black-box and microscopic-level simulators to perform system-level analysis directly, without the need to pass through an intermediate, coarsegrained, macroscopic-level, "conventional" description of the system dynamics. The backbone of the method is the 'on-demand' identification of the quantities required for continuum numerics (coarse residuals, the action of coarse slow Jacobians, eigenvalues, Hessians, etc). These are obtained by repeated, appropriately initialized calls to an existing time-stepping routine, which is treated as a black box. The key assumption is that deterministic, macroscopic, coarse models exist and close for the expected behavior of a few macroscopic system observables, yet they are unavailable in closed form. In this work the Equation-free concept is exploited for the approximation of the coarse-grained center manifold on co-dimension one bifurcations of black-box legacy simulators or microscopic/stochastic simulators. The simulator is treated as a constantly evolving experiment and the approximation of a local center manifold is provided in a polynomial form whose coefficients are computed by wrapping around the available simulator an optimization algorithm. This is a three-step protocol including [7] (a) the detection of the (coarse-grained) nonhyperbolic equilibrium, (b) the stability analysis of the critical point(s), and (c) the approximation of a local center manifold by identifying the quantities required for the optimization algorithm (coarse residuals, Hessians, etc). The proposed method is demonstrated through kinetic Monte Carlo simulations- of simple reactions taking place on catalytic surfaces- whose dynamics exhibit coarse-grained turning points and Andronov-Hopf criticalities.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.