The object of this paper is to transform a set of local chain complexes to a single global complex using an equivalence relation of congruence of cells, solving topologically the numerical inaccuracies of floating-point arithmetics. While computing the space arrangement generated by a collection of cellular complexes, one may start from independently and efficiently computing the intersection of each single input 2-cell with the others. The topology of these intersections is codified within a set of (0-2)-dimensional chain complexes. The target of this paper is to merge the local chains by using the equivalence relations of {epsilon}-congruence between 0-, 1-, and 2-cells (elementary chains). In particular, we reduce the block-diagonal coboundary matrices [Delta_0] and [Delta_1], used as matrix accumulators of the local coboundary chains, to the global matrices [delta_0] and [delta_1], representative of congruence topology, i.e., of congruence quotients between all 0-,1-,2-cells, via elementary algebraic operations on their columns. This algorithm is codified using the Julia porting of the SuiteSparse:GraphBLAS implementation of the GraphBLAS standard, conceived to efficiently compute algorithms on large graphs using linear algebra and sparse matrices [1, 2].

Delmonte, G., Onofri, E., Scorzelli, G., Paoluzzi, A. (2020). Local congruence of chain complexes.

Local congruence of chain complexes

Elia Onofri;Giorgio Scorzelli;Alberto Paoluzzi
2020

Abstract

The object of this paper is to transform a set of local chain complexes to a single global complex using an equivalence relation of congruence of cells, solving topologically the numerical inaccuracies of floating-point arithmetics. While computing the space arrangement generated by a collection of cellular complexes, one may start from independently and efficiently computing the intersection of each single input 2-cell with the others. The topology of these intersections is codified within a set of (0-2)-dimensional chain complexes. The target of this paper is to merge the local chains by using the equivalence relations of {epsilon}-congruence between 0-, 1-, and 2-cells (elementary chains). In particular, we reduce the block-diagonal coboundary matrices [Delta_0] and [Delta_1], used as matrix accumulators of the local coboundary chains, to the global matrices [delta_0] and [delta_1], representative of congruence topology, i.e., of congruence quotients between all 0-,1-,2-cells, via elementary algebraic operations on their columns. This algorithm is codified using the Julia porting of the SuiteSparse:GraphBLAS implementation of the GraphBLAS standard, conceived to efficiently compute algorithms on large graphs using linear algebra and sparse matrices [1, 2].
Delmonte, G., Onofri, E., Scorzelli, G., Paoluzzi, A. (2020). Local congruence of chain complexes.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11590/379203
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