Abstract
This article addresses the weak convergence of numerical methods for Brownian dynamics. Typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic (stepsize h → 0 ) convergence behaviour of the error of finite-time averages. Recently, it has been demonstrated, by study of Fokker–Planck operators, that a non-Markovian numerical method generates approximations in the long-time limit with higher accuracy order (second order) than would be expected from its weak convergence analysis (finite-time averages are first-order accurate). In this article, we describe the transition from the transient to the steady-state regime of this numerical method by estimating the time-dependency of the coefficients in an asymptotic expansion for the weak error, demonstrating that the convergence to second order is exponentially rapid in time. Moreover, we provide numerical tests of the theory, including comparisons of the efficiencies of the Euler–Maruyama method, the popular second-order Heun method, and the non-Markovian method.
Highlights
Stochastic gradient systems are stochastic differential equations in d dimensions having the form dX = a(X) dt + σ dw, X(0) = X0, (1.1)where a(x) := −∇V(x), (1.2)V(x), x ∈ Rd, is a potential energy function, and σ > 0 is a constant which characterizes the strength of the additive rspa.royalsocietypublishing.org Proc
Noise, here described by a standard d-dimensional Wiener process w(t). These systems originate in the work of Einstein [1,2] to describe the motion of Brownian particles. They arise in mathematical models for chemistry, physics, biology and other areas, when the cumulative effect of unresolved degrees of freedom must be incorporated into a model to ensure its physical relevance
The error in the numerical solution is typically quantified in either a strong sense (accuracy with respect to a particular stochastic path associated to (1.1)) or by reference to an evolving distribution; the latter is the focus of this article
Summary
The result of [10] relies on study of the invariant distribution and a Baker–Campbell–Hausdorff expansion of the generator of the process Such an operator-based approach does not clarify the progression from finite-time averaging to infinitetime averaging and, in particular, nothing is demonstrated in [9,10] about the weak accuracy of the method. We note that there are several recent papers (see [11,12,13] and references therein), where the idea of modified differential equations is exploited in order to construct higher order schemes for computing ergodic limits This approach provides the possibility of modifying schemes which are of weak order one on finite-time intervals to provide second-order approximations in ergodic limits. The theoretical approaches in our paper and in [11] share some similarities, the results of [11] are not applicable to the nonMarkovian approximation (1.7) and they do not include an analysis demonstrating that the leading term in the error of their modified schemes goes to zero exponentially fast
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