380 research outputs found
Delaunay Hodge Star
We define signed dual volumes at all dimensions for circumcentric dual
meshes. We show that for pairwise Delaunay triangulations with mild boundary
assumptions these signed dual volumes are positive. This allows the use of such
Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge
star operator can now be correctly defined for such meshes. This operator is
crucial for DEC and is a diagonal matrix with the ratio of primal and dual
volumes along the diagonal. A correct definition requires that all entries be
positive. DEC is a framework for numerically solving differential equations on
meshes and for geometry processing tasks and has had considerable impact in
computer graphics and scientific computing. Our result allows the use of DEC
with a much larger class of meshes than was previously considered possible.Comment: Corrected error in Figure 1 (columns 3 and 4) and Figure 6 and a
formula error in Section 2. All mathematical statements (theorems and lemmas)
are unchanged. The previous arXiv version v3 (minus the Appendix) appeared in
the journal Computer-Aided Desig
Numerical Experiments for Darcy Flow on a Surface Using Mixed Exterior Calculus Methods
There are very few results on mixed finite element methods on surfaces. A
theory for the study of such methods was given recently by Holst and Stern,
using a variational crimes framework in the context of finite element exterior
calculus. However, we are not aware of any numerical experiments where mixed
finite elements derived from discretizations of exterior calculus are used for
a surface domain. This short note shows results of our preliminary experiments
using mixed methods for Darcy flow (hence scalar Poisson's equation in mixed
form) on surfaces. We demonstrate two numerical methods. One is derived from
the primal-dual Discrete Exterior Calculus and the other from lowest order
finite element exterior calculus. The programming was done in the language
Python, using the PyDEC package which makes the code very short and easy to
read. The qualitative convergence studies seem to be promising.Comment: 14 pages, 11 figure
Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties
The main purpose of this paper is to present a conceptual approach to
understanding the extension of the Prym map from the space of admissible double
covers of stable curves to different toroidal compactifications of the moduli
space of principally polarized abelian varieties. By separating the
combinatorial problems from the geometric aspects we can reduce this to the
computation of certain monodromy cones. In this way we not only shed new light
on the extension results of Alexeev, Birkenhake, Hulek, and Vologodsky for the
second Voronoi toroidal compactification, but we also apply this to other
toroidal compactifications, in particular the perfect cone compactification,
for which we obtain a combinatorial characterization of the indeterminacy
locus, as well as a geometric description up to codimension six, and an
explicit toroidal resolution of the Prym map up to codimension four.Comment: 53 pages, AMS LaTeX, Appendix by Mathieu Dutour Sikiric, minor
revisions, to appear in JEM
PyDEC: Software and Algorithms for Discretization of Exterior Calculus
This paper describes the algorithms, features and implementation of PyDEC, a
Python library for computations related to the discretization of exterior
calculus. PyDEC facilitates inquiry into both physical problems on manifolds as
well as purely topological problems on abstract complexes. We describe
efficient algorithms for constructing the operators and objects that arise in
discrete exterior calculus, lowest order finite element exterior calculus and
in related topological problems. Our algorithms are formulated in terms of
high-level matrix operations which extend to arbitrary dimension. As a result,
our implementations map well to the facilities of numerical libraries such as
NumPy and SciPy. The availability of such libraries makes Python suitable for
prototyping numerical methods. We demonstrate how PyDEC is used to solve
physical and topological problems through several concise examples.Comment: Revised as per referee reports. Added information on scalability,
removed redundant text, emphasized the role of matrix based algorithms,
shortened length of pape
Discrete Riemann Surfaces and the Ising model
We define a new theory of discrete Riemann surfaces and present its basic
results. The key idea is to consider not only a cellular decomposition of a
surface, but the union with its dual. Discrete holomorphy is defined by a
straightforward discretisation of the Cauchy-Riemann equation. A lot of
classical results in Riemann theory have a discrete counterpart, Hodge star,
harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of
holomorphic forms with prescribed holonomies. Giving a geometrical meaning to
the construction on a Riemann surface, we define a notion of criticality on
which we prove a continuous limit theorem. We investigate its connection with
criticality in the Ising model. We set up a Dirac equation on a discrete
universal spin structure and we prove that the existence of a Dirac spinor is
equivalent to criticality
- …