Geodesic
Probabilistic and numerical validation of homology computations for nodal domains
Day, Sarah1 ; Kalies, William2 ; Mischaikow, Konstantin3 ; Wanner, Thomas4
1 College of William and Mary, Department of Mathematics, P.O. Box 8795, Williamsburg, VA 23187
2 Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431
3 Department of Mathematics, Rutgers University, 110 Frelinghusen Road, Piscataway, NJ 08854
4 Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS 3F2, Fairfax, VA 22030
Electronic research announcements of the American Mathematical Society, Tome 13 (2007), pp. 60-73 Cet article a éte moissonné depuis la source American Mathematical Society

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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.
DOI : 10.1090/S1079-6762-07-00175-8

Day, Sarah 1 ; Kalies, William 2 ; Mischaikow, Konstantin 3 ; Wanner, Thomas 4

1 College of William and Mary, Department of Mathematics, P.O. Box 8795, Williamsburg, VA 23187
2 Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431
3 Department of Mathematics, Rutgers University, 110 Frelinghusen Road, Piscataway, NJ 08854
4 Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS 3F2, Fairfax, VA 22030 
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Day, Sarah; Kalies, William; Mischaikow, Konstantin; Wanner, Thomas. Probabilistic and numerical validation of homology computations for nodal domains. Electronic research announcements of the American Mathematical Society, Tome 13 (2007), pp. 60-73. doi: 10.1090/S1079-6762-07-00175-8

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