Probabilistic and numerical validation of homology computations for nodal domains

40. Sarah Day, William D. Kalies, Konstantin Mischaikow, Thomas Wanner:
    Probabilistic and numerical validation of homology computations for nodal domains
    Electronic Research Announcements of the American Mathematical Society 13, pp. 60-73, 2007.

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 Fourier series in one and two space dimensions, which furnishes explicit probabilistic a-priori bounds for the suitability of certain discretization sizes. In addition, we introduce a numerical method for verifying the homology computation using interval arithmetic.

Links

The published version of the paper can be found at https://doi.org/10.1090/S1079-6762-07-00175-8.

Bibtex

@article{day:etal:07a,
   author = {Sarah Day and William D. Kalies and Konstantin Mischaikow
             and Thomas Wanner},
   title = {Probabilistic and numerical validation of homology computations
            for nodal domains},
   journal = {Electronic Research Announcements of the American
              Mathematical Society},
   year = 2007,
   volume = 13,
   pages = {60--73},
   doi = {10.1090/S1079-6762-07-00175-8}
   }