Efficient generation of Gaussian random fields on metric graphs via domain decomposition and mass matrix lumping

TL;DR

This paper introduces an efficient method for generating Gaussian random fields on metric graphs by combining domain decomposition and mass matrix lumping, significantly improving speed and memory efficiency.

Contribution

It proposes a novel approach that preserves convergence rates while reducing computational complexity and memory usage for sampling Gaussian fields on large graphs.

Findings

Achieves multi-order speedups in sampling process.

Reduces memory usage significantly compared to traditional methods.

Maintains theoretical convergence rates in empirical tests.

Abstract

We consider Gaussian Random Fields on metric graphs defined implicitly as the stationary solution to a fractional SPDE driven by Gaussian white noise. Sampling from the finite element approximation requires the Cholesky factorization of the mass matrix, causing non-linear execution time explosions and massive memory fill-in on large graphs. Hence, we combine Neumann-Neumann graph decomposition with mass matrix lumping and demonstrate empirically, that our approach preserves exact theoretical convergence rates established in [8] while achieving multi-order speedups and massive memory reductions.