Probabilistic validation of homology computations for nodal domains

Details Probabilistic validation of homology computations for nodal domains Probabilistic validation of homology computations for nodal domains

2007-07-31

arxiv.org/abs/0707.4588v1

Annals of Applied Probability 2007, Vol. 17, No. 3, 980-1018

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051607000000050 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051607000000050

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.

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