Laing, C. R. ; Zou, Y. ; Smith, B. ; Kevrekidis, I. G. Managing heterogeneity in the study of neural oscillator dynamics.
Journal of Mathematical Neuroscience 2012,
2.
AbstractWe consider a coupled, heterogeneous population of relaxation oscillators used to model rhythmic oscillations in the pre-Bötzinger complex. By choosing specific values of the parameter used to describe the heterogeneity, sampled from the probability distribution of the values of that parameter, we show how the effects of heterogeneity can be studied in a computationally efficient manner. When more than one parameter is heterogeneous, full or sparse tensor product grids are used to select appropriate parameter values. The method allows us to effectively reduce the dimensionality of the model, and it provides a means for systematically investigating the effects of heterogeneity in coupled systems, linking ideas from uncertainty quantification to those for the study of network dynamics.
Zou, Y. ; Torrens, P. M. ; Ghanem, R. G. ; Kevrekidis, I. G. Accelerating agent-based computation of complex urban systems.
International Journal of Geographical Information Science 2012,
26, 1917-1937.
AbstractDespite its popularity, agent-based modeling is limited by serious barriers that constrain its usefulness as an exploratory tool. In particular, there is a paucity of systematic approaches for extracting coarse-grained, system-level information as it emerges in direct simulation. This is particularly problematic for agent-based models (ABMs) of complex urban systems in which macroscopic phenomena, such as sprawl, may manifest themselves coarsely from bottom-up dynamics among diverse agent-actors interacting across scales. Often these connections are not known, but treating them is nevertheless crucial in enabling prediction, in supporting decisions, and in facilitating the design, control, and optimization of urban systems. In this article, we describe and implement a metasimulation scheme for extracting macroscopic information from local dynamics of agent-based simulation, which allows acceleration of coarse-scale computing and which may also serve as a precursor to handle emergence in complex urban simulation. We compare direct ABM simulation, population-level equation solutions, and coarse projective integration. We apply the scheme to the simulation of urban sprawl from local drivers of urbanization, urban growth, and population dynamics. Numerical examples of the three approaches are provided to compare their accuracy and efficiency. We find that our metasimulation scheme can significantly accelerate complex urban simulations while maintaining faithful representation of the original model.
Punckt, C. ; Jan, L. ; Jiang, P. ; Frewen, T. A. ; Saville, D. A. ; Kevrekidis, I. G. ; Aksay, I. A. Autonomous colloidal crystallization in a galvanic microreactor.
Journal of Applied Physics 2012,
112, 074905.
AbstractWe report on a technique that utilizes an array of galvanic microreactors to guide the assembly of two-dimensional colloidal crystals with spatial and orientational order. Our system is comprised of an array of copper and gold electrodes in a coplanar arrangement, immersed in a dilute hydrochloric acid solution in which colloidal micro-spheres of polystyrene and silica are suspended. Under optimized conditions, two-dimensional colloidal crystals form at the anodic copper with patterns and crystal orientation governed by the electrode geometry. After the aggregation process, the colloidal particles are cemented to the substrate by co-deposition of reaction products. As we vary the electrode geometry, the dissolution rate of the copper electrodes is altered. This way, we control the colloidal motion as well as the degree of reaction product formation. We show that particle motion is governed by a combination of electrokinetic effects acting directly on the colloidal particles and bulk electrolyte flow generated at the copper-gold interface. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4755807]
Laing, C. R. ; Rajendran, K. ; Kevrekidis, I. G. Chimeras in random non-complete networks of phase oscillators.
Chaos 2012,
22, 013132.
AbstractWe consider the simplest network of coupled non-identical phase oscillators capable of displaying a "chimera" state (namely, two subnetworks with strong coupling within the subnetworks and weaker coupling between them) and systematically investigate the effects of gradually removing connections within the network, in a random but systematically specified way. We average over ensembles of networks with the same random connectivity but different intrinsic oscillator frequencies and derive ordinary differential equations (ODEs), whose fixed points describe a typical chimera state in a representative network of phase oscillators. Following these fixed points as parameters are varied we find that chimera states are quite sensitive to such random removals of connections, and that oscillations of chimera states can be either created or suppressed in apparent bifurcation points, depending on exactly how the connections are gradually removed. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3694118]
Tsoumanis, A. C. ; Rajendran, K. ; Siettos, C. I. ; Kevrekidis, I. G. Coarse-graining the dynamics of network evolution: the rise and fall of a networked society.
New Journal of Physics 2012,
14, 083037.
AbstractWe explore a systematic approach to studying the dynamics of evolving networks at a coarse-grained, system level. We emphasize the importance of finding good observables (network properties) in terms of which coarse-grained models can be developed. We illustrate our approach through a particular social network model: the 'rise and fall' of a networked society (Marsili M et al 2004 Proc. Natl Acad. Sci. USA 101 1439). We implement our low-dimensional description computationally using the equation-free approach and show how it can be used to (i) accelerate simulations and (ii) extract system-level stability/bifurcation information from the detailed dynamic model. We discuss other system-level tasks that can be enabled through such a computer-assisted coarse-graining approach.
Gauthier, E. ; Hellstern, T. ; Kevrekidis, I. G. ; Benziger, J. Drop Detachment and Motion on Fuel Cell Electrode Materials.
Acs Applied Materials & Interfaces 2012,
4 761-771.
AbstractLiquid water is pushed through flow channels of fuel cells, where one surface is a porous carbon electrode made up of carbon fibers. Water drops grow on the fibrous carbon surface in the gas flow channel. The drops adhere to the superficial fiber surfaces but exhibit little penetration into the voids between the fibers. The fibrous surfaces are hydrophobic, but there is. a substantial threshold force necessary to initiate water drop, motion. Once the water drops, begin to move, however, the adhesive force decreases and drops move with minimal friction, similar to motion on superhydrophobic materials. We report here studies of water wetting and water drop motion on typical porous carbon materials (carbon paper and carbon cloth) employed in fuel cells. The static coefficient of friction on these textured surfaces is comparable to that for smooth Teflon. But the dynamic coefficient of-friction-is several orders of magnitude smaller on the textured surfaces than on smooth Teflon. Carbon cloth displays a much smaller static contact angle hysteresis than carbon paper due to its two-scale roughness. The dynamic contact angle hysteresis for carbon paper is greatly reduced compared to the static contact angle hysteresis. Enhanced dynamic hydrophobicity is suggested to result from the extent to which a dynamic contact line can track topological heterogeneities of the liquid/solid interface.
Chauviere, A. ; Hatzikirou, H. ; Kevrekidis, I. G. ; Lowengrub, J. S. ; Cristini, V. Dynamic density functional theory of solid tumor growth: Preliminary models.
Aip Advances 2012,
2 011210.
AbstractCancer is a disease that can be seen as a complex system whose dynamics and growth result from nonlinear processes coupled across wide ranges of spatio-temporal scales. The current mathematical modeling literature addresses issues at various scales but the development of theoretical methodologies capable of bridging gaps across scales needs further study. We present a new theoretical framework based on Dynamic Density Functional Theory (DDFT) extended, for the first time, to the dynamics of living tissues by accounting for cell density correlations, different cell types, phenotypes and cell birth/death processes, in order to provide a biophysically consistent description of processes across the scales. We present an application of this approach to tumor growth. Copyright 2012 Author(s). This article is distributed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.3699065]
Kavousanakis, M. E. ; Liu, P. ; Boudouvis, A. G. ; Lowengrub, J. ; Kevrekidis, I. G. Efficient coarse simulation of a growing avascular tumor.
Physical Review E 2012,
85, 031912.
AbstractThe subject of this work is the development and implementation of algorithms which accelerate the simulation of early stage tumor growth models. Among the different computational approaches used for the simulation of tumor progression, discrete stochastic models (e. g., cellular automata) have been widely used to describe processes occurring at the cell and subcell scales (e. g., cell-cell interactions and signaling processes). To describe macroscopic characteristics (e. g., morphology) of growing tumors, large numbers of interacting cells must be simulated. However, the high computational demands of stochastic models make the simulation of large-scale systems impractical. Alternatively, continuum models, which can describe behavior at the tumor scale, often rely on phenomenological assumptions in place of rigorous upscaling of microscopic models. This limits their predictive power. In this work, we circumvent the derivation of closed macroscopic equations for the growing cancer cell populations; instead, we construct, based on the so-called "equation-free" framework, a computational superstructure, which wraps around the individual-based cell-level simulator and accelerates the computations required for the study of the long-time behavior of systems involving many interacting cells. The microscopic model, e. g., a cellular automaton, which simulates the evolution of cancer cell populations, is executed for relatively short time intervals, at the end of which coarse-scale information is obtained. These coarse variables evolve on slower time scales than each individual cell in the population, enabling the application of forward projection schemes, which extrapolate their values at later times. This technique is referred to as coarse projective integration. Increasing the ratio of projection times to microscopic simulator execution times enhances the computational savings. Crucial accuracy issues arising for growing tumors with radial symmetry are addressed by applying the coarse projective integration scheme in a cotraveling (cogrowing) frame. As a proof of principle, we demonstrate that the application of this scheme yields highly accurate solutions, while preserving the computational savings of coarse projective integration.
Siettos, C. I. ; Gear, C. W. ; Kevrekidis, I. G. An equation-free approach to agent-based computation: Bifurcation analysis and control of stationary states.
Epl 2012,
99, 48007.
AbstractWe show how the equation-free approach can be exploited to enable agent-based simulators to perform system-level computations such as bifurcation, stability analysis and controller design. We illustrate these tasks through an event-driven agent-based model describing the dynamic behaviour of many interacting investors in the presence of mimesis. Using short bursts of appropriately initialized runs of the detailed, agent-based simulator, we construct the coarse-grained bifurcation diagram of the (expected) density of agents and investigate the stability of its multiple solution branches. When the mimetic coupling between agents becomes strong enough, the stable stationary state loses its stability at a coarse turning point bifurcation. We also demonstrate how the framework can be used to design a wash-out dynamic controller that stabilizes open-loop unstable stationary states even under model uncertainty. Copyright (C) EPLA, 2012
Makeev, A. G. ; Kurkina, E. S. ; Kevrekidis, I. G. Kinetic Monte Carlo simulations of travelling pulses and spiral waves in the lattice Lotka-Volterra model.
Chaos 2012,
22, 023141.
AbstractKinetic Monte Carlo simulations are used to study the stochastic two-species Lotka-Volterra model on a square lattice. For certain values of the model parameters, the system constitutes an excitable medium: travelling pulses and rotating spiral waves can be excited. Stable solitary pulses travel with constant (modulo stochastic fluctuations) shape and speed along a periodic lattice. The spiral waves observed persist sometimes for hundreds of rotations, but they are ultimately unstable and break-up (because of fluctuations and interactions between neighboring fronts) giving rise to complex dynamic behavior in which numerous small spiral waves rotate and interact with each other. It is interesting that travelling pulses and spiral waves can be exhibited by the model even for completely immobile species, due to the non-local reaction kinetics. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4729141]
Bavarian, M. ; Soroush, M. ; Kevrekidis, I. G. ; Benziger, J. B. Mathematical Modeling and Steady-State Analysis of a Co-Ionic-Conducting Solid Oxide Fuel Cell. In
2012 American Control Conference (acc); Ieee Computer Soc: Los Alamitos, 2012; pp. 4269-4274.
AbstractA mathematical model of a solid oxide fuel cell (SOFC) with a BaCe1-xSmxO3-alpha type electrolyte is developed. This class of electrolytes exhibits both proton and oxygen-anion conductivity. To develop the model, heat transfer, mass transfer and electrochemical processes are taken into account. The existence of steady-state multiplicity in this class of fuel cells is investigated under three operation modes: constant ohmic load, potentiostatic and galavanostatic. The cell has up to three steady states under the constant ohmic load and potentiostatic modes, and a unique steady state under the galvanostatic mode. This same steady state behavior has been observed in oxygen-anion conducting and proton conducting SOFCs. Interestingly, this study shows that in this class of SOFCs, thermal and concentration multiplicities can coexist; ignition in the solid temperature is accompanied by extinction in the fuel and oxygen concentrations, and ignition and extinction in concentrations of water in the anode and cathode sides, respectively.
Kavousanakis, M. E. ; Colosqui, C. E. ; Kevrekidis, I. G. ; Papathanasiou, A. G. Mechanisms of wetting transitions on patterned surfaces: continuum and mesoscopic analysis.
Soft Matter 2012,
8 7928-7936.
AbstractMicro-or nano-structurally roughened solid surfaces exhibit a rich variety of wetting behavior types, ranging from superhydro- or superoleophobicity to superhydro- or superoleophilicity. Depending on their material chemistry, the scale and morphology of their roughness or even the application of external electric fields, their apparent wettability can be significantly modified giving rise to challenging technological applications by exploiting the associated capillary phenomena at the micrometer scale. Certain applications, however, are limited by hysteretic wetting transitions, which inhibit spontaneous switching between wetting states, requiring external stimuli or actuation like thermal heating. The presence of surface roughness, necessary for the manifestation of the superhydrophobicity, induces multiplicity of wetting states and the inevitable hysteresis appears due to considerable energy barriers separating the equilibrium states. Here, by using continuum as well as mesoscopic computational analysis we perform a systems level study of the mechanisms of wetting transitions on model structured solid surfaces. By tracing entire equilibrium solution families and determining their relative stability we are able to illuminate mechanisms of wetting transitions and compute the corresponding energy barriers. The implementation of our analysis to 'real world' structured or unstructured surfaces is straightforward, rendering our computational tools valuable not only for the realization of surfaces with addressable wettability through roughness design, but also for the design of suitable actuation for optimal switching between wetting states.
Zou, Y. ; Fonoberov, V. A. ; Fonoberova, M. ; Mezic, I. ; Kevrekidis, I. G. Model reduction for agent-based social simulation: Coarse-graining a civil violence model.
Physical Review E 2012,
85, 066106.
AbstractAgent-based modeling (ABM) constitutes a powerful computational tool for the exploration of phenomena involving emergent dynamic behavior in the social sciences. This paper demonstrates a computer-assisted approach that bridges the significant gap between the single-agent microscopic level and the macroscopic (coarse-grained population) level, where fundamental questions must be rationally answered and policies guiding the emergent dynamics devised. Our approach will be illustrated through an agent-based model of civil violence. This spatiotemporally varying ABM incorporates interactions between a heterogeneous population of citizens [active (insurgent), inactive, or jailed] and a population of police officers. Detailed simulations exhibit an equilibrium punctuated by periods of social upheavals. We show how to effectively reduce the agent-based dynamics to a stochastic model with only two coarse-grained degrees of freedom: the number of jailed citizens and the number of active ones. The coarse-grained model captures the ABM dynamics while drastically reducing the computation time (by a factor of approximately 20).
Zagaris, A. ; Vandekerckhove, C. ; Gear, W. C. ; Kaper, T. J. ; Kevrekidis, I. G. STABILITY AND STABILIZATION OF THE CONSTRAINED RUNS SCHEMES FOR EQUATION-FREE PROJECTION TO A SLOW MANIFOLD.
Discrete and Continuous Dynamical Systems 2012,
32, 2759-2803.
AbstractIn [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, ...) 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.