1 research outputs found
Local congruence of chain complexes
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].Comment: to submi