1. What are the contributions mentioned in the paper "Extracting macroscopic dynamics: model problems & algorithms" ?
This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them.. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of time-scales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum.. The general problem may be described as follows: let Z be a Hilbert space, and consider the noise-driven differential equation for z ∈ Z: dz dt = h ( z ) + γ ( z ) dW dt, ( 1. 1 ) where W ( t ) is a noise process, chosen so that z ( t ) is Markovian.. Anticipating this, the authors introduce the projection P: Z 7→ X and the orthogonal complement of X in Z, Y = ( I − P ) Z.. The authors will study situations where the y variable can be eliminated, and an effective, approximate equation for x alone is obtained.. The authors consider cases where the original model ( 1. 1 ) for z is either an autonomous ODE or a noise-driven differential equation, such as an SDE, and where the effective dynamics ( 1. 3 ) for X is either an ODE or an SDE.. The ideas the authors describe have discrete time analogues, and some of the algorithms they overview extract a discrete time model in X, such as a Markov chain, rather than a continuous time model.. The authors will also examine situations where the effective dimension reduction can be carried out in the space of probability densities propagated by paths of ( 1. 1 ) ; this requires consideration of the Master Equation for probability densities.. The primary motivation for this paper is to overview the wealth of recent work concerning algorithms which attempt to find the effective dynamics in X.. This work is, at present, not very unified and their aim is to highlight the similarities and differences among the approaches currently emerging.. In order to do this the authors will spend a substantial fraction of the paper explaining situations in which it is possible to find closed equations for X which adequately approximate the dynamics of x ∈ X.. Thus most of the paper will be devoted to the development of model problems, and the underlying theoretical context in which they lie.. The authors do not state theorems and give proofs— they present the essential ideas and reference the literature for details and rigorous analysis.. On uncountable state spaces, and for W ( t ) Brownian motion in ( 1. 1 ), the Master Equation is a partial differential equation ( PDE ) —the Fokker-Planck equation—and its adjoint—the ChapmanKolmogorov equation—propagates expectations ; the authors describe these PDEs.. In Section 3 the authors outline the Mori-Zwanzig projection operator approach which describes the elimination of variables at the level of the Master Equation.. The following provides an overview of a number of important themes 4 D. GIVON, R. KUPFERMAN & A. M. STUART running throughout this paper.. In conceptualizing these algorithms it is important to appreciate that any algorithm aimed at extracting dynamics in X, given the equations of motion ( 1. 1 ) in Z, has two essential components: ( i ) determining the projection P which defines X through X = PZ ; ( ii ) determining the effective dynamics in X.. In this section the authors consider dynamical systems of the form ( 1. 1 ) within a probabilistic setting, by considering the evolution of probability measures induced by the dynamics of paths of ( 1. 1 ).. In spite of the increased complexity due to the infinite dimensionality of the system, the linearity enables the use of numerous techniques adapted for linear systems, such as projection methods, and perturbation expansions.. A useful example illustrating the second point is passive tracer advection: the position of a particle advected in a velocity field, and subject to molecular diffusion, can then be modelled by a nonlinear SDE ; collections of such particles have density satisfying a linear advection-diffusion equation.. In Subsection 2. 1 the authors describe the derivation of the equation governing probability measures for countable state space Markov chains ; in Subsection 6 D. GIVON, R. KUPFERMAN & A. M. STUART 2. 2 they generalize this to the case of Itô SDEs, which give rise to Markov processes on uncountable state spaces.. Model problems are of primary importance in order to make clear statements about the situations in which the authors expect given algorithms to be of use, and in order to develop examples which can be used to test these algorithms.
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