2,021 research outputs found
On the two-dimensional rotational body of maximal Newtonian resistance
We investigate, by means of computer simulations, shapes of nonconvex bodies
that maximize resistance to their motion through a rarefied medium, considering
that bodies are moving forward and at the same time slowly rotating. A
two-dimensional geometric shape that confers to the body a resistance very
close to the theoretical supremum value is obtained, improving previous
results.Comment: This is a preprint version of the paper published in J. Math. Sci.
(N. Y.), Vol. 161, no. 6, 2009, 811--819. DOI:10.1007/s10958-009-9602-
How to Walk Your Dog in the Mountains with No Magic Leash
We describe a -approximation algorithm for computing the
homotopic \Frechet distance between two polygonal curves that lie on the
boundary of a triangulated topological disk. Prior to this work, algorithms
were known only for curves on the Euclidean plane with polygonal obstacles.
A key technical ingredient in our analysis is a -approximation
algorithm for computing the minimum height of a homotopy between two curves. No
algorithms were previously known for approximating this parameter.
Surprisingly, it is not even known if computing either the homotopic \Frechet
distance, or the minimum height of a homotopy, is in NP
Two-dimensional body of maximum mean resistance
A two-dimensional body, exhibiting a slight rotational movement, moves in a
rarefied medium of particles which collide with it in a perfectly elastic way.
In previously realized investigations by the first two authors, Plakhov &
Gouveia (2007, Nonlinearity, 20), shapes of nonconvex bodies were sought which
would maximize the braking force of the medium on their movement. Giving
continuity to this study, new investigations have been undertaken which
culminate in an outcome which represents a large qualitative advance relative
to that which was achieved earlier. This result, now presented, consists of a
two-dimensional shape which confers on the body a resistance which is very
close to its theoretical supremum value. But its interest does not lie solely
in the maximization of Newtonian resistance; on regarding its characteristics,
other areas of application are seen to begin to appear which are thought to be
capable of having great utility. The optimal shape which has been encountered
resulted from numerical studies, thus it is the object of additional study of
an analytical nature, where it proves some important properties which explain
in great part its effectiveness.Comment: Accepted (April 16, 2009) for publication in the journal "Applied
Mathematics and Computation
Shapes From Pixels
Continuous-domain visual signals are usually captured as discrete (digital)
images. This operation is not invertible in general, in the sense that the
continuous-domain signal cannot be exactly reconstructed based on the discrete
image, unless it satisfies certain constraints (\emph{e.g.}, bandlimitedness).
In this paper, we study the problem of recovering shape images with smooth
boundaries from a set of samples. Thus, the reconstructed image is constrained
to regenerate the same samples (consistency), as well as forming a shape
(bilevel) image. We initially formulate the reconstruction technique by
minimizing the shape perimeter over the set of consistent binary shapes. Next,
we relax the non-convex shape constraint to transform the problem into
minimizing the total variation over consistent non-negative-valued images. We
also introduce a requirement (called reducibility) that guarantees equivalence
between the two problems. We illustrate that the reducibility property
effectively sets a requirement on the minimum sampling density. One can draw
analogy between the reducibility property and the so-called restricted isometry
property (RIP) in compressed sensing which establishes the equivalence of the
minimization with the relaxed minimization. We also evaluate
the performance of the relaxed alternative in various numerical experiments.Comment: 13 pages, 14 figure
Dislocation microstructures and strain-gradient plasticity with one active slip plane
We study dislocation networks in the plane using the vectorial phase-field
model introduced by Ortiz and coworkers, in the limit of small lattice spacing.
We show that, in a scaling regime where the total length of the dislocations is
large, the phase field model reduces to a simpler model of the strain-gradient
type. The limiting model contains a term describing the three-dimensional
elastic energy and a strain-gradient term describing the energy of the
geometrically necessary dislocations, characterized by the tangential gradient
of the slip. The energy density appearing in the strain-gradient term is
determined by the solution of a cell problem, which depends on the line tension
energy of dislocations. In the case of cubic crystals with isotropic elasticity
our model shows that complex microstructures may form, in which dislocations
with different Burgers vector and orientation react with each other to reduce
the total self energy
Conditional gradients for total variation regularization with PDE constraints: a graph cuts approach
Total variation regularization has proven to be a valuable tool in the
context of optimal control of differential equations. This is particularly
attributed to the observation that TV-penalties often favor piecewise constant
minimizers with well-behaved jumpsets. On the downside, their intricate
properties significantly complicate every aspect of their analysis, from the
derivation of first-order optimality conditions to their discrete approximation
and the choice of a suitable solution algorithm. In this paper, we investigate
a general class of minimization problems with TV-regularization, comprising
both continuous and discretized control spaces, from a convex geometry
perspective. This leads to a variety of novel theoretical insights on
minimization problems with total variation regularization as well as tools for
their practical realization. First, by studying the extremal points of the
respective total variation unit balls, we enable their efficient solution by
geometry exploiting algorithms, e.g. fully-corrective generalized conditional
gradient methods. We give a detailed account on the practical realization of
such a method for piecewise constant finite element approximations of the
control on triangulations of the spatial domain. Second, in the same setting
and for suitable sequences of uniformly refined meshes, it is shown that
minimizers to discretized PDE-constrained optimal control problems approximate
solutions to a continuous limit problem involving an anisotropic total
variation reflecting the fine-scale geometry of the mesh.Comment: 47 pages, 12 figure
Variational approximation of functionals defined on 1-dimensional connected sets: the planar case
In this paper we consider variational problems involving 1-dimensional
connected sets in the Euclidean plane, such as the classical Steiner tree
problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal
partition problems and provide a variational approximation through
Modica-Mortola type energies proving a -convergence result. We also
introduce a suitable convex relaxation and develop the corresponding numerical
implementations. The proposed methods are quite general and the results we
obtain can be extended to -dimensional Euclidean space or to more general
manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure
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