3,990 research outputs found
Aggregation-diffusion equations: dynamics, asymptotics, and singular limits
Given a large ensemble of interacting particles, driven by nonlocal
interactions and localized repulsion, the mean-field limit leads to a class of
nonlocal, nonlinear partial differential equations known as
aggregation-diffusion equations. Over the past fifteen years,
aggregation-diffusion equations have become widespread in biological
applications and have also attracted significant mathematical interest, due to
their competing forces at different length scales. These competing forces lead
to rich dynamics, including symmetrization, stabilization, and metastability,
as well as sharp dichotomies separating well-posedness from finite time blowup.
In the present work, we review known analytical results for
aggregation-diffusion equations and consider singular limits of these
equations, including the slow diffusion limit, which leads to the constrained
aggregation equation, as well as localized aggregation and vanishing diffusion
limits, which lead to metastability behavior. We also review the range of
numerical methods available for simulating solutions, with special attention
devoted to recent advances in deterministic particle methods. We close by
applying such a method -- the blob method for diffusion -- to showcase key
properties of the dynamics of aggregation-diffusion equations and related
singular limits
A blob method for diffusion
As a counterpoint to classical stochastic particle methods for diffusion, we
develop a deterministic particle method for linear and nonlinear diffusion. At
first glance, deterministic particle methods are incompatible with diffusive
partial differential equations since initial data given by sums of Dirac masses
would be smoothed instantaneously: particles do not remain particles. Inspired
by classical vortex blob methods, we introduce a nonlocal regularization of our
velocity field that ensures particles do remain particles, and we apply this to
develop a numerical blob method for a range of diffusive partial differential
equations of Wasserstein gradient flow type, including the heat equation, the
porous medium equation, the Fokker-Planck equation, the Keller-Segel equation,
and its variants. Our choice of regularization is guided by the Wasserstein
gradient flow structure, and the corresponding energy has a novel form,
combining aspects of the well-known interaction and potential energies. In the
presence of a confining drift or interaction potential, we prove that
minimizers of the regularized energy exist and, as the regularization is
removed, converge to the minimizers of the unregularized energy. We then
restrict our attention to nonlinear diffusion of porous medium type with at
least quadratic exponent. Under sufficient regularity assumptions, we prove
that gradient flows of the regularized energies converge to solutions of the
porous medium equation. As a corollary, we obtain convergence of our numerical
blob method, again under sufficient regularity assumptions. We conclude by
considering a range of numerical examples to demonstrate our method's rate of
convergence to exact solutions and to illustrate key qualitative properties
preserved by the method, including asymptotic behavior of the Fokker-Planck
equation and critical mass of the two-dimensional Keller-Segel equation
Quantitative Assessment of Robotic Swarm Coverage
This paper studies a generally applicable, sensitive, and intuitive error
metric for the assessment of robotic swarm density controller performance.
Inspired by vortex blob numerical methods, it overcomes the shortcomings of a
common strategy based on discretization, and unifies other continuous notions
of coverage. We present two benchmarks against which to compare the error
metric value of a given swarm configuration: non-trivial bounds on the error
metric, and the probability density function of the error metric when robot
positions are sampled at random from the target swarm distribution. We give
rigorous results that this probability density function of the error metric
obeys a central limit theorem, allowing for more efficient numerical
approximation. For both of these benchmarks, we present supporting theory,
computation methodology, examples, and MATLAB implementation code.Comment: Proceedings of the 15th International Conference on Informatics in
Control, Automation and Robotics (ICINCO), Porto, Portugal, 29--31 July 2018.
11 pages, 4 figure
Mean-field instabilities and cluster formation in nuclear reactions
We review recent results on intermediate mass cluster production in heavy ion
collisions at Fermi energy and in spallation reactions. Our studies are based
on modern transport theories, employing effective interactions for the nuclear
mean-field and incorporating two-body correlations and fluctuations. Namely we
will consider the Stochastic Mean Field (SMF) approach and the recently
developed Boltzmann-Langevin One Body (BLOB) model. We focus on cluster
production emerging from the possible occurrence of low-density mean-field
instabilities in heavy ion reactions. Within such a framework, the respective
role of one and two-body effects, in the two models considered, will be
carefully analysed. We will discuss, in particular, fragment production in
central and semi-peripheral heavy ion collisions, which is the object of many
recent experimental investigations. Moreover, in the context of spallation
reactions, we will show how thermal expansion may trigger the development of
mean-field instabilities, leading to a cluster formation process which competes
with important re-aggregation effects
SuperPoint: Self-Supervised Interest Point Detection and Description
This paper presents a self-supervised framework for training interest point
detectors and descriptors suitable for a large number of multiple-view geometry
problems in computer vision. As opposed to patch-based neural networks, our
fully-convolutional model operates on full-sized images and jointly computes
pixel-level interest point locations and associated descriptors in one forward
pass. We introduce Homographic Adaptation, a multi-scale, multi-homography
approach for boosting interest point detection repeatability and performing
cross-domain adaptation (e.g., synthetic-to-real). Our model, when trained on
the MS-COCO generic image dataset using Homographic Adaptation, is able to
repeatedly detect a much richer set of interest points than the initial
pre-adapted deep model and any other traditional corner detector. The final
system gives rise to state-of-the-art homography estimation results on HPatches
when compared to LIFT, SIFT and ORB.Comment: Camera-ready version for CVPR 2018 Deep Learning for Visual SLAM
Workshop (DL4VSLAM2018
Dynamical Monte Carlo Study of Equilibrium Polymers (II): The Role of Rings
We investigate by means of a number of different dynamical Monte Carlo
simulation methods the self-assembly of equilibrium polymers in dilute,
semidilute and concentrated solutions under good-solvent conditions. In our
simulations, both linear chains and closed loops compete for the monomers,
expanding on earlier work in which loop formation was disallowed. Our findings
show that the conformational properties of the linear chains, as well as the
shape of their size distribution function, are not altered by the formation of
rings. Rings only seem to deplete material from the solution available to the
linear chains. In agreement with scaling theory, the rings obey an algebraic
size distribution, whereas the linear chains conform to a Schultz--Zimm type of
distribution in dilute solution, and to an exponentional distribution in
semidilute and concentrated solution. A diagram presenting different states of
aggregation, including monomer-, ring- and chain-dominated regimes, is given
Formation and Equilibrium Properties of Living Polymer Brushes
Polydisperse brushes obtained by reversible radical chain polymerization
reaction onto a solid substrate with surface-attached initiators, are studied
by means of an off-lattice Monte Carlo algorithm of living polymers (LP).
Various properties of such brushes, like the average chain length and the
conformational orientation of the polymers, or the force exerted by the brush
on the opposite container wall, reveal power-law dependence on the relevant
parameters. The observed molecular weight distribution (MWD) of the grafted LP
decays much more slowly than the corresponding LP bulk system due to the
gradient of the monomer density within the dense pseudo-brush which favors
longer chains. Both MWD and the density profiles of grafted polymers and chain
ends are well fitted by effective power laws whereby the different exponents
turn out to be mutually self-consistent for a pseudo-brush in the
strong-stretching regime.Comment: 33 pages, 11 figues, J.Chem. Phys. accepted Oct. 199
Quantifying Robotic Swarm Coverage
In the field of swarm robotics, the design and implementation of spatial
density control laws has received much attention, with less emphasis being
placed on performance evaluation. This work fills that gap by introducing an
error metric that provides a quantitative measure of coverage for use with any
control scheme. The proposed error metric is continuously sensitive to changes
in the swarm distribution, unlike commonly used discretization methods. We
analyze the theoretical and computational properties of the error metric and
propose two benchmarks to which error metric values can be compared. The first
uses the realizable extrema of the error metric to compute the relative error
of an observed swarm distribution. We also show that the error metric extrema
can be used to help choose the swarm size and effective radius of each robot
required to achieve a desired level of coverage. The second benchmark compares
the observed distribution of error metric values to the probability density
function of the error metric when robot positions are randomly sampled from the
target distribution. We demonstrate the utility of this benchmark in assessing
the performance of stochastic control algorithms. We prove that the error
metric obeys a central limit theorem, develop a streamlined method for
performing computations, and place the standard statistical tests used here on
a firm theoretical footing. We provide rigorous theoretical development,
computational methodologies, numerical examples, and MATLAB code for both
benchmarks.Comment: To appear in Springer series Lecture Notes in Electrical Engineering
(LNEE). This book contribution is an extension of our ICINCO 2018 conference
paper arXiv:1806.02488. 27 pages, 8 figures, 2 table
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