Probabilistic validation of homology computations for nodal domains

TL;DR

This paper introduces a probabilistic method to assess the accuracy of homology computations for nodal domains of random fields, providing explicit bounds for discretization validity in one and two dimensions.

Contribution

It develops a novel probabilistic framework for validating homology calculations of nodal domains, with explicit bounds and applications to specific random field models.

Findings

Probabilistic bounds for homology computation accuracy.

Application to random periodic fields and trigonometric polynomials.

Quantitative assessment of discretization size effects.

Abstract

Homology has long been accepted as an important computable tool for quantifying complex structures. In many applications, these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurate the resulting homology computations are. In this paper, we present a probabilistic approach to quantifying the validity of homology computations for nodal domains of random fields in one and two space dimensions, which furnishes explicit probabilistic a priori bounds for the suitability of certain discretization sizes. We illustrate our results for the special cases of random periodic fields and random trigonometric polynomials.