Physics inspired algorithms for (co)homology computation

TL;DR

This paper introduces a physics-inspired algorithm for computing the first cohomology group of 3D complexes, achieving significant speedups and enabling solutions to longstanding problems in computational electromagnetics.

Contribution

A novel physics-inspired algorithm for cohomology computation that outperforms existing methods and incorporates the concept of lazy cohomology generators.

Findings

Orders of magnitude speed-up over existing algorithms

Enables solving longstanding problems in low-frequency electromagnetics

Introduces the concept of lazy cohomology generators

Abstract

The issue of computing (co)homology generators of a cell complex is gaining a pivotal role in various branches of science. While this issue can be rigorously solved in polynomial time, it is still overly demanding for large scale problems. Drawing inspiration from low-frequency electrodynamics, this paper presents a physics inspired algorithm for first cohomology group computations on three-dimensional complexes. The algorithm is general and exhibits orders of magnitude speed up with respect to competing ones, allowing to handle problems not addressable before. In particular, when generators are employed in the physical modeling of magneto-quasistatic problems, this algorithm solves one of the most long-lasting problems in low-frequency computational electromagnetics. In this case, the effectiveness of the algorithm and its ease of implementation may be even improved by introducing the novel concept of \textit{lazy cohomology generators}.