754 research outputs found
Postprocessed integrators for the high order integration of ergodic SDEs
The concept of effective order is a popular methodology in the deterministic
literature for the construction of efficient and accurate integrators for
differential equations over long times. The idea is to enhance the accuracy of
a numerical method by using an appropriate change of variables called the
processor. We show that this technique can be extended to the stochastic
context for the construction of new high order integrators for the sampling of
the invariant measure of ergodic systems. The approach is illustrated with
modifications of the stochastic -method applied to Brownian dynamics,
where postprocessors achieving order two are introduced. Numerical experiments,
including stiff ergodic systems, illustrate the efficiency and versatility of
the approach.Comment: 21 pages, to appear in SIAM J. Sci. Compu
Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs
We introduce a new algebraic framework based on a modification (called
exotic) of aromatic Butcher-series for the systematic study of the accuracy of
numerical integrators for the invariant measure of a class of ergodic
stochastic differential equations (SDEs) with additive noise. The proposed
analysis covers Runge-Kutta type schemes including the cases of partitioned
methods and postprocessed methods. We also show that the introduced exotic
aromatic B-series satisfy an isometric equivariance property.Comment: 33 page
Pathwise Accuracy and Ergodicity of Metropolized Integrators for SDEs
Metropolized integrators for ergodic stochastic differential equations (SDE)
are proposed which (i) are ergodic with respect to the (known) equilibrium
distribution of the SDE and (ii) approximate pathwise the solutions of the SDE
on finite time intervals. Both these properties are demonstrated in the paper
and precise strong error estimates are obtained. It is also shown that the
Metropolized integrator retains these properties even in situations where the
drift in the SDE is nonglobally Lipschitz, and vanilla explicit integrators for
SDEs typically become unstable and fail to be ergodic.Comment: 46 pages, 5 figure
Ergodic SDEs on submanifolds and related numerical sampling schemes
In many applications, it is often necessary to sample the mean value of
certain quantity with respect to a probability measure {\mu} on the level set
of a smooth function , .
A specially interesting case is the so-called conditional probability measure,
which is useful in the study of free energy calculation and model reduction of
diffusion processes. By Birkhoff's ergodic theorem, one approach to estimate
the mean value is to compute the time average along an infinitely long
trajectory of an ergodic diffusion process on the level set whose invariant
measure is {\mu}. Motivated by the previous work of Ciccotti, Leli\`evre, and
Vanden-Eijnden [11], as well as the work of Leli\`evre, Rousset, and Stoltz
[33], in this paper we construct a family of ergodic diffusion processes on the
level set of whose invariant measures coincide with the given one. For
the conditional measure, in particular, we show that the corresponding SDEs of
the constructed ergodic processes have relatively simple forms, and, moreover,
we propose a consistent numerical scheme which samples the conditional measure
asymptotically. The numerical scheme doesn't require computing the second
derivatives of and the error estimates of its long time sampling
efficiency are obtained.Comment: 45 pages. Accepted versio
A decreasing step method for strongly oscillating stochastic models
We propose an algorithm for approximating the solution of a strongly
oscillating SDE, that is, a system in which some ergodic state variables evolve
quickly with respect to the other variables. The algorithm profits from
homogenization results and consists of an Euler scheme for the slow scale
variables coupled with a decreasing step estimator for the ergodic averages of
the quick variables. We prove the strong convergence of the algorithm as well
as a C.L.T. like limit result for the normalized error distribution. In
addition, we propose an extrapolated version that has an asymptotically lower
complexity and satisfies the same properties as the original version.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1016 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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