A trajectory-free framework for analysing multiscale systems

https://doi.org/10.1016/j.physd.2016.04.010Get rights and content

Highlights

  • A trajectory-free method to test for multiscale dynamics.

  • Isolates the fast dynamics and determines the reduced slow dynamics.

  • Based on the spectral properties of the transfer and Koopman operators.

Abstract

We develop algorithms built around properties of the transfer operator and Koopman operator which (1) test for possible multiscale dynamics in a given dynamical system, (2) estimate the magnitude of the time-scale separation, and finally (3) distill the reduced slow dynamics on a suitably designed subspace. By avoiding trajectory integration, the developed techniques are highly computationally efficient. We corroborate our findings with numerical simulations of a test problem.

Introduction

Effective numerical simulation of multiscale systems constitutes a formidable challenge. Consider a system which has slow dynamics on a time-scale of order one and fast dynamics on the scale of order 1/ϵ for some parameter ϵ1. To accurately simulate orbits numerically and to assure numerical stability, the time step of the integrator must be of the order of ϵ. To capture the relevant slow dynamics a total number of integration steps of the order of 1/ϵ is required, making direct numerical simulations of orbits computationally impractical.

Numerical integrators are subject to two main sources of error. The first is truncation error, which is the inability of the numerical method (Runge–Kutta, Euler–Maruyama, etc.) to fully capture the actual dynamics of the system. The second is round-off error, due to implementing the numerical method on a computer with finite precision arithmetic. While truncation error decreases with a smaller time step, round-off error increases  [1], [2]. In a multiscale system, if the time-scale separation is large, it may be impossible to find a time step which is simultaneously small enough to avoid significant truncation error for the fast dynamics and sufficiently large to avoid detrimental accumulation of round-off error for the slow dynamics.

Even if orbits could be computed exactly, analysing a multiscale system using a time series extracted from a true orbit can still yield incorrect data about the diffusion process of the slow variables  [3]. To avoid this problem, the time series must be sampled at a rate intermediate between the slow and fast variables and these rates might not be known in advance.

There exists a variety of numerical methods dealing with one or more aspects of these numerical difficulties (see  [4] and references therein). These methods rely on producing trajectories of the dynamical system via some form of time-integration with some of the issues mentioned above remaining. In this paper, we develop algorithms which avoid trajectory integration altogether. Besides the advantages relating to the issues of time-integration mentioned above, the algorithm allows for a huge reduction in computational time. Our main objective is to develop numerical algorithms which, given a dynamical system,

  • 1.

    test whether the system exhibits multiscale behaviour, and if so

  • 2.

    determine the order of the time-scale separation, and then

  • 3.

    construct effective reduced equations for the slow dynamics allowing for the application of large time steps.

The framework we adopt for this integration-free approach is based on the infinitesimal generator associated to the underlying continuous-time dynamical system.

The construction of effective reduced equations (point 3 above), requires the estimation of coordinates in which the fast and slow dynamics operate. In the situation where there is an attracting slow manifold, existing numerical methods to determine the slow manifold include  [5], [6], [7], [8], [9] (see  [10] for a recent review). Most of these methods rely on the existence of some attracting slow manifold towards which transient fast dynamics is approaching along fast fibres. Here we consider the situation of multiscale systems whose asymptotic behaviour does not necessarily occur on an attracting slow manifold. In contrast to methods which determine the fast fibres locally, we instead globally estimate the nonlinear foliation of fast fibres.

Once slow and fast coordinates are established, it is a further challenge to identify their dynamics. There are two approaches; either to devise an effective numerical method which allows for the reliable simulation of the slow coordinates or to construct closed equations for the slow coordinates. The first avenue has been successfully pursued by numerical methods such as the equation-free method  [8] and the heterogeneous multiscale method  [11], [12] which employ short finely resolved bursts of the full dynamics to numerically estimate the averaged slow vector field which then subsequently may be propagated with a large time step. Here we tackle the second avenue of determining the slow dynamics explicitly without the need for temporally resolving the fast dynamics at each step. To compute reduced equations on the (in general, non-unique) slow coordinates, we nonlinearly project local computations along the fast fibres. Our approach does not rely on any temporal integration to estimate the reduced equations. Hence it does not suffer from possible sensitivity of these estimates to the choice of the length of the fast bursts. For example, in the case where the fast dynamics itself involves transitions between metastable states, a short temporal sampling of the full dynamics might not be sufficient to capture the fast invariant measure. This would then bias the averaged slow vector field.

In Section  2 we briefly review the notion of generators of transfer and Koopman operators. Section  3 introduces a trajectory-free test for multiscale behaviour. The degree of time-scale separation is estimated in algorithms described in Sections  4 Estimating the time-scale separation, 5 Parameterizing by arc length. A method to determine the reduced slow dynamics from a multiscale system without relying on statistics obtained from long time-integrations is given in Section  6. The algorithms are tested in numerical simulations in Section  7. We conclude with a discussion in Section  8.

Section snippets

Generators

We describe our methodology for Itō drift–diffusion processes, as these are a large and flexible class of dynamical systems, and the spectral properties of the corresponding transfer operators are relatively straightforward. Consider a drift–diffusion process dζi=μidt+k=1σikdWkwithi=1,,d defined on a subset Z of Rd where ld and each Wk for k=1,, represents an independent Wiener process. Given a probability density function at time t=0, the density at future times is determined by the

An algorithm to test for multiscale behaviour

We now use these properties of the Fokker–Planck operator and its adjoint to develop an algorithm to test for multiscale behaviour. We focus on the case of stochastic differential equations (SDEs) where the slow dynamics is one-dimensional. The only information used by the algorithm are functions μ and D defining the drift and diffusion of the process. In particular, we do not assume any a priori knowledge of the slow or fast directions (if they exist) or that these directions align with the

Estimating the time-scale separation

We now propose a further test for multiscale behaviour which also estimates the magnitude of the time-scale separation between the slow and fast dynamics. This estimation is achieved by comparing the spectrum of the operator L on the full system to the spectrum of an operator defined by dynamics along the fibre F.

Recall from Section  2 that the leading eigenvalues of L (which are the same as those of its adjoint L) correspond to the slowest rates of decay for observables under the dynamics of

Parameterizing by arc length

In the case that the fast fibre is one-dimensional (and the full system is therefore two-dimensional), an alternative to expressing the fibre as a graph is to parameterize the fibre by arc length. This is always possible, as topologically a one-dimensional fibre F will either be a line or a circle.

Given a representation of the curve F, compute a sequence of points {qn} such that each point is at a uniform distance from the previous. To analyse the drift and diffusion at qn, first approximate

Determining the reduced dynamics

Once the existence of multiscale dynamics has been established, one further step is to compute the reduced slow dynamics. We give here a simple technique to perform the reduction to a lower dimensional SDE. As with the other methods developed in this paper, the algorithm does not rely on trajectory integration.

First, consider a multiscale system where model reduction is applicable. That is, following the formalism given in  [25], [22], there is a projection P:ZX from the domain of the full

Numerical example

We now apply these techniques to an example studied in  [27] and given by ẋ=siny+1+12sinyẆx,ẏ=1ϵ[y+sinx]+1ϵẆy where Wx and Wy are independent Wiener processes. This SDE is defined on [0,2π]×R with periodic boundary conditions on x. In order to have an example with nonlinear fast fibres, we consider the system after a change of coordinates taking (x,y) to (x+sin(y),y). After this change, the system is given by the (admittedly much uglier) equations ẋ=[siny+cosyϵ(sin(xsiny)y)siny2ϵ]+1+12

Discussion

To conclude, we look at the computational overhead of the techniques introduced in this paper in comparison to other methods for analysing multiscale systems. In the example in Section  7, building the matrices used in steps 1 and 2 of Algorithm 1 required evaluating the functions for the drift and diffusion at every point of a 50×50 grid. Other steps in Algorithms 1, 2A, and 2B consider the drift and diffusion on lower-dimensional fibres and require even fewer evaluations. Overall, the

References (29)

  • Christian Kuehn
  • E. Weinan et al.

    The heterogeneous multiscale methods

    Commun. Math. Sci.

    (2003)
  • Eric Vanden-Eijnden

    On HMM-like integrators and projective integration methods for systems with multiple time scales

    Commun. Math. Sci.

    (2007)
  • Viviane Baladi

    Correlation spectrum of quenched and annealed equilibrium states for random expanding maps

    Comm. Math. Phys.

    (1997)
  • Cited by (0)

    This project was funded by the Australian Research Council Grant DP120104514.

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