New Monte Carlo schemes for simulating diffusions in discontinuous media

International audienceWe introduce new Monte Carlo simulation schemes for diffusions in a discontinuous media divided in subdomains with piecewise constant diffusivity. These schemes are higher order extensions of the usual schemes and take into account the two dimensional aspects of the diffusion at the interface between subdomains. This is achieved using either stochastic processes techniques or an approach based on finite differences. Numerical tests on elliptic, parabolic and eigenvalue problems involving an operator in divergence form show the efficiency of these new schemes

How does variability in cells aging and growth rates influence the malthus parameter?

The aim of this study is to compare the growth speed of different cell populations measured by their Malthus parameter. We focus on both the age-structured and size-structured equations. A first population (of reference) is composed of cells all aging or growing at the same rate $\bar v$. A second population (with variability) is composed of cells each aging or growing at a rate $v$ drawn according to a non-degenerated distribution $\rho$ with mean $\bar v$. In a first part, analytical answers -- based on the study of an eigenproblem -- are provided for the age-structured model. In a second part, numerical answers -- based on stochastic simulations -- are derived for the size-structured model. It appears numerically that the population with variability proliferates more slowly than the population of reference (for experimentally plausible division rates). The decrease in the Malthus parameter we measure, around 2% for distributions $\rho$ with realistic coefficients of variations around 15-20\%, is determinant since it controls the {\it exponential} growth of the whole population

Numerical Analysis of Parallel Replica Dynamics

Parallel replica dynamics is a method for accelerating the computation of processes characterized by a sequence of infrequent events. In this work, the processes are governed by the overdamped Langevin equation. Such processes spend much of their time about the minima of the underlying potential, occasionally transitioning into different basins of attraction. The essential idea of parallel replica dynamics is that the exit time distribution from a given well for a single process can be approximated by the minimum of the exit time distributions of $N$ independent identical processes, each run for only 1/N-th the amount of time. While promising, this leads to a series of numerical analysis questions about the accuracy of the exit distributions. Building upon the recent work in Le Bris et al., we prove a unified error estimate on the exit distributions of the algorithm against an unaccelerated process. Furthermore, we study a dephasing mechanism, and prove that it will successfully complete.Comment: 37 pages, 4 figures, revised and new estimates from the previous versio

Simulating diffusions with piecewise constant coefficients using a kinetic approximation

International audienceUsing a kinetic approximation of a linear diffusion operator, we propose an algorithm that allows one to deal with the simulation of a multi-dimensional stochastic process in a media which is locally isotropic except on some surface where the diffusion coefficient presents some discontinuities. Numerical examples are given in dimensions one to three on PDEs or stochastic PDEs with or without source terms

Simulation of diffusions by means of importance sampling paradigm

The aim of this paper is to introduce a new Monte Carlo method based on importance sampling techniques for the simulation of stochastic differential equations. The main idea is to combine random walk on squares or rectangles methods with importance sampling techniques. The first interest of this approach is that the weights can be easily computed from the density of the one-dimensional Brownian motion. Compared to the Euler scheme this method allows one to obtain a more accurate approximation of diffusions when one has to consider complex boundary conditions. The method provides also an interesting alternative to performing variance reduction techniques and simulating rare events.Comment: Published in at http://dx.doi.org/10.1214/09-AAP659 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Estimation of the mean residence time in cells surrounded by semi-permeable membranes by a Monte Carlo method

This report aims at validating a Monte Carlo algorithm to simulate the behavior of diffusive particles in a mediawith semi-permeables membranes seen as approximationsof a thin layer problems. Following some homogenization approachfor solving a diffusion Magnetic Resonance Imaging problem (dMRI), we estimate themean residence time inside a cell living a in one-dimensional periodicmedia and compare the estimated value with the one computedby solving an eigenvalue problem. The numerical analysis shows a good agreement, unless the strength of the membraneis too strong

exitbm: a library for simulating Brownian motion's exit times and positions from simple domains

This document presents some algorithms and formulae that are spread out in the literature regarding the Brownian motions's first exit time and position.This C library aims at computing and simulating various quantities and random variables related to where and when a the Brownian motion hit the boundary of an interval, a square or a rectangle. We present here the algorithms used in this library.Cette bibliothèque en langage C calcule et simule diverses quantités et variables aléatoire reliées au temps et positions de sortie pour un mouvement brownien d'un intervalle, d'un carré ou d'un rectangle. Nous présentons ici les algorithmes utilisés dans cette bibliothèque

PDE Solvers for Hybrid CPU-GPU Architectures

Many problems of scientific and industrial interest are investigated through numerically solving partial differential equations (PDEs). For some of these problems, the scope of the investigation is limited by the costs of computational resources. A new approach to reducing these costs is the use of coprocessors, such as graphics processing units (GPUs) and Many Integrated Core (MIC) cards, which can execute floating point operations at a higher rate than a central processing unit (CPU) of the same cost. This is achieved through the use of a large number of processors in a single device, each with very limited dedicated memory per thread. Codes for a number of continuum methods, such as boundary element methods (BEM), finite element methods (FEM) and finite difference methods (FDM) have already been implemented on coprocessor architectures. These methods were designed before the adoption of coprocessor architectures, so implementing them efficiently with reduced thread-level memory can be challenging. There are other methods that do operate efficiently with limited thread-level memory, such as Monte Carlo methods (MCM) and lattice Boltzmann methods (LBM) for kinetic formulations of PDEs, but they are not competitive on CPUs and generally have poorer convergence than the continuum methods. In this work, we introduce a class of methods in which the parallelism of kinetic formulations on GPUs is combined with the better convergence of continuum methods on CPUs. We first extend an existing Feynman-Kac formulation for determining the principal eigenpair of an elliptic operator to create a version that can retrieve arbitrarily many eigenpairs. This new method is implemented for multiple GPUs, and combined with a standard deflation preconditioner on multiple CPUs to create a hybrid concurrent method with superior convergence to that of the deflation preconditioner alone. The hybrid method exhibits good parallelism, with an efficiency of 80% on a problem with 300 million unknowns, run on a configuration of 324 CPU cores and 54 GPUs.Doctor of Philosoph

Untangling hotel industry’s inefficiency: An SFA approach applied to a renowned Portuguese hotel chain

The present paper explores the technical efficiency of four hotels from Teixeira Duarte Group - a renowned Portuguese hotel chain. An efficiency ranking is established from these four hotel units located in Portugal using Stochastic Frontier Analysis. This methodology allows to discriminate between measurement error and systematic inefficiencies in the estimation process enabling to investigate the main inefficiency causes. Several suggestions concerning efficiency improvement are undertaken for each hotel studied.info:eu-repo/semantics/publishedVersio