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.
Publié le : 2007-06-15
Classification:
Homology,
random fields,
nodal domains,
60G60,
55N99,
60G15,
60G17
@article{1179839180,
author = {Mischaikow, Konstantin and Wanner, Thomas},
title = {Probabilistic validation of homology computations for nodal domains},
journal = {Ann. Appl. Probab.},
volume = {17},
number = {1},
year = {2007},
pages = { 980-1018},
language = {en},
url = {http://dml.mathdoc.fr/item/1179839180}
}
Mischaikow, Konstantin; Wanner, Thomas. Probabilistic validation of homology computations for nodal domains. Ann. Appl. Probab., Tome 17 (2007) no. 1, pp. 980-1018. http://gdmltest.u-ga.fr/item/1179839180/