arXiv subjects

Ioannis G. Kevrekidis

Publications and source records attributed to Ioannis G. Kevrekidis.

At least 163 records · Page 9Linked to original sources

arXiv2006

Projective and Coarse Projective Integration for Problems with Continuous Symmetries

Temporal integration of equations possessing continuous symmetries (e.g. systems with translational invariance associated with traveling solutions and scale invariance associated with self-similar solutions) in a ``co-evolving'' frame (i.e. a frame which is co-traveling, co-collapsing or co-exploding with the evolving solution) leads to improved accuracy because of the smaller time derivative in the new spatial frame. The slower time behavior permits the use of {\it projective} and {\it coarse projective} integration with longer projective steps in the computation of the time evolution of partial differential equations and multiscale systems, respectively. These methods are also demonstrated to be effective for systems which only approximately or asymptotically possess continuous symmetries. The ideas of projective integration in a co-evolving frame are illustrated on the one-dimensional, translationally invariant Nagumo partial differential equation (PDE). A corresponding kinetic Monte Carlo model, motivated from the Nagumo kinetics, is used to illustrate the coarse-grained method. A simple, one-dimensional diffusion problem is used to illustrate the scale invariant case. The efficiency of projective integration in the co-evolving frame for both the macroscopic diffusion PDE and for a random-walker particle based model is again demonstrated.

math.DS↗

arXiv2006

Coarse Molecular-Dynamics Determination of the Onset of Structural Transitions: Melting of Crystalline Solids

Using a coarse molecular-dynamics (CMD) approach with an appropriate choice of coarse variable (order parameter), we map the underlying effective free-energy landscape for the melting of a crystalline solid. Implementation of this approach provides a means for constructing effective free-energy landscapes of structural transitions in condensed matter. The predictions of the approach for the thermodynamic melting point of a model silicon system are in excellent agreement with those of ''traditional'' techniques for melting-point calculations, as well as with literature values.

cond-mat.mtrl-sci↗

arXiv2006

Dynamics of polydisperse irreversible adsorption: a pharmacological example

Many drug delivery systems suffer from undesirable interactions with the host immune system. It has been experimentally established that covalent attachment (irreversible adsorption) of suitable macromolecules to the surface of the drug carrier can reduce such undesirable interactions. A fundamental understanding of the adsorption process is still lacking. In this paper, the classical random irreversible adsorption model is generalized to capture certain essential processes involved in pharmacological applications, allowing for macromolecules of different sizes, partial overlapping of the tails of macromolecules, and the influence of reactions with the solvent on the adsorption process. Working in one dimension, an integro-differential evolution equation for the adsorption process is derived and the asymptotic behaviour of the surface area covered and the number of molecules attached to the surface is studied. Finally, equation-free dynamic renormalization tools are applied to study the asymptotically self-similar behaviour of the adsorption statistics.

physics.bio-ph↗

arXiv2005

Equation-Free Particle-Based Computations: Coarse Projective Integration and Coarse Dynamic Renormalization in 2D

Equation-free approaches have been proposed in recent years for the computational study of multiscale phenomena in engineering problems where evolution equations for the coarse-grained, system-level behavior are not explicitly available. In this paper we study the dynamics of a diffusive particle system in a laminar shear flow, described by a two-dimensional Brownian motion; in particular, we perform coarse projective integration and demonstrate the particle-based computation of coarse self-similar and asymptotically self-similar solutions for this problem. We use marginal and conditional Inverse Cumulative Distribution Functions (ICDFs) as the macroscopic observables of the evolving particle distribution.

math.DS↗

arXiv2005

Coarse Nonlinear Dynamics and Metastability of Filling-Emptying Transitions: Water in Carbon Nanotubes

Using a Coarse-grained Molecular Dynamics (CMD) approach we study the apparent nonlinear dynamics of water molecules filling/emptying carbon nanotubes as a function of system parameters. Different levels of the pore hydrophobicity give rise to tubes that are empty, water-filled, or fluctuate between these two long-lived metastable states. The corresponding coarse-grained free energy surfaces and their hysteretic parameter dependence are explored by linking MD to continuum fixed point and bifurcation algorithms. The results are validated through equilibrium MD simulations.

cond-mat.stat-mech↗

arXiv2005

Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation

We present computer-assisted methods for analyzing stochastic models of gene regulatory networks. The main idea that underlies this equation-free analysis is the design and execution of appropriately-initialized short bursts of stochastic simulations; the results of these are processed to estimate coarse-grained quantities of interest, such as mesoscopic transport coefficients. In particular, using a simple model of a genetic toggle switch, we illustrate the computation of an effective free energy and of a state-dependent effective diffusion coefficient that characterize an unavailable effective Fokker-Planck equation. Additionally we illustrate the linking of equation-free techniques with continuation methods for performing a form of stochastic "bifurcation analysis"; estimation of mean switching times in the case of a bistable switch is also implemented in this equation-free context. The accuracy of our methods is tested by direct comparison with long-time stochastic simulations. This type of equation-free analysis appears to be a promising approach to computing features of the long-time, coarse-grained behavior of certain classes of complex stochastic models of gene regulatory networks, circumventing the need for long Monte Carlo simulations.

physics.bio-ph↗

arXiv2005

Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck operators

This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency matrix of all points, we define a diffusion distance between any two data points and show that the low dimensional representation of the data by the first few eigenvectors of the corresponding Markov matrix is optimal under a certain mean squared error criterion. Furthermore, assuming that data points are random samples from a density $p(\x) = e^{-U(\x)}$ we identify these eigenvectors as discrete approximations of eigenfunctions of a Fokker-Planck operator in a potential $2U(\x)$ with reflecting boundary conditions. Finally, applying known results regarding the eigenvalues and eigenfunctions of the continuous Fokker-Planck operator, we provide a mathematical justification for the success of spectral clustering and dimensional reduction algorithms based on these first few eigenvectors. This analysis elucidates, in terms of the characteristics of diffusion processes, many empirical findings regarding spectral clustering algorithms.

math.NA↗

arXiv2005

Equation-free Dynamic Renormalization of a KPZ-type Equation

In the context of equation-free computation, we devise and implement a procedure for using short-time direct simulations of a KPZ type equation to calculate the self-similar solution for its ensemble averaged correlation function. The method involves "lifting" from candidate pair-correlation functions to consistent realization ensembles, short bursts of KPZ-type evolution, and appropriate rescaling of the resulting averaged pair correlation functions. Both the self-similar shapes and their similarity exponents are obtained at a computational cost significantly reduced to that required to reach saturation in such systems.

cond-mat.stat-mech↗

arXiv2005

An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal

The movement of many organisms can be described as a random walk at either or both the individual and population level. The rules for this random walk are based on complex biological processes and it may be difficult to develop a tractable, quantitatively-accurate, individual-level model. However, important problems in areas ranging from ecology to medicine involve large collections of individuals, and a further intellectual challenge is to model population-level behavior based on a detailed individual-level model. Because of the large number of interacting individuals and because the individual-level model is complex, classical direct Monte Carlo simulations can be very slow, and often of little practical use. In this case, an equation-free approach may provide effective methods for the analysis and simulation of individual-based models. In this paper we analyze equation-free coarse projective integration. For analytical purposes, we start with known partial differential equations describing biological random walks and we study the projective integration of these equations. In particular, we illustrate how to accelerate explicit numerical methods for solving these equations. Then we present illustrative kinetic Monte Carlo simulations of these random walks and show a decrease in computational time by as much as a factor of a thousand can be obtained by exploiting the ideas developed by analysis of the closed form PDEs. The illustrative biological example here is chemotaxis, but it could be any random walker which biases its movement in response to environmental cues.

physics.bio-ph↗

arXiv2005

Core Collapse via Coarse Dynamic Renormalization

In the context of the recently developed "equation-free" approach to computer-assisted analysis of complex systems, we extract the self-similar solution describing core collapse of a stellar system from numerical experiments. The technique allows us to side-step the core "bounce" that occurs in direct N-body simulations due to the small-N correlations that develop in the late stages of collapse, and hence to follow the evolution well into the self-similar regime.

astro-ph↗

arXiv2005

Diffusion maps, spectral clustering and reaction coordinates of dynamical systems

A central problem in data analysis is the low dimensional representation of high dimensional data, and the concise description of its underlying geometry and density. In the analysis of large scale simulations of complex dynamical systems, where the notion of time evolution comes into play, important problems are the identification of slow variables and dynamically meaningful reaction coordinates that capture the long time evolution of the system. In this paper we provide a unifying view of these apparently different tasks, by considering a family of {\em diffusion maps}, defined as the embedding of complex (high dimensional) data onto a low dimensional Euclidian space, via the eigenvectors of suitably defined random walks defined on the given datasets. Assuming that the data is randomly sampled from an underlying general probability distribution $p(\x)=e^{-U(\x)}$, we show that as the number of samples goes to infinity, the eigenvectors of each diffusion map converge to the eigenfunctions of a corresponding differential operator defined on the support of the probability distribution. Different normalizations of the Markov chain on the graph lead to different limiting differential operators. One normalization gives the Fokker-Planck operators with the same potential U(x), best suited for the study of stochastic differential equations as well as for clustering. Another normalization gives the Laplace-Beltrami (heat) operator on the manifold in which the data resides, best suited for the analysis of the geometry of the dataset, regardless of its possibly non-uniform density.

math.NA↗

arXiv2005

An equation-free approach to coupled oscillator dynamics: the Kuramoto model example

We present an equation-free multi-scale approach to the computational study of the collective dynamics of the Kuramoto model [{\it Chemical Oscillations, Waves, and Turbulence}, Springer-Verlag (1984)], a prototype model for coupled oscillator populations. Our study takes place in a reduced phase space of coarse-grained "observables" of the system: the first few moments of the oscillator phase angle distribution. We circumvent the derivation of explicit dynamical equations (approximately) governing the evolution of these coarse-grained macroscopic variables; instead we use the {\it equation-free framework} [Kevrekidis {\it et al.}, {\it Comm. Math. Sci.} {\bf 1}(4), 715 (2003)] to computationally solve these equations without obtaining them in closed form. In this approach, the numerical tasks for the conceptually existing but unavailable coarse-grained equations are implemented through short bursts of appropriately initialized simulations of the "fine-scale", detailed coupled oscillator model. Coarse projective integration and coarse fixed point computations are illustrated.

nlin.AO↗

arXiv2004

Patch dynamics with buffers for homogenization problems

An important class of problems exhibits smooth behaviour on macroscopic space and time scales, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, an "equation-free" framework has been proposed, of which patch dynamics is an essential component. Patch dynamics is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes (patches), which cover only a fraction of the space-time domain. To reduce the effect of the artificially introduced box boundaries, we use buffer regions to "shield" the boundary artefacts from the interior of the domain for short time intervals. We analyze the accuracy of this scheme for a diffusion homogenization problem with periodic heterogeneity, and propose a simple heuristic to determine a sufficient buffer size. The algorithm performance is illustrated through a set of numerical examples, which include a non-linear reaction-diffusion equation and the Kuramoto--Sivashinsky equation.

physics.comp-ph↗

arXiv2004

Equation-free optimal switching policies for bistable reacting systems using coarse time-steppers

We present a computer-assisted approach to locating approximate coarse optimal switching policies between stationary states of chemically reacting systems described by microscopic/stochastic evolution rules. The ``coarse time-stepper" constitutes a bridge between the underlying kinetic Monte Carlo simulation and traditional, continuum numerical optimization techniques formulated in discrete time. The approach is illustrated through two simple catalytic surface reaction models, implemented through kinetic Monte Carlo: NO reduction on Pt, and CO oxidation on Pt. The objective sought in both cases is to switch between two coexisting stable stationary states by minimal manipulation of a macroscopic system parameter.

math.OC↗

arXiv2004

Coarse Grained Computations for a Micellar System

We establish, through coarse-grained computation, a connection between traditional, continuum numerical algorithms (initial value problems as well as fixed point algorithms) and atomistic simulations of the Larson model of micelle formation. The procedure hinges on the (expected) evolution of a few slow, coarse-grained mesoscopic observables of the MC simulation, and on (computational) time scale separation between these and the remaining "slaved", fast variables. Short bursts of appropriately initialized atomistic simulation are used to estimate the (coarse-grained, deterministic) local dynamics of the evolution of the observables. These estimates are then in turn used to accelerate the evolution to computational stationarity through traditional continuum algorithms (forward Euler integration, Newton-Raphson fixed point computation). This "equation-free" framework, bypassing the derivation of explicit, closed equations for the observables (e.g. equations of state) may provide a computational bridge between direct atomistic / stochastic simulation and the analysis of its macroscopic, system-level consequences.

cond-mat.soft↗

arXiv2004

Coarse-Grained Kinetic Computations for Rare Events: Application to Micelle Formation

We discuss a coarse-grained approach to the computation of rare events in the context of grand canonical Monte Carlo (GCMC) simulations of self-assembly of surfactant molecules into micelles. The basic assumption is that the {\it computational} system dynamics can be decomposed into two parts -- fast (noise) and slow (reaction coordinates) dynamics, so that the system can be described by an effective, coarse grained Fokker-Planck (FP) equation. While such an assumption may be valid in many circumstances, an explicit form of FP equation is not always available. In our computations we bypass the analytic derivation of such an effective FP equation. The effective free energy gradient and the state-dependent magnitude of the random noise, which are necessary to formulate the effective Fokker-Planck equation, are obtained from ensembles of short bursts of microscopic simulations {\it with judiciously chosen initial conditions}. The reaction coordinate in our micelle formation problem is taken to be the size of a cluster of surfactant molecules. We test the validity of the effective FP description in this system and reconstruct a coarse-grained free energy surface in good agreement with full-scale GCMC simulations. We also show that, for very small clusters, the cluster size seizes to be a good reaction coordinate for a one-dimensional effective description. We discuss possible ways to improve the current model and to take higher-dimensional coarse-grained dynamics into account.

cond-mat.soft↗

arXiv2003

The gap-tooth scheme for homogenization problems

An important class of problems exhibits smooth behaviour in space and time on a macroscopic scale, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, an ``equation-free framework'' has been proposed, of which the gap-tooth scheme is an essential component. The gap-tooth scheme is designed to approximate a time-stepper for an unavailable macroscopic equation in a macroscopic domain; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes, which cover only a fraction of the domain. We analyze the convergence of this scheme for a parabolic homogenization problem with non-linear reaction. In this case, the microscopic model is a partial differential equation with rapidly oscillating coefficients, while the unknown macroscopic model is approximated by the homogenized equation. We show that our method approximates a finite difference scheme of arbitrary (even) order for the homogenized equation when we appropriately constrain the microscopic problem in the boxes. We illustrate this theoretical result with numerical tests on several model problems. We also demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffer regions around the computational boxes.

physics.comp-ph↗

arXiv2003

Equation-free modeling of evolving diseases: Coarse-grained computations with individual-based models

We demonstrate how direct simulation of stochastic, individual-based models can be combined with continuum numerical analysis techniques to study the dynamics of evolving diseases. % Sidestepping the necessity of obtaining explicit population-level models, the approach analyzes the (unavailable in closed form) `coarse' macroscopic equations, estimating the necessary quantities through appropriately initialized, short `bursts' of individual-based dynamic simulation. % We illustrate this approach by analyzing a stochastic and discrete model for the evolution of disease agents caused by point mutations within individual hosts. % Building up from classical SIR and SIRS models, our example uses a one-dimensional lattice for variant space, and assumes a finite number of individuals. % Macroscopic computational tasks enabled through this approach include stationary state computation, coarse projective integration, parametric continuation and stability analysis.

nlin.AO↗