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".

Submitted to Journal of Computational Physics

References in corpus (6)

• Coarse Molecular Dynamics of a Peptide Fragment: Free Energy, Kinetics, and Long-Time Dynamics Computations
• Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples
• Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation Diagrams via Stochastic Simulation
• Microscopic/stochastic timesteppers and coarse control: a kinetic Monte Carlo example
• Coarse Bifurcation Studies of Bubble Flow Microscopic Simulations
• Reverse Integration for Computing Stationary Points of Unstable Stiff Systems

Cited by in corpus (18)

• Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation
• Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit
• Coarse-graining the dynamics of coupled oscillators
• Coarse Grained Computations for a Micellar System
• Projective and Coarse Projective Integration for Problems with Continuous Symmetries
• A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations
• Exploration of Effective Potential Landscapes using Coarse Reverse Integration
• Multiscale analysis of re-entrant production lines: An equation-free approach
• Projective and Telescopic Projective Integration for Non-Linear Kinetic Mixtures
• Scalable Bayesian reduced-order models for high-dimensional multiscale dynamical systems
• Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes
• Telescopic projective integration for kinetic equations with multiple relaxation times
• A micro/macro algorithm to accelerate Monte Carlo simulation of stochastic differential equations
• Variance reduction for particle filters of systems with time-scale separation
• On the acceleration of spatially distributed agent-based computations: a patch dynamics scheme
• Modeling Heterogeneity in Networks using Uncertainty Quantification Tools
• Equation-free, multiscale computation for unsteady random diffusion
• A toolbox of Equation-Free functions in Matlab\Octave for efficient system level simulation