arXiv subjects

Ioannis G. Kevrekidis

Publications and source records attributed to Ioannis G. Kevrekidis.

At least 145 records · Page 8Linked to original sources

arXiv2012

An equation-free approach to coarse-graining the dynamics of networks

We propose and illustrate an approach to coarse-graining the dynamics of evolving networks (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 coarsegrained fixed point algorithms (through matrix-free Newton-Krylov GMRES). The approach is illustrated through a 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.

cs.SI↗

arXiv2011

Coarse Graining the Dynamics of Heterogeneous Oscillators in Networks with Spectral Gaps

We present a computer-assisted approach to coarse-graining the evolutionary dynamics of a system of nonidentical oscillators coupled through a (fixed) network structure. The existence of a spectral gap for the coupling network graph Laplacian suggests that the graph dynamics may quickly become low-dimensional. Our first choice of coarse variables consists of the components of the oscillator states -their (complex) phase angles- along the leading eigenvectors of this Laplacian. We then use the equation-free framework [1], circumventing the derivation of explicit coarse-grained equations, to perform computational tasks such as coarse projective integration, coarse fixed point and coarse limit cycle computations. In a second step, we explore an approach to incorporating oscillator heterogeneity in the coarse-graining process. The approach is based on the observation of fastdeveloping correlations between oscillator state and oscillator intrinsic properties, and establishes a connection with tools developed in the context of uncertainty quantification.

cond-mat.stat-mech↗

arXiv2011

Noisy dynamic simulations in the presence of symmetry: data alignment and model reduction

We process snapshots of trajectories of evolution equations with intrinsic symmetries, and demonstrate the use of recently developed eigenvector-based techniques to successfully quotient out the degrees of freedom associated with the symmetries in the presence of noise. Our illustrative examples include a one-dimensional evolutionary partial differential (the Kuramoto-Sivashinsky) equation with periodic boundary conditions, as well as a stochastic simulation of nematic liquid crystals which can be effectively modeled through a nonlinear Smoluchowski equation on the surface of a sphere. This is a useful first step towards data mining the "symmetry-adjusted" ensemble of snapshots in search of an accurate low-dimensional parametrization (and the associated reduction of the original dynamical system). We also demonstrate a technique ("vector diffusion maps") that combines, in a single formulation, the symmetry removal step and the dimensionality reduction step.

physics.comp-ph↗

arXiv2010

Manifold learning techniques and model reduction applied to dissipative PDEs

We link nonlinear manifold learning techniques for data analysis/compression with model reduction techniques for evolution equations with time scale separation. In particular, we demonstrate a `"nonlinear extension" of the POD-Galerkin approach to obtaining reduced dynamic models of dissipative evolution equations. The approach is illustrated through a reaction-diffusion PDE, and the performance of different simulators on the full and the reduced models is compared. We also discuss the relation of this nonlinear extension with the so-called "nonlinear Galerkin" methods developed in the context of Approximate Inertial Manifolds.

physics.comp-ph↗

arXiv2010

Ergodic directional switching in mobile insect groups

We obtain a Fokker-Planck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including non-monotonic behavior of several order/disorder transition indicators in noise intensity. This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the group's direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.

q-bio.PE↗

arXiv2010

An approach for both the computation of coarse-scale steady state solutions and initialization on a slow manifold

We present a simple technique for the computation of coarse-scale steady states of dynamical systems with time scale separation in the form of a "wrapper" around a fine-scale simulator. We discuss how this approach alleviates certain problems encountered by comparable existing approaches, and illustrate its use by computing coarse-scale steady states of a lattice Boltzmann fine scale code. Interestingly, in the same context of multiple time scale problems, the approach can be slightly modified to provide initial conditions (on the slow manifold) with prescribed coarse-scale observables. The approach is based on appropriately designed short bursts of the fine-scale simulator whose results are used to track changes in the coarse variables of interest, a core component of the equation-free framework.

physics.comp-ph↗

arXiv2009

Analysis and Computation of a Discrete KdV-Burgers Type Equation with Fast Dispersion and Slow Diffusion

The long time behavior of the dynamics of a fast-slow system of ordinary differential equations is examined. The system is derived from a spatial discretization of a Korteweg-de Vries-Burgers type equation, with fast dispersion and slow diffusion. The discretization is based on a model developed by Goodman and Lax, that is composed of a fast system drifted by a slow forcing term. A natural split to fast and slow state variables is, however, not available. Our approach views the limit behavior as an invariant measure of the fast motion drifted by the slow component, where the known constants of motion of the fast system are employed as slowly evolving observables; averaging equations for the latter lead to computation of characteristic features of the motion. Such computations are presented in the paper.

math.NA↗

arXiv2009

Analysis of a stochastic chemical system close to a SNIPER bifurcation of its mean-field model

A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs for example in the modelling of cell-cycle regulation. It is shown that the stochastic system possesses oscillatory solutions even for parameter values for which the mean-field model does not oscillate. The dependence of the mean period of these oscillations on the parameters of the model (kinetic rate constants) and the size of the system (number of molecules present) is studied. Our approach is based on the chemical Fokker-Planck equation. To get some insights into advantages and disadvantages of the method, a simple one-dimensional chemical switch is first analyzed, before the chemical SNIPER problem is studied in detail. First, results obtained by solving the Fokker-Planck equation numerically are presented. Then an asymptotic analysis of the Fokker-Planck equation is used to derive explicit formulae for the period of oscillation as a function of the rate constants and as a function of the system size.

physics.chem-ph↗

arXiv2009

Coarse-graining the Dynamics of a Driven Interface in the Presence of Mobile Impurities: Effective Description via Diffusion Maps

Developing effective descriptions of the microscopic dynamics of many physical phenomena can both dramatically enhance their computational exploration and lead to a more fundamental understanding of the underlying physics. Previously, an effective description of a driven interface in the presence of mobile impurities, based on an Ising variant model and a single empirical coarse variable, was partially successful; yet it underlined the necessity of selecting additional coarse variables in certain parameter regimes. In this paper we use a data mining approach to help identify the coarse variables required. We discuss the implementation of this diffusion map approach, the selection of a similarity measure between system snapshots required in the approach, and the correspondence between empirically selected and automatically detected coarse variables. We conclude by illustrating the use of the diffusion map variables in assisting the atomistic simulations, and we discuss the translation of information between fine and coarse descriptions using lifting and restriction operators.

cond-mat.stat-mech↗

arXiv2008

Exploration of Effective Potential Landscapes using Coarse Reverse Integration

We describe a reverse integration approach for the exploration of low-dimensional effective potential landscapes. Coarse reverse integration initialized on a ring of coarse states enables efficient "navigation" on the landscape terrain: escape from local effective potential wells, detection of saddle points, and identification of significant transition paths between wells. We consider several distinct ring evolution modes: backward stepping in time, solution arc--length, and effective potential. The performance of these approaches is illustrated for a deterministic problem where the energy landscape is known explicitly. Reverse ring integration is then applied to "noisy" problems where the ring integration routine serves as an outer "wrapper" around a forward-in-time inner simulator. Three versions of such inner simulators are considered: a system of stochastic differential equations, a Gillespie--type stochastic simulator, and a molecular dynamics simulator. In these "equation-free" computational illustrations, estimation techniques are applied to the results of short bursts of "inner" simulation to obtain the unavailable (in closed form) quantities (local drift and diffusion coefficient estimates) required for reverse ring integration; this naturally leads to approximations of the effective landscape.

physics.chem-ph↗

arXiv2008

Robust oscillations in SIS epidemics on adaptive networks: Coarse-graining by automated moment closure

We investigate the dynamics of an epidemiological susceptible-infected-susceptible (SIS) model on an adaptive network. This model combines epidemic spreading (dynamics on the network) with rewiring of network connections (topological evolution of the network). We propose and implement a computational approach that enables us to study the dynamics of the network directly on an emergent, coarse-grained level. The approach sidesteps the derivation of closed low-dimensional approximations. Our investigations reveal that global coupling, which enters through the awareness of the population to the disease, can result in robust large-amplitude oscillations of the state and topology of the network.

nlin.AO↗

arXiv2007

Equation-free implementation of statistical moment closures

We present a general numerical scheme for the practical implementation of statistical moment closures suitable for modeling complex, large-scale, nonlinear systems. Building on recently developed equation-free methods, this approach numerically integrates the closure dynamics, the equations of which may not even be available in closed form. Although closure dynamics introduce statistical assumptions of unknown validity, they can have significant computational advantages as they typically have fewer degrees of freedom and may be much less stiff than the original detailed model. The closure method can in principle be applied to a wide class of nonlinear problems, including strongly-coupled systems (either deterministic or stochastic) for which there may be no scale separation. We demonstrate the equation-free approach for implementing entropy-based Eyink-Levermore closures on a nonlinear stochastic partial differential equation.

cond-mat.stat-mech↗

arXiv2006

Periodically-forced finite networks of heterogeneous coupled oscillators: a low-dimensional approach

We study a network of 500 coupled modified van der Pol oscillators. The value of a parameter associated with each oscillator is drawn from a normal distribution, giving a heterogeneous network. For strong enough coupling the oscillators all have the same period, and we consider periodic forcing of the network when it is in this state. By exploiting the correlations that quickly develop between the state of an oscillator and the value of its parameter we obtain an approximate low-dimensional description of the system in terms of the first few coefficients in a polynomial chaos expansion. Standard bifurcation analysis can then be performed on this low-dimensional system, and the results obtained from this predict very well the behaviour of the high-dimensional system for any set of realisations of the random parameter. Situations in which the method begins to fails are also discussed.

math.DS↗

arXiv2006

Variable-free exploration of stochastic models: a gene regulatory network example

Finding coarse-grained, low-dimensional descriptions is an important task in the analysis of complex, stochastic models of gene regulatory networks. This task involves (a) identifying observables that best describe the state of these complex systems and (b) characterizing the dynamics of the observables. In a previous paper [13], we assumed that good observables were known a priori, and presented an equation-free approach to approximate coarse-grained quantities (i.e, effective drift and diffusion coefficients) that characterize the long-time behavior of the observables. Here we use diffusion maps [9] to extract appropriate observables ("reduction coordinates") in an automated fashion; these involve the leading eigenvectors of a weighted Laplacian on a graph constructed from network simulation data. We present lifting and restriction procedures for translating between physical variables and these data-based observables. These procedures allow us to perform equation-free coarse-grained, computations characterizing the long-term dynamics through the design and processing of short bursts of stochastic simulation initialized at appropriate values of the data-based observables.

q-bio.QM↗

arXiv2006

Heterogeneous Cell Population Dynamics: Equation-Free Uncertainty Quantification Computations

We propose a computational approach to modeling the collective dynamics of populations of coupled heterogeneous biological oscillators. In contrast to Monte Carlo simulation, this approach utilizes generalized Polynomial Chaos (gPC) to represent random properties of the population, thus reducing the dynamics of ensembles of oscillators to dynamics of their (typically significantly fewer) representative gPC coefficients. Equation-Free (EF) methods are employed to efficiently evolve these gPC coefficients in time and compute their coarse-grained stationary state and/or limit cycle solutions, circumventing the derivation of explicit, closed-form evolution equations. Ensemble realizations of the oscillators and their statistics can be readily reconstructed from these gPC coefficients. We apply this methodology to the synchronization of yeast glycolytic oscillators coupled by the membrane exchange of an intracellular metabolite. The heterogeneity consists of a single random parameter, which accounts for glucose influx into a cell, with a Gaussian distribution over the population. Coarse projective integration is used to accelerate the evolution of the population statistics in time. Coarse fixed-point algorithms in conjunction with a Poincaré return map are used to compute oscillatory solutions for the cell population and to quantify their stability.

q-bio.QM↗

arXiv2006

Coarse molecular-dynamics analysis of stress-induced structural transitions in crystals

We present a coarse molecular-dynamics (CMD) approach for the study of stress-induced structural transformations in crystals at finite temperatures. The method relies on proper choice of a coarse variable (order parameter, observable), which parameterizes the changes in effective free energy during the transformation. Results are reported for bcc-to-hcp lattice transitions under pressure. We explore coarse-variable space to reconstruct an effective free-energy landscape quantifying the relative stability of different metastable basins and locate the onset, at a critical pressure, of the bcc-to-hcp transformation.

cond-mat.mtrl-sci↗

arXiv2006

Uncertainty Quantification for Atomistic Reaction Models: An Equation-Free Stochastic Simulation Algorithm Example

We describe a computational framework linking Uncertainty Quantification (UQ) methods for continuum problems depending on random parameters with Equation-Free (EF) methods for performing continuum deterministic numerics by acting directly on atomistic/stochastic simulators. Our illustrative example is a heterogeneous catalytic reaction mechanism with an uncertain atomistic kinetic parameter; the "inner" dynamic simulator of choice is a Gillespie Stochastic Simulation Algorithm (SSA). We demonstrate UQ computations at the coarse-grained level in a nonintrusive way, through the design of brief, appropriately initialized computational experiments with the SSA code. The system is thus observed at three levels: (a) a fine scale for each stochastic simulation at each value of the uncertain parameter; (b) an intermediate coarse-grained state for the expected behavior of the SSA at each value of the uncertain parameter; and (c) the desired fully coarse-grained level: distributions of the coarse-grained behavior over the range of uncertain parameter values. The latter are computed in the form of generalized Polynomial Chaos (gPC) coefficients in terms of the random parameter. Coarse projective integration and coarse fixed point computation are employed to accelerate the computational evolution of these desired observables, to converge on random stable/unstable steady states, and to perform parametric studies with respect to other (nonrandom) system parameters.

math.DS↗

arXiv2006

Spatially Distributed Stochastic Systems: equation-free and equation-assisted preconditioned computation

Spatially distributed problems are often approximately modelled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g. concentrations). The derivation of accurate such PDEs starting from finer scale, atomistic models, and using suitable averaging, is often a challenging task; approximate PDEs are typically obtained through mathematical closure procedures (e.g. mean-field approximations). In this paper, we show how such approximate macroscopic PDEs can be exploited in constructing preconditioners to accelerate stochastic simulations for spatially distributed particle-based process models. We illustrate how such preconditioning can improve the convergence of equation-free coarse-grained methods based on coarse timesteppers. Our model problem is a stochastic reaction-diffusion model capable of exhibiting Turing instabilities.

q-bio.QM↗