6 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
Proto-Plasm: parallel language for adaptive and scalable modelling of biosystems
This paper discusses the design goals and the first developments of
Proto-Plasm, a novel computational environment to produce libraries
of executable, combinable and customizable computer models of natural and
synthetic biosystems, aiming to provide a supporting framework for predictive
understanding of structure and behaviour through multiscale geometric modelling
and multiphysics simulations. Admittedly, the Proto-Plasm platform is
still in its infancy. Its computational framework—language, model library,
integrated development environment and parallel engine—intends to provide
patient-specific computational modelling and simulation of organs and biosystem,
exploiting novel functionalities resulting from the symbolic combination of
parametrized models of parts at various scales. Proto-Plasm may define
the model equations, but it is currently focused on the symbolic description of
model geometry and on the parallel support of simulations. Conversely, CellML
and SBML could be viewed as defining the behavioural functions (the model
equations) to be used within a Proto-Plasm program. Here we exemplify
the basic functionalities of Proto-Plasm, by constructing a schematic
heart model. We also discuss multiscale issues with reference to the geometric
and physical modelling of neuromuscular junctions
Chain-Based Representations for Solid and Physical Modeling
In this paper we show that the (co)chain complex associated with a
decomposition of the computational domain, commonly called a mesh in
computational science and engineering, can be represented by a block-bidiagonal
matrix that we call the Hasse matrix. Moreover, we show that
topology-preserving mesh refinements, produced by the action of (the simplest)
Euler operators, can be reduced to multilinear transformations of the Hasse
matrix representing the complex. Our main result is a new representation of the
(co)chain complex underlying field computations, a representation that provides
new insights into the transformations induced by local mesh refinements. Our
approach is based on first principles and is general in that it applies to most
representational domains that can be characterized as cell complexes, without
any restrictions on their type, dimension, codimension, orientability,
manifoldness, connectedness
Solid and Physical Modeling with Chain Complexes
Copyright © 2007 by the Association for Computing Machinery, Inc. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Reques
Abstract Solid and Physical Modeling with Chain Complexes
In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topologypreserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multi-linear transformations of the Hasse matrix representing the complex. Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. This paper is a further contribution towards bridging the subject of computer representations for solid and physical modeling—which flourished border-line between computer graphics, engineering mechanic