# Random domain decomposition for parabolic PDEs on graphs

## TL;DR

This paper introduces a stochastic domain decomposition method using the Random Batch Method for efficiently solving parabolic PDEs on complex graph structures, reducing computational resources while maintaining accuracy.

## Contribution

The paper applies RBM directly at the PDE level on graphs without prior discretization, proving convergence and demonstrating significant computational and memory savings.

## Key findings

- Convergence of the RBM scheme with first-order accuracy.
- Substantial reductions in memory and computational time.
- Method effectiveness across various time discretization schemes.

## Abstract

The simulation of complex systems, such as gas transport in large pipeline networks, often involves solving PDEs posed on intricate graph structures. Such problems require considerable computational and memory resources. The Random Batch Method (RBM) has shown promise in addressing these challenges via stochastic decomposition techniques. In this paper, we apply the RBM at the PDE level for parabolic equations on graphs, without assuming any preliminary discretization in space or time. We consider a non-overlapping domain decomposition in which the PDE coefficients and source terms are randomized. We prove that the resulting RBM-based scheme converges, in the mean-square sense and uniformly in time, to the true PDE solution with first-order accuracy in the RBM step size. Numerical experiments confirm this convergence rate and demonstrate substantial reductions in both memory usage and computational time compared to solving on the full graph. Moreover, these advantages persist across different time discretization schemes.