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
DOI:
https://doi.org/10.1090/S1079-6762-07-00175-8
Published electronically:
July 11, 2007
MathSciNet review:
2320683
Full-text PDF Free Access
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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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adler:81a R. J. Adler. The Geometry of Random Fields. John Wiley & Sons Ltd., Chichester, 1981.
bharuchareid:s:86a A. T. Bharucha-Reid and M. Sambandham. Random Polynomials. Academic Press, Orlando, 1986.
bloemker:etal:01b D. Blömker, S. Maier-Paape, and T. Wanner. Spinodal decomposition for the Cahn-Hilliard-Cook equation. Communications in Mathematical Physics, 223(3):553–582, 2001.
bloemker:etal:p02a D. Blömker, S. Maier-Paape, and T. Wanner. Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation. Transactions of the American Mathematical Society, to appear.
cahn:hilliard:58a J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system I. Interfacial free energy. Journal of Chemical Physics, 28:258–267, 1958.
carlsson:desilva:03 G. Carlsson and V. de Silva. Topological approximation by small simplicial complexes. Preprint, 2003.
cook:70a H. Cook. Brownian motion in spinodal decomposition. Acta Metallurgica, 18:297–306, 1970.
day:etal:p06a S. Day, W. D. Kalies, and T. Wanner. Homology computations of nodal domains: Accuracy estimates and validation. In preparation, 2007.
desilva:carlsson:04 V. de Silva and G. Carlsson. Topological estimation using witness complexes. In M. Alexa and S. Rusinkiewicz, editors, Eurographics Symposium on Point-Based Graphics. The Eurographics Association, 2004.
donoho:grimes:03 D. L. Donoho and C. Grimes. Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. Proceedings of the National Academy of Science, 100(10):5591–5596, 2003.
dunnage:66a J. E. A. Dunnage. The number of real zeros of a random trigonometric polynomial. Proceedings of the London Mathematical Society, 16:53–84, 1966.
farahmand:98a K. Farahmand. Topics in Random Polynomials, volume 393 of Pitman Research Notes in Mathematics. Longman, Harlow, 1998.
gameiro:etal:04a M. Gameiro, K. Mischaikow, and W. Kalies. Topological characterization of spatial-temporal chaos. Physical Review E, 70(3):035203, 4, 2004.
gameiro:etal:05a M. Gameiro, K. Mischaikow, and T. Wanner. Evolution of pattern complexity in the Cahn-Hilliard theory of phase separation. Acta Materialia, 53(3):693–704, 2005.
kaczynski:mischaikow:mrozek:04 T. Kaczynski, K. Mischaikow, and M. Mrozek. Computational homology, volume 157 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.
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chomp W. Kalies, M. Mrozek, and P. Pilarczyk. Computational homology project. http://www.math.gatech.edu/~chomp/, 2006.
krishan:etal:p05a K. Krishan, M. Gameiro, K. Mischaikow, and M. F. Schatz. Homological characterization of spiral defect chaos in Rayleigh-Benard convection. Preprint, 2005.
marcus:pisier:81a M. B. Marcus and G. Pisier. Random Fourier Series with Applications to Harmonic Analysis, volume 101 of Annals of Mathematics Studies. Princeton University Press, Princeton, 1981.
mischaikow:wanner:p06a K. Mischaikow and T. Wanner. Probabilistic validation of homology computations for nodal domains. Annals of Applied Probability 17(3):980-1018, 2007.
niyogi:smale:weinberger:p04a P. Niyogi, S. Smale, and S. Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete and Computational Geometry, 2007. To appear.
roweis:saul:00 S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290:2323–2326, 2000.
sander:wanner:99a E. Sander and T. Wanner. Monte Carlo simulations for spinodal decomposition. Journal of Statistical Physics, 95(5–6):925–948, 1999.
sander:wanner:00a E. Sander and T. Wanner. Unexpectedly linear behavior for the Cahn-Hilliard equation. SIAM Journal on Applied Mathematics, 60(6):2182–2202, 2000.
tenenbaum:etal:00 J. B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290:2319–2323, 2000.
wanner:04a T. Wanner. Maximum norms of random sums and transient pattern formation. Transactions of the American Mathematical Society, 356(6):2251–2279, 2004.
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Additional Information
Sarah Day
Affiliation:
College of William and Mary, Department of Mathematics, P.O. Box 8795, Williamsburg, VA 23187
MR Author ID:
646696
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
MR Author ID:
249919
Email:
mischaik@math.rutgers.edu
Thomas Wanner
Affiliation:
Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS 3F2, Fairfax, VA 22030
MR Author ID:
262105
Email:
wanner@math.gmu.edu
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
