1,565 research outputs found
Integration over curves and surfaces defined by the closest point mapping
We propose a new formulation for integrating over smooth curves and surfaces
that are described by their closest point mappings. Our method is designed for
curves and surfaces that are not defined by any explicit parameterization and
is intended to be used in combination with level set techniques. However,
contrary to the common practice with level set methods, the volume integrals
derived from our formulation coincide exactly with the surface or line
integrals that one wish to compute. We study various aspects of this
formulation and provide a geometric interpretation of this formulation in terms
of the singular values of the Jacobian matrix of the closest point mapping.
Additionally, we extend the formulation - initially derived to integrate over
manifolds of codimension one - to include integration along curves in three
dimensions. Some numerical examples using very simple discretizations are
presented to demonstrate the efficacy of the formulation.Comment: Revised the pape
Basic Types of Coarse-Graining
We consider two basic types of coarse-graining: the Ehrenfests'
coarse-graining and its extension to a general principle of non-equilibrium
thermodynamics, and the coarse-graining based on uncertainty of dynamical
models and Epsilon-motions (orbits). Non-technical discussion of basic notions
and main coarse-graining theorems are presented: the theorem about entropy
overproduction for the Ehrenfests' coarse-graining and its generalizations,
both for conservative and for dissipative systems, and the theorems about
stable properties and the Smale order for Epsilon-motions of general dynamical
systems including structurally unstable systems. Computational kinetic models
of macroscopic dynamics are considered. We construct a theoretical basis for
these kinetic models using generalizations of the Ehrenfests' coarse-graining.
General theory of reversible regularization and filtering semigroups in
kinetics is presented, both for linear and non-linear filters. We obtain
explicit expressions and entropic stability conditions for filtered equations.
A brief discussion of coarse-graining by rounding and by small noise is also
presented.Comment: 60 pgs, 11 figs., includes new analysis of coarse-graining by
filtering. A talk given at the research workshop: "Model Reduction and
Coarse-Graining Approaches for Multiscale Phenomena," University of
Leicester, UK, August 24-26, 200
Elastic principal manifolds and their practical applications
Principal manifolds serve as useful tool for many practical applications.
These manifolds are defined as lines or surfaces passing through "the middle"
of data distribution. We propose an algorithm for fast construction of grid
approximations of principal manifolds with given topology. It is based on
analogy of principal manifold and elastic membrane. The first advantage of this
method is a form of the functional to be minimized which becomes quadratic at
the step of the vertices position refinement. This makes the algorithm very
effective, especially for parallel implementations. Another advantage is that
the same algorithmic kernel is applied to construct principal manifolds of
different dimensions and topologies. We demonstrate how flexibility of the
approach allows numerous adaptive strategies like principal graph constructing,
etc. The algorithm is implemented as a C++ package elmap and as a part of
stand-alone data visualization tool VidaExpert, available on the web. We
describe the approach and provide several examples of its application with
speed performance characteristics.Comment: 26 pages, 10 figures, edited final versio
Mathematics and Algorithms in Tomography
This was the ninth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, emission tomography, X-ray CT, and vector tomography along with a wide range of mathematical analysis
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