We present an efficient algorithm for the construction of a basis of (Formula presented.) via the Poincaré-Lefschetz duality theorem. Denoting by g the first Betti number of (Formula presented.) the idea is to find, first g different 1-boundaries of (Formula presented.) with supports contained in ∂Ω whose homology classes in (Formula presented.) form a basis of (Formula presented.), and then to construct a set of 2-chains in (Formula presented.) having these 1-boundaries as their boundaries. The Poincaré-Lefschetz duality theorem ensures that the relative homology classes of these 2-chains in (Formula presented.) modulo ∂Ω form a basis of (Formula presented.). We devise a simple procedure for the construction of the required set of 1-boundaries of (Formula presented.) that, combined with a fast algorithm for the construction of 2-chains with prescribed boundary, allows the efficient computation of a basis of (Formula presented.) via this very natural approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms.
Efficient construction of 2-chains representing a basis of $H_2(overline Omega , partial Omega ; mathbb Z)$ / Alonso Rodríguez, Ana; Bertolazzi, Enrico; Ghiloni, Riccardo; Specogna, Ruben. - In: ADVANCES IN COMPUTATIONAL MATHEMATICS. - ISSN 1019-7168. - STAMPA. - 44:5(2018), pp. 1411-1440. [10.1007/s10444-018-9588-6]
Efficient construction of 2-chains representing a basis of $H_2(overline Omega , partial Omega ; mathbb Z)$
Alonso Rodríguez, Ana;Bertolazzi, Enrico;Ghiloni, Riccardo;
2018-01-01
Abstract
We present an efficient algorithm for the construction of a basis of (Formula presented.) via the Poincaré-Lefschetz duality theorem. Denoting by g the first Betti number of (Formula presented.) the idea is to find, first g different 1-boundaries of (Formula presented.) with supports contained in ∂Ω whose homology classes in (Formula presented.) form a basis of (Formula presented.), and then to construct a set of 2-chains in (Formula presented.) having these 1-boundaries as their boundaries. The Poincaré-Lefschetz duality theorem ensures that the relative homology classes of these 2-chains in (Formula presented.) modulo ∂Ω form a basis of (Formula presented.). We devise a simple procedure for the construction of the required set of 1-boundaries of (Formula presented.) that, combined with a fast algorithm for the construction of 2-chains with prescribed boundary, allows the efficient computation of a basis of (Formula presented.) via this very natural approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms.File | Dimensione | Formato | |
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AlonsoRodríguez2018_Article_EfficientConstructionOf2-chain.pdf
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