85 research outputs found
Measuring the irreversibility of numerical schemes for reversible stochastic differential equations
Abstract. For a Markov process the detailed balance condition is equivalent to the time-reversibility of the process. For stochastic differential equations (SDE’s) time discretization numerical schemes usually destroy the property of time-reversibility. Despite an extensive literature on the numerical analysis for SDE’s, their stability properties, strong and/or weak error estimates, large deviations and infinite-time estimates, no quantitative results are known on the lack of reversibility of the discrete-time approximation process. In this paper we provide such quantitative estimates by using the concept of entropy production rate, inspired by ideas from non-equilibrium statistical mechanics. The entropy production rate for a stochastic process is defined as the relative entropy (per unit time) of the path measure of the process with respect to the path measure of the time-reversed process. By construction the entropy production rate is nonnegative and it vanishes if and only if the process is reversible. Crucially, from a numerical point of view, the entropy production rate is an a posteriori quantity, hence it can be computed in the course of a simulation as the ergodic average of a certain functional of the process (the so-called Gallavotti-Cohen (GC) action functional). We compute the entropy production for various numerical schemes such as explicit Euler-Maruyama and explicit Milstein’s for reversible SDEs with additive or multiplicative noise. Additionally, we analyze the entropy production for th
Statistical equilibrium measures in micromagnetics
We derive an equilibrium statistical theory for the macroscopic description
of a ferromagnetic material at positive finite temperatures. Our formulation
describes the most-probable equilibrium macrostates that yield a coherent
deterministic large-scale picture varying at the size of the domain, as well as
it captures the effect of random spin fluctuations caused by the thermal noise.
We discuss connections of the proposed formulation to the Landau-Lifschitz
theory and to the studies of domain formation based on Monte Carlo lattice
simulations.Comment: 5 pages (2-column format
Spatial multi-level interacting particle simulations and information theory-based error quantification
We propose a hierarchy of multi-level kinetic Monte Carlo methods for
sampling high-dimensional, stochastic lattice particle dynamics with complex
interactions. The method is based on the efficient coupling of different
spatial resolution levels, taking advantage of the low sampling cost in a
coarse space and by developing local reconstruction strategies from
coarse-grained dynamics. Microscopic reconstruction corrects possibly
significant errors introduced through coarse-graining, leading to the
controlled-error approximation of the sampled stochastic process. In this
manner, the proposed multi-level algorithm overcomes known shortcomings of
coarse-graining of particle systems with complex interactions such as combined
long and short-range particle interactions and/or complex lattice geometries.
Specifically, we provide error analysis for the approximation of long-time
stationary dynamics in terms of relative entropy and prove that information
loss in the multi-level methods is growing linearly in time, which in turn
implies that an appropriate observable in the stationary regime is the
information loss of the path measures per unit time. We show that this
observable can be either estimated a priori, or it can be tracked
computationally a posteriori in the course of a simulation. The stationary
regime is of critical importance to molecular simulations as it is relevant to
long-time sampling, obtaining phase diagrams and in studying metastability
properties of high-dimensional complex systems. Finally, the multi-level nature
of the method provides flexibility in combining rejection-free and null-event
implementations, generating a hierarchy of algorithms with an adjustable number
of rejections that includes well-known rejection-free and null-event
algorithms.Comment: 34 page
Error analysis of coarse-grained kinetic Monte Carlo method
In this paper we investigate the approximation properties of the
coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice
stochastic dynamics. We provide both analytical and numerical evidence that the
hierarchy of the coarse models is built in a systematic way that allows for
error control in both transient and long-time simulations. We demonstrate that
the numerical accuracy of the CGMC algorithm as an approximation of stochastic
lattice spin flip dynamics is of order two in terms of the coarse-graining
ratio and that the natural small parameter is the coarse-graining ratio over
the range of particle/particle interactions. The error estimate is shown to
hold in the weak convergence sense. We employ the derived analytical results to
guide CGMC algorithms and we demonstrate a CPU speed-up in demanding
computational regimes that involve nucleation, phase transitions and
metastability.Comment: 30 page
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