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Probabilistic and numerical validation of homology computations for nodal domains
Author(s):
Sarah
Day;
William
D.
Kalies;
Konstantin
Mischaikow;
Thomas
Wanner
Journal:
Electron. Res. Announc. Amer. Math. Soc.
13
(2007),
60-73.
MSC (2000):
Primary 60G60, 55-04;
Secondary 55N99, 60G15
Posted:
July 11, 2007
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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:
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
Posted:
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
Copyright of article:
Copyright
2007,
American Mathematical Society
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