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Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations.

Liang Qiao¹, Radek Erban, C T Kelley

¹Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA.

The Journal of Chemical Physics

|December 6, 2006

Summary

This study introduces a novel preconditioning method using approximate macroscopic partial differential equations (PDEs) to accelerate stochastic computations in particle-based models. This approach enhances the convergence of equation-free coarse-graining methods for complex systems.

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Area of Science:

Computational Science
Mathematical Modeling
Chemical Physics

Background:

Spatially distributed problems are often modeled using partial differential equations (PDEs).
Deriving accurate PDEs from finer-scale models via averaging is challenging, often relying on approximations like mean-field theory.
Stochastic, particle-based models are crucial for simulating complex processes but can be computationally intensive.

Purpose of the Study:

To develop a preconditioning strategy for accelerating stochastic computations in spatially distributed particle-based models.
To demonstrate the effectiveness of using approximate macroscopic PDEs for this preconditioning.
To improve the convergence of equation-free coarse-graining methods.

Main Methods:

Constructing preconditioners from approximate macroscopic partial differential equations (PDEs).
Applying these preconditioners to accelerate stochastic computations in particle-based process models.
Utilizing coarse timesteppers within equation-free coarse-graining frameworks.
Employing a stochastic reaction-diffusion model exhibiting Turing instabilities as a model problem.

Main Results:

Demonstrated that approximate macroscopic PDEs can be effectively used to construct preconditioners.
Showed significant acceleration of stochastic computations for spatially distributed particle-based models.
Illustrated improved convergence rates for equation-free coarse-graining methods through preconditioning.

Conclusions:

Approximate macroscopic PDEs are valuable for developing efficient preconditioners in computational modeling.
The proposed preconditioning technique offers a robust way to accelerate stochastic simulations.
This method holds promise for studying complex phenomena like Turing instabilities in reaction-diffusion systems.