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28
Time series analysis via mechanistic models. In review; prepublished at arxiv.org/abs/0802.0021
, 2008
"... The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consi ..."
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Cited by 36 (10 self)
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The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plugandplay property. Our work builds on recently developed plugandplay inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae. 1. Introduction. A
The moment map: Nonlinear dynamics of density evolution via a few moments, submitted to
 SIAM Journal on Applied Dynamical Systems
, 2005
"... We explore situations in which certain stochastic and highdimensional deterministic systems behave effectively as lowdimensional dynamical systems. We define and study moment maps, maps on spaces of loworder moments of evolving distributions, as a means of understanding equationsfree multiscale ..."
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Cited by 12 (1 self)
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We explore situations in which certain stochastic and highdimensional deterministic systems behave effectively as lowdimensional dynamical systems. We define and study moment maps, maps on spaces of loworder moments of evolving distributions, as a means of understanding equationsfree multiscale algorithms for these systems. We demonstrate how nonlinearity arises in these maps and how this results in the stabilization of metastable states. Examples are shown for a hierarchy of models, ranging from simple stochastic differential equations to molecular dynamics simulations of a particle in contact with a heat bath. Key words. Equationfree, multiscale, moment map, metastable states. AMS subject classifications. 1. Introduction. An
Decentralised autonomic computing: Analysing selforganising emergent behaviour using advanced numerical methods
 in Proceedings of the Second International Conference on Autonomic Computing, (Los Alamitos
, 2005
"... When designing decentralised autonomic computing systems, a fundamental engineering issue is to assess systemwide behaviour. Such decentralised systems are characterised by the lack of global control, typically consist of autonomous cooperating entities, and often rely on selforganised emergent ..."
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Cited by 12 (6 self)
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When designing decentralised autonomic computing systems, a fundamental engineering issue is to assess systemwide behaviour. Such decentralised systems are characterised by the lack of global control, typically consist of autonomous cooperating entities, and often rely on selforganised emergent behaviour to achieve the requirements. A wellfounded and practically feasible approach to study overall system behaviour is a prerequisite for successful deployment. On one hand, formal proofs of correct behaviour and even predictions of the exact systemwide behaviour are practically infeasible due to the complex, dynamic, and often nondeterministic nature of selforganising emergent systems. On the other hand, simple simulations give no convincing arguments for guaranteeing systemwide properties. We describe an alternative approach that allows to analyse and assess trends in systemwide behaviour, based on socalled “equationfree ” macroscopic analysis. This technique yields more reliable results about the systemwide behaviour, compared to mere observation of simulation results, at an affordable computational cost. Numerical algorithms act at the systemwide level and steer the simulations. This allows to limit the amount of simulations considerably. We illustrate the approach by studying a particular systemwide property of a decentralised control system for Automated Guided Vehicles and we outline a road map towards a general methodology for studying decentralised autonomic computing systems. 1.
Engineering SelfOrganising Emergent Systems with Simulationbased Scientific Analysis
 In: Proceedings of the Fourth International Workshop on Engineering SelfOrganising Applications, Universiteit Utrecht
, 2005
"... The goal of engineering selforganising emergent systems is to acquire a macroscopic system behaviour solely from autonomous local activity and interaction. Due to the nondeterministic nature of such systems, it is hard to guarantee that the required macroscopic behaviour is achieved and maintained ..."
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Cited by 7 (0 self)
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The goal of engineering selforganising emergent systems is to acquire a macroscopic system behaviour solely from autonomous local activity and interaction. Due to the nondeterministic nature of such systems, it is hard to guarantee that the required macroscopic behaviour is achieved and maintained. Before even considering a selforganising emergent system in an industrial context, e.g. for Automated Guided Vehicle (AGV) transportation systems, such guarantees are needed. An empirical analysis approach is proposed that combines realistic agentbased simulations with existing scientific numerical algorithms for analysing the macroscopic behaviour. The numerical algorithm itself obtains the analysis results on the fly by steering and accelerating the simulation process according to the algorithms goal. The approach is feasible, compared to formal proofs, and leads to more reliable and valuable results, compared to mere observation of simulation results. Also, the approach allows to systematically analyse the macroscopic behaviour to acquire macroscopic guarantees and feedback that can be used by an engineering process to iteratively shape a selforganising emergent solution.
An equationfree approach to coupled oscillator dynamics: the Kuramoto model example
, 2005
"... We present an equationfree multiscale approach to the computational study of the collective dynamics of the Kuramoto model [Chemical Oscillations, Waves, and Turbulence, SpringerVerlag (1984)], a prototype model for coupled oscillator populations. Our study takes place in a reduced phase space of ..."
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Cited by 4 (1 self)
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We present an equationfree multiscale approach to the computational study of the collective dynamics of the Kuramoto model [Chemical Oscillations, Waves, and Turbulence, SpringerVerlag (1984)], a prototype model for coupled oscillator populations. Our study takes place in a reduced phase space of coarsegrained “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 coarsegrained macroscopic variables; instead we use the equationfree framework [Kevrekidis et al., Comm. Math. Sci. 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 coarsegrained equations are implemented through short bursts of appropriately initialized simulations of the “finescale”, detailed coupled oscillator model. Coarse projective integration and coarse fixed point computations are illustrated. Coupled nonlinear oscillators can exhibit spontaneous emergence of order, a fundamental qualitative feature of many complex dynamical systems [Manrubia et al., 2004]. The collective,
Equationfree, multiscale computation for unsteady random diffusion. Multiscale Model
 Simul
, 2005
"... Abstract. We present an “equationfree ” multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast in the form of stochastic differential equations. A ..."
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Abstract. We present an “equationfree ” multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast in the form of stochastic differential equations. A detailed finescale computation of such a problem requires discretization and solution of a large system of equations, and can be prohibitively timeconsuming. To circumvent this difficulty, we propose an equationfree approach, where the finescale computation is conducted only for a (small) fraction of the overall time. The evolution of a set of appropriately defined coarsegrained variables (observables) is evaluated during the finescale computation, and “projective integration” is used to accelerate the integration. The choice of these coarse variables is an important part of the approach: they are the coefficients of pointwise polynomial expansions of the random solutions. Such a choice of coarse variables allows us to reconstruct representative ensembles of finescale solutions with “correct ” correlation structures, which is a key to algorithm efficiency. Numerical examples demonstrating accuracy and efficiency of the approach are presented. Key words. multiscale problem, diffusion in random media, stochastic modeling, equationfree.
Necessary Conditions for Optimality for a Distributed Optimal Control Problem
"... Abstract—This paper presents a novel optimal control problem formulation and new optimality conditions, referred to as distributed optimal control, for systems comprised of many dynamic agents that can each be described by the same ordinary differential equations (ODEs). The macroscopic system per ..."
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Abstract—This paper presents a novel optimal control problem formulation and new optimality conditions, referred to as distributed optimal control, for systems comprised of many dynamic agents that can each be described by the same ordinary differential equations (ODEs). The macroscopic system performance is represented by an integral cost function of a restriction operator comprised of the probability density function of the individual agents ’ state variables, and of their control laws. It is shown that, under proper assumptions, the macroscopic cost can be optimized subject to a hyperbolic partial differential equation (PDE) that describes the evolution of the macroscopic state over larger spatial and temporal scales. This methodology extends the capabilities of optimal control to complex systems described by numerous interacting dynamical systems. The approach is demonstrated on a simulated network of distributed sensors installed on autonomous underwater vehicles, and deployed to provide track coverage over a region of interest. I.
Ed Ionides Infectious disease dynamics: a statistical perspective 1 Infectious disease dynamics: a statistical perspective CCMB/Bioinformatics Seminar
, 2009
"... Why do we seek to quantify and understand disease dynamics? • Prevention and control of emerging infectious diseases (SARS, HIV/AIDS, H5N1 influenza “bird flu”) • Understanding the development and spread of drug resistant strains (malaria, tuberculosis, MRSA “the hospital superbug”) Ed Ionides Infe ..."
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Why do we seek to quantify and understand disease dynamics? • Prevention and control of emerging infectious diseases (SARS, HIV/AIDS, H5N1 influenza “bird flu”) • Understanding the development and spread of drug resistant strains (malaria, tuberculosis, MRSA “the hospital superbug”) Ed Ionides Infectious disease dynamics: a statistical perspective 4 Disease dynamics: epidemiology or ecology, or both? • Environmental host/pathogen dynamics are close to predator/prey relationships which are a central topic of ecology. • Analysis of diseases as ecosystems complements more traditional epidemiology (risk factors etc). • Ecologists typically seek to avoid extinctions, whereas epidemiologists typically seek the reverse. Things are not always this simple... – Helicobacter pylori bacteria used to live in the stomach of most humans. Some strains cause stomach ulcers and cancer. It is almost extinct in the developed world due to widespread use of
AN EQUATION FREE, REDUCEDORDER MODELING APPROACH TO TROPICAL PACIFIC SIMULATION
"... The “equationfree ” (EF) method is often used in complex, multiscale problems. In such cases it is necessary to know the closed form of the required evolution equations about macroscopic variables within some applied fields. Conceptually such equations exist, however, they are not available in clo ..."
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The “equationfree ” (EF) method is often used in complex, multiscale problems. In such cases it is necessary to know the closed form of the required evolution equations about macroscopic variables within some applied fields. Conceptually such equations exist, however, they are not available in closed form. The EF method can bypass this difficulty. This method can obtain macroscopic information by implementing models at a microscopic level. Given an initial macroscopic variable, through lifting we can obtain the associated microscopic variable, which may be evolved using Direct Numerical Simulations (DNS) and by restriction, we can obtain the necessary macroscopic information and the projective integration to obtain the desired quantities. In this paper we apply the EF PODassisted method to the reduced modeling of a largescale upper ocean circulation in the tropical Pacific domain. The computation cost is reduced dramatically. Compared with the POD method, the method provided more accurate results and it did not require the availability of any explicit equations or the righthandside (RHS) of the evolution equation.