2,430 research outputs found

    An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal

    Full text link
    The movement of many organisms can be described as a random walk at either or both the individual and population level. The rules for this random walk are based on complex biological processes and it may be difficult to develop a tractable, quantitatively-accurate, individual-level model. However, important problems in areas ranging from ecology to medicine involve large collections of individuals, and a further intellectual challenge is to model population-level behavior based on a detailed individual-level model. Because of the large number of interacting individuals and because the individual-level model is complex, classical direct Monte Carlo simulations can be very slow, and often of little practical use. In this case, an equation-free approach may provide effective methods for the analysis and simulation of individual-based models. In this paper we analyze equation-free coarse projective integration. For analytical purposes, we start with known partial differential equations describing biological random walks and we study the projective integration of these equations. In particular, we illustrate how to accelerate explicit numerical methods for solving these equations. Then we present illustrative kinetic Monte Carlo simulations of these random walks and show a decrease in computational time by as much as a factor of a thousand can be obtained by exploiting the ideas developed by analysis of the closed form PDEs. The illustrative biological example here is chemotaxis, but it could be any random walker which biases its movement in response to environmental cues.Comment: 30 pages, submitted to Physica

    Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation

    Get PDF
    We present computer-assisted methods for analyzing stochastic models of gene regulatory networks. The main idea that underlies this equation-free analysis is the design and execution of appropriately-initialized short bursts of stochastic simulations; the results of these are processed to estimate coarse-grained quantities of interest, such as mesoscopic transport coefficients. In particular, using a simple model of a genetic toggle switch, we illustrate the computation of an effective free energy and of a state-dependent effective diffusion coefficient that characterize an unavailable effective Fokker-Planck equation. Additionally we illustrate the linking of equation-free techniques with continuation methods for performing a form of stochastic "bifurcation analysis"; estimation of mean switching times in the case of a bistable switch is also implemented in this equation-free context. The accuracy of our methods is tested by direct comparison with long-time stochastic simulations. This type of equation-free analysis appears to be a promising approach to computing features of the long-time, coarse-grained behavior of certain classes of complex stochastic models of gene regulatory networks, circumventing the need for long Monte Carlo simulations.Comment: 33 pages, submitted to The Journal of Chemical Physic

    Rotated sphere packing designs

    Full text link
    We propose a new class of space-filling designs called rotated sphere packing designs for computer experiments. The approach starts from the asymptotically optimal positioning of identical balls that covers the unit cube. Properly scaled, rotated, translated and extracted, such designs are excellent in maximin distance criterion, low in discrepancy, good in projective uniformity and thus useful in both prediction and numerical integration purposes. We provide a fast algorithm to construct such designs for any numbers of dimensions and points with R codes available online. Theoretical and numerical results are also provided
    • …
    corecore