Coarse projective kMC integration: forward/reverse initial and boundary value problems

Abstract

In ''equation-free'' multiscale computation a dynamic model is given at a fine, microscopic level; yet we believe that its coarse-grained, macroscopic dynamics can be described by closed equations involving only coarse variables. These variables are typically various low-order moments of the distributions evolved through the microscopic model. We consider the problem of integrating these unavailable equations by acting directly on kinetic Monte Carlo microscopic simulators, thus circumventing their derivation in closed form. In particular, we use projective multi-step integration to solve the coarse initial value problem forward in time as well as backward in time (under certain conditions). Macroscopic trajectories are thus traced back to unstable, source-type, and even sometimes saddle-like stationary points, even though the microscopic simulator only evolves forward in time. We also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of ''coarse limit cycles'' of the macroscopic behavior, and the approximation of their stability through estimates of the leading ''coarse Floquet multipliers''.

Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes

“…Such initial conditions are essential for the implementation of equation-free algorithms: algorithms that solve the reduced problem without ever deriving it in closed form [4,25,27]). Indeed, short bursts of appropriately initialized simulations can be used to perform long term prediction (projective and coarse projective integration) for the reduced problem, its stability and bifurcation analysis, as well as tasks like control and optimization.…”

Local diffusion models for stochastic reacting systems: estimation issues in equation-free numerics

“…When the ODEs in the above system are not explicitly available (as will be the case when we use the output of a noisy atomistic simulator to estimate temporal derivatives by using regression in time), the RHS of the second set of ODEs is sometimes estimated using a finite difference stencil [56]. In such cases, the monodromy matrix associated with the coarse limit cycle can be approximated starting at the initial conditions above and performing numerical integration of the above ODEs through some standard temporal discretization scheme.…”

“…Coarse-grained limit cycle calculations using the projective integration formalism were first reported in Ref. [56]. In Ref.…”

“…In previous studies [27,37,38,40,56] such so-called equation-free numerical methods appealed to regression in time in order to estimate quantities associated with effective equations "ondemand" from fine scale simulation data. Here we describe and implement a local estimation strategy that focuses on state-space estimation [31].…”

Strong Convergence Rate for Two-Time-Scale Jump-Diffusion Stochastic Differential Systems

“…This model consists of no drift, only jumps. In the past five years extensive progress has been made in describing the effective dynamics for chemical kinetic systems that take place on vastly different time scales [13,14,15,16].…”