Abstract
How heterogeneous multiscale methods (HMM) handle fluctuations acting on the slow variables in fast–slow systems is investigated. In particular, it is shown via analysis of central limit theorem (CLT) and large deviation principle (LDP) that the standard version of HMM artificially amplifies these fluctuations. A simple modification of HMM, termed parallel HMM, is introduced and is shown to remedy this problem, capturing fluctuations correctly both at the level of the CLT and the LDP. All results in this article assume the HMM speedup factor lambda to be constant and in particular independent of the scale parameter varepsilon . Similar type of arguments can also be used to justify that the tau -leaping method used in the context of Gillespie’s stochastic simulation algorithm for Markov jump processes also captures the right CLT and LDP for these processes.
Highlights
The heterogeneous multiscale methods (HMM) [1,21,23,24] provide an efficient strategy for integrating fast–slow systems of the type dXε = f (Xε, Y ε), dt dY ε dt = 1 ε g (X ε Y ε ). (1.1)The method relies on an averaging principle that holds under some assumption of ergodicity and states that as ε → 0 the slow variables Xε can be uniformly approximated by the solution to the following averaged equation X ̄ = F (X ). (1.2)Here F (x) = f (x, y)μx(dy) is the averaged vector field, with μx(dy) being the ergodic invariant measure of the fast variables Yx with a frozen x variable
Both theoretically (Sect. 4) and numerically (Sect. 7), that the amplitude of fluctuations is enhanced by an HMM-type method
In particular with an HMM speedup factor λ, in the central limit theorem (CLT) the variance of Gaussian fluctuations about the average is increased by a factor λ as well
Summary
By comparing (1.5) with (1.8), we see that the first approximation is essentially replacing a sum of λ weakly correlated random variables with one random variable, multiplied by λ This introduces correlations that should not be there and in particular results in enhanced fluctuations. We stress that the theoretical results of this article are all obtained under the approximation scenario above, namely that we have discretized the slow variables and worked with fast variables Yxε that solve the exact evolution equations, but with frozen x variables. We briefly recall the averaging principle for stochastic fast–slow systems and discuss two results that characterize the fluctuations about the average, the central limit theorem (CLT) and the large deviations principle (LDP). Details of this convergence result in the setting above are given in (for instance) [15, Chapter 7.2]
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