46,114 research outputs found
Engineering Resilient Collective Adaptive Systems by Self-Stabilisation
Collective adaptive systems are an emerging class of networked computational
systems, particularly suited in application domains such as smart cities,
complex sensor networks, and the Internet of Things. These systems tend to
feature large scale, heterogeneity of communication model (including
opportunistic peer-to-peer wireless interaction), and require inherent
self-adaptiveness properties to address unforeseen changes in operating
conditions. In this context, it is extremely difficult (if not seemingly
intractable) to engineer reusable pieces of distributed behaviour so as to make
them provably correct and smoothly composable.
Building on the field calculus, a computational model (and associated
toolchain) capturing the notion of aggregate network-level computation, we
address this problem with an engineering methodology coupling formal theory and
computer simulation. On the one hand, functional properties are addressed by
identifying the largest-to-date field calculus fragment generating
self-stabilising behaviour, guaranteed to eventually attain a correct and
stable final state despite any transient perturbation in state or topology, and
including highly reusable building blocks for information spreading,
aggregation, and time evolution. On the other hand, dynamical properties are
addressed by simulation, empirically evaluating the different performances that
can be obtained by switching between implementations of building blocks with
provably equivalent functional properties. Overall, our methodology sheds light
on how to identify core building blocks of collective behaviour, and how to
select implementations that improve system performance while leaving overall
system function and resiliency properties unchanged.Comment: To appear on ACM Transactions on Modeling and Computer Simulatio
Discrete Lie Advection of Differential Forms
In this paper, we present a numerical technique for performing Lie advection
of arbitrary differential forms. Leveraging advances in high-resolution finite
volume methods for scalar hyperbolic conservation laws, we first discretize the
interior product (also called contraction) through integrals over Eulerian
approximations of extrusions. This, along with Cartan's homotopy formula and a
discrete exterior derivative, can then be used to derive a discrete Lie
derivative. The usefulness of this operator is demonstrated through the
numerical advection of scalar fields and 1-forms on regular grids.Comment: Accepted version; to be published in J. FoC
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
Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds
We develop exterior calculus approaches for partial differential equations on
radial manifolds. We introduce numerical methods that approximate with spectral
accuracy the exterior derivative , Hodge star , and their
compositions. To achieve discretizations with high precision and symmetry, we
develop hyperinterpolation methods based on spherical harmonics and Lebedev
quadrature. We perform convergence studies of our numerical exterior derivative
operator and Hodge star operator
showing each converge spectrally to and . We show how the
numerical operators can be naturally composed to formulate general numerical
approximations for solving differential equations on manifolds. We present
results for the Laplace-Beltrami equations demonstrating our approach.Comment: 22 pages, 13 figure
Map Calculus in GIS: a proposal and demonstration
This paper provides a new representation for fields (continuous surfaces) in Geographical Information Systems (GIS), based on the notion of spatial functions and their combinations. Following Tomlin's (1990) Map Algebra, the term 'Map Calculus' is used for this new representation. In Map Calculus, GIS layers are stored as functions, and new layers can be created by combinations of other functions. This paper explains the principles of Map Calculus and demonstrates the creation of function-based layers and their supporting management mechanism. The proposal is based on Church's (1941) Lambda Calculus and elements of functional computer languages (such as Lisp or Scheme)
- …