Digital atlases and difference forms

ABSTRACT

When integrating a differential equation numerically, it can be important for the solution method to reflect the geometric properties of the original model. These include

conservation laws and first integrals, symmetries, and symplectic or variational structures. Thus there is an increasingly sophisticated subject of "geometric integration" concentrating mostly on local properties of the equation.

This talk is concerned with ways of ensuring that finite difference schemes accurately mirror global properties. To this end, lattice varieties are introduced on which finite difference schemes amongst others may be defined. There is no assumption of continuity, or that either the lattice variety or the difference systems have a continuum limit; our theory is more general than that of cubical complexes, and the proofs require a different foundation.

We show that the global structure of a lattice variety can be determined from its digital atlas. This is important for two reasons. First, if the digital atlas has the same "system of intersections" as that of the smooth model it approximates, you are guaranteed the same global information. Secondly, since our proofs are independent of any continuum limit, global information for inherently discrete models may be obtained. The techniques used are algebraic, specifically homological algebra, which amounts to linear algebra.

This talk has two meta-messages: 1) Continuity is an illusion. 2) If you want to capture analytic structures in discrete models successfully, cherchez l'algebre.

No particular expertise is assumed for this talk, which is based on the paper, Difference Forms by Elizabeth L. Mansfield and Peter E. Hydon, to appear in Foundations of Computational Mathematics.

REFERENCES

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• 5: Marian Mrozek , Paweł Pilarczyk , Natalia Żelazna, Homology algorithm based on acyclic subspace, Computers & Mathematics with Applications, v.55 n.11, p.2395-2412, June, 2008 [doi>10.1016/j.camwa.2007.08.044]
• 6: S. Suuriniemi and L. Kettunen, Trade-off between information and complexity: a technique for automated topological computations, COMPEL, 22 (2003), 481--494.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.0 General

General Terms:
Algorithms, Theory

Keywords:
cohomology, difference chains, difference forms, lattice variety, local difference potentials, local exactness, symbolic numeric methods