Probabilistic validation of homology computations for nodal domains

• 成果类型: Article
• 署名作者: Mischaikow, Konstantin; Wanner, Thomas
• 署名单位: Rutgers University System; Rutgers University New Brunswick; George Mason University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051607000000050
• 发表日期: 2007
• 页码: 980-1018
• 关键词: fe-cr alloys; spinodal decomposition; computer-models; atomic-level; zeros

摘要

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.