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ISSN 1079-6762

 

Probabilistic and numerical validation of homology computations for nodal domains


Authors: Sarah Day, William D. Kalies, Konstantin Mischaikow and Thomas Wanner
Journal: Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 60-73
MSC (2000): Primary 60G60, 55-04; Secondary 55N99, 60G15
Published electronically: July 11, 2007
MathSciNet review: 2320683
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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.


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Additional Information

Sarah Day
Affiliation: College of William and Mary, Department of Mathematics, P.O. Box 8795, Williamsburg, VA 23187
Email: sday@math.wm.edu

William D. Kalies
Affiliation: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431
Email: wkalies@fau.edu

Konstantin Mischaikow
Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghusen Road, Piscataway, NJ 08854
Email: mischaik@math.rutgers.edu

Thomas Wanner
Affiliation: Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS 3F2, Fairfax, VA 22030
Email: wanner@math.gmu.edu

DOI: http://dx.doi.org/10.1090/S1079-6762-07-00175-8
PII: S 1079-6762(07)00175-8
Keywords: Homology, random Fourier series, nodal domain
Received by editor(s): November 15, 2006
Published electronically: July 11, 2007
Additional Notes: Sarah Day was partially supported by NSF grant DMS-9983660 at Cornell University and NSF grant DMS-0441170 at MSRI
William Kalies was partially supported by NSF grant DMS-0511208 and DOE grant DE-FG02-05ER25713.
Konstantin Mischaikow was partially supported by NSF grants DMS-0511115 and DMS-0107396, by DARPA, and by DOE grant DE-FG02-05ER25711.
Thomas Wanner was partially supported by NSF grant DMS-0406231 and DOE grant DE-FG02-05ER25712.
Communicated by: Carlos Kenig
Article copyright: © Copyright 2007 American Mathematical Society