3,678 research outputs found
Accurate stationary densities with partitioned numerical methods for stochastic partial differential equations
We consider the numerical solution, by finite differences, of second-order-in-time stochastic partial differential equations (SPDEs) in one space dimension. New timestepping methods are introduced by generalising recently-introduced methods for second-order-in-time stochastic differential equations to multidimensional systems. These stochastic methods, based on leapfrog and RungeâKutta methods, are designed to give good approximations to the stationary variances and the correlations in the position and velocity variables. In particular, we introduce the reverse leapfrog method and stochastic RungeâKutta Leapfrog methods, analyse their performance applied to linear SPDEs and perform numerical experiments to examine their accuracy applied to a type of nonlinear SPDE
Fractional Spectral Moments for Digital Simulation of Multivariate Wind Velocity Fields
In this paper, a method for the digital simulation of wind velocity fields by
Fractional Spectral Moment function is proposed. It is shown that by
constructing a digital filter whose coefficients are the fractional spectral
moments, it is possible to simulate samples of the target process as
superposition of Riesz fractional derivatives of a Gaussian white noise
processes. The key of this simulation technique is the generalized Taylor
expansion proposed by the authors. The method is extended to multivariate
processes and practical issues on the implementation of the method are
reported.Comment: 12 pages, 2 figure
Many-server queues with customer abandonment: numerical analysis of their diffusion models
We use multidimensional diffusion processes to approximate the dynamics of a
queue served by many parallel servers. The queue is served in the
first-in-first-out (FIFO) order and the customers waiting in queue may abandon
the system without service. Two diffusion models are proposed in this paper.
They differ in how the patience time distribution is built into them. The first
diffusion model uses the patience time density at zero and the second one uses
the entire patience time distribution. To analyze these diffusion models, we
develop a numerical algorithm for computing the stationary distribution of such
a diffusion process. A crucial part of the algorithm is to choose an
appropriate reference density. Using a conjecture on the tail behavior of a
limit queue length process, we propose a systematic approach to constructing a
reference density. With the proposed reference density, the algorithm is shown
to converge quickly in numerical experiments. These experiments also show that
the diffusion models are good approximations for many-server queues, sometimes
for queues with as few as twenty servers
General order conditions for stochastic partitioned Runge-Kutta methods
In this paper stochastic partitioned Runge-Kutta (SPRK) methods are
considered. A general order theory for SPRK methods based on stochastic
B-series and multicolored, multishaped rooted trees is developed. The theory is
applied to prove the order of some known methods, and it is shown how the
number of order conditions can be reduced in some special cases, especially
that the conditions for preserving quadratic invariants can be used as
simplifying assumptions
Rational Construction of Stochastic Numerical Methods for Molecular Sampling
In this article, we focus on the sampling of the configurational
Gibbs-Boltzmann distribution, that is, the calculation of averages of functions
of the position coordinates of a molecular -body system modelled at constant
temperature. We show how a formal series expansion of the invariant measure of
a Langevin dynamics numerical method can be obtained in a straightforward way
using the Baker-Campbell-Hausdorff lemma. We then compare Langevin dynamics
integrators in terms of their invariant distributions and demonstrate a
superconvergence property (4th order accuracy where only 2nd order would be
expected) of one method in the high friction limit; this method, moreover, can
be reduced to a simple modification of the Euler-Maruyama method for Brownian
dynamics involving a non-Markovian (coloured noise) random process. In the
Brownian dynamics case, 2nd order accuracy of the invariant density is
achieved. All methods considered are efficient for molecular applications
(requiring one force evaluation per timestep) and of a simple form. In fully
resolved (long run) molecular dynamics simulations, for our favoured method, we
observe up to two orders of magnitude improvement in configurational sampling
accuracy for given stepsize with no evident reduction in the size of the
largest usable timestep compared to common alternative methods
Going from microscopic to macroscopic on nonuniform growing domains
Throughout development, chemical cues are employed to guide the functional specification of underlying tissues while the spatiotemporal distributions of such chemicals can be influenced by the growth of the tissue itself. These chemicals, termed morphogens, are often modeled using partial differential equations (PDEs). The connection between discrete stochastic and deterministic continuum models of particle migration on growing domains was elucidated by Baker, Yates, and Erban [ Bull. Math. Biol. 72 719 (2010)] in which the migration of individual particles was modeled as an on-lattice position-jump process. We build on this work by incorporating a more physically reasonable description of domain growth. Instead of allowing underlying lattice elements to instantaneously double in size and divide, we allow incremental element growth and splitting upon reaching a predefined threshold size. Such a description of domain growth necessitates a nonuniform partition of the domain. We first demonstrate that an individual-based stochastic model for particle diffusion on such a nonuniform domain partition is equivalent to a PDE model of the same phenomenon on a nongrowing domain, providing the transition rates (which we derive) are chosen correctly and we partition the domain in the correct manner. We extend this analysis to the case where the domain is allowed to change in size, altering the transition rates as necessary. Through application of the master equation formalism we derive a PDE for particle density on this growing domain and corroborate our findings with numerical simulations
GPU accelerated Monte Carlo simulation of Brownian motors dynamics with CUDA
This work presents an updated and extended guide on methods of a proper
acceleration of the Monte Carlo integration of stochastic differential
equations with the commonly available NVIDIA Graphics Processing Units using
the CUDA programming environment. We outline the general aspects of the
scientific computing on graphics cards and demonstrate them with two models of
a well known phenomenon of the noise induced transport of Brownian motors in
periodic structures. As a source of fluctuations in the considered systems we
selected the three most commonly occurring noises: the Gaussian white noise,
the white Poissonian noise and the dichotomous process also known as a random
telegraph signal. The detailed discussion on various aspects of the applied
numerical schemes is also presented. The measured speedup can be of the
astonishing order of about 3000 when compared to a typical CPU. This number
significantly expands the range of problems solvable by use of stochastic
simulations, allowing even an interactive research in some cases.Comment: 21 pages, 5 figures; Comput. Phys. Commun., accepted, 201
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