Abstract. The collective behavior of bacterial populations provides an example of how cell-level decision making translates into population-level behavior and illustrates clearly the difficult multiscale mathematical problem of incorporating individual-level behavior into population-level models. Here we focus on the flagellated bacterium E. coli, for which a great deal is known about signal detection, transduction, and cell-level swimming behavior. We review the biological background on individual- and population-level processes and discuss the velocity-jump approach used for describing population-level behavior based on individual-level intracellular processes. In particular, we generalize the moment-based approach to macroscopic equations used earlier [R. Erban and H. G. Othmer, SIAM J. Appl. Math., 65 (2004), pp. 361–391] to higher dimensions and show how aspects of the signal transduction and response enter into the macroscopic equations. We also discuss computational issues surrounding the bacterial pattern formation problem and technical issues involved in the derivation of macroscopic equations.

Abstract not found

An important class of problems exhibits macroscopically smooth behaviour in space and time, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, the gap-tooth scheme has recently been proposed. The scheme approximates the evolution of an unavailable (in closed form) macroscopic equation in a macroscopic domain; it only uses appropriately initialized simulations of the available microscopic model in a number of small boxes. For some model problems, including numerical homogenization, the scheme is essentially equivalent to a finite difference scheme, provided we repeatedly impose appropriate algebraic constraints on the solution for each box. Here, we demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffers that “shield” over relatively short times the dynamics inside each box from boundary effects. We explore and quantify the behavior of these schemes systematically through the numerical computation of damping factors of the corresponding coarse time-stepper, for which no closed formula is available. 1

Abstract. We present an “equation-free ” multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast in the form of stochastic differential equations. A detailed fine-scale computation of such a problem requires discretization and solution of a large system of equations, and can be prohibitively time-consuming. To circumvent this difficulty, we propose an equation-free approach, where the fine-scale computation is conducted only for a (small) fraction of the overall time. The evolution of a set of appropriately defined coarse-grained variables (observables) is evaluated during the fine-scale computation, and “projective integration” is used to accelerate the integration. The choice of these coarse variables is an important part of the approach: they are the coefficients of pointwise polynomial expansions of the random solutions. Such a choice of coarse variables allows us to reconstruct representative ensembles of fine-scale solutions with “correct ” correlation structures, which is a key to algorithm efficiency. Numerical examples demonstrating accuracy and efficiency of the approach are presented. Key words. multiscale problem, diffusion in random media, stochastic modeling, equation-free.

Summary. We consider problems in which there is a separation between the (mi-croscopic) scale at which the available model is defined, and the (macroscopic) scale of interest. For time-dependent multi-scale problems of this type, an “equation-free” framework has been proposed, of which patch dynamics is an essential component. Patch dynamics is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales; it only uses appro-priately initialized simulations of the available microscopic model in a number of small boxes (patches), which cover a fraction of the space-time domain. We review some recent convergence results and demonstrate that the method allows to simulate advection-dominated problems accurately. 1

Holistic discretisation ensures fidelity to dynamics in two spatial dimensions

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equation-free computational approach for extracting population-level

Abstract not found

The engineering analysis and microscopic simulations required for predicting materials’ properties from atomistic descriptions require new approaches for predicting macroscopic properties. Patch dynamics bridges the gap between the time and space scales at which the microscopic models operate, helping predict system-level behavior.