A network partition method for solving large-scale complex nonlinear processes

TL;DR

This paper introduces a network partition and operator splitting framework to efficiently and robustly solve large-scale nonlinear differential equations in physics, chemistry, and biology, enabling parallel computation and overcoming traditional numerical difficulties.

Contribution

The paper presents a novel network partition method that reduces splitting steps and facilitates parallel solutions for large-scale nonlinear systems, improving stability and efficiency.

Findings

Convergent solutions for complex nonlinear processes.

Efficient parallel simulation of large-scale systems.

Overcomes numerical instability and convergence issues.

Abstract

A numerical framework based on network partition and operator splitting is developed to solve nonlinear differential equations of large-scale dynamic processes encountered in physics, chemistry and biology. Under the assumption that those dynamic processes can be characterized by sparse networks, we minimize the number of splitting for constructing subproblems by network partition. Then the numerical simulation of the original system is simplified by solving a small number of subproblems, with each containing uncorrelated elementary processes. In this way, numerical difficulties of conventional methods encountered in large-scale systems such as numerical instability, negative solutions, and convergence issue are avoided. In addition, parallel simulations for each subproblem can be achieved, which is beneficial for large-scale systems. Examples with complex underlying nonlinear processes, including chemical reactions and reaction-diffusion on networks, demonstrate that this method generates convergent solution in a efficient and robust way.