1,771 research outputs found
Control of harmonic map heat flow with an external field
We investigate the control problem of harmonic map heat flow by means of an
external magnetic field. In contrast to the situation of a parabolic system
with internal or boundary control, the magnetic field acts as the coefficients
of the lower order terms of the equation. We show that for initial data whose
image stays in a hemisphere, with one control acting on a subset of the domain
plus a spatial-independent control acting on the whole domain, the state of the
system can be steered to any ground state, i.e. any given unit vector, within
any short time. To achieve this, in the first step a spatial independent
control is applied to steer the solution into a small neighborhood of the peak
of the hemisphere. Then under stereographic projection, the original system is
reduced to an internal parabolic control system with initial data sufficiently
close to such that the existing method for local controllability can be
applied. The key process in this step is to give an explicit solution of an
underdetermined algebraic system such that the affine type control can be
converted into an internal control.Comment: a few details are added in this new versio
Patchwork: A Patch-wise Attention Network for Efficient Object Detection and Segmentation in Video Streams
Recent advances in single-frame object detection and segmentation techniques
have motivated a wide range of works to extend these methods to process video
streams. In this paper, we explore the idea of hard attention aimed for
latency-sensitive applications. Instead of reasoning about every frame
separately, our method selects and only processes a small sub-window of the
frame. Our technique then makes predictions for the full frame based on the
sub-windows from previous frames and the update from the current sub-window.
The latency reduction by this hard attention mechanism comes at the cost of
degraded accuracy. We made two contributions to address this. First, we propose
a specialized memory cell that recovers lost context when processing
sub-windows. Secondly, we adopt a Q-learning-based policy training strategy
that enables our approach to intelligently select the sub-windows such that the
staleness in the memory hurts the performance the least. Our experiments
suggest that our approach reduces the latency by approximately four times
without significantly sacrificing the accuracy on the ImageNet VID video object
detection dataset and the DAVIS video object segmentation dataset. We further
demonstrate that we can reinvest the saved computation into other parts of the
network, and thus resulting in an accuracy increase at a comparable
computational cost as the original system and beating other recently proposed
state-of-the-art methods in the low latency range.Comment: ICCV 2019 Camera Ready + Supplementar
Phase-field approximation of the Willmore flow
We investigate the phase-field approximation of the Willmore flow. This is a
fourth-order diffusion equation with a parameter that is
proportional to the thickness of the diffuse interface. We show rigorously that
for well-prepared initial data, as trends to zero the level-set of
solution will converge to motion by Willmore flow before the singularity of the
later occurs. This is done by constructing an approximate solution from the
limiting flow via matched asymptotic expansions, and then estimating its
difference with the real solution. The crucial step and also the major
contribution of this work is to show a spectrum condition of the linearized
operator at the optimal profile. This is a fourth-order operator written as the
sum of the squared Allen-Cahn operator and a singular linear perturbation.Comment: convergence rate is improved and more details are provided for the
matched asymptotical expansion
The Oseen-Frank limit of Onsager's molecular theory for liquid crystals
We study the relationship between Onsager's molecular theory and the
Oseen-Frank theory for nematic liquid crystals. Under the molecular setting, we
consider the free energy that includes the effects of nonlocal molecular
interactions. By imposing the strong anchoring boundary condition on the second
moment of the number density function, we prove the existence of global
minimizers for the free energy. Moreover, when the re-scaled interaction
distance tends to zero, the corresponding global minimizers will converge to a
uniaxial distribution whose orientation is described by a minimizer of
Oseen-Frank energy.Comment: 24 page
The small Deborah number limit of the Doi-Onsager equation without hydrodynamics
We study the small Deborah number limit of the Doi-Onsager equation for the
dynamics of nematic liquid crystals without hydrodynamics. This is a
Smoluchowski-type equation that characterizes the evolution of a number density
function, depending upon both particle position and
orientation vector (the unit sphere). We prove that, when
the Deborah number tends to zero, the family of solutions with rough initial
data near local equilibria will converge strongly to a local equilibrium
distribution prescribed by a weak solution of the harmonic map heat flow into
. This flow is a special case of the gradient flow to the
Oseen-Frank energy functional for nematic liquid crystals. The key ingredient
is to show the strong compactness of the family of number density functions and
the proof relies on the strong compactness of the corresponding second moment
(or the -tensor), a spectral decomposition of the linearized operator near
the limit local equilibrium distribution, as well as the energy dissipation
estimate.Comment: 38 pages. Comments are welcom
Well-posedness of a fully-coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data
We prove short-time well-posedness and existence of global weak solutions of
the Beris--Edwards model for nematic liquid crystals in the case of a bounded
domain with inhomogeneous mixed Dirichlet and Neumann boundary conditions. The
system consists of the Navier-Stokes equations coupled with an evolution
equation for the -tensor. The solutions possess higher regularity in time of
order one compared to the class of weak solutions with finite energy. This
regularity is enough to obtain Lipschitz continuity of the non-linear terms in
the corresponding function spaces. Therefore the well-posedness is shown with
the aid of the contraction mapping principle using that the linearized system
is an isomorphism between the associated function spaces.Comment: 25 page
Global Well-posedness of the Two Dimensional Beris-Edwards System with General Laudau-de Gennes Free Energy
In this paper, we consider the Beris-Edwards system for incompressible
nematic liquid crystal flows. The system under investigation consists of the
Navier-Stokes equations for the fluid velocity coupled with an
evolution equation for the order parameter -tensor. One important feature of
the system is that its elastic free energy takes a general form and in
particular, it contains a cubic term that possibly makes it unbounded from
below. In the two dimensional periodic setting, we prove that if the initial
-norm of the -tensor is properly small, then the system admits a
unique global weak solution. The proof is based on the construction of a
specific approximating system that preserves the -norm of the
-tensor along the time evolution
Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions
Existence and uniqueness of local strong solution for the Beris--Edwards
model for nematic liquid crystals, which couples the Navier-Stokes equations
with an evolution equation for the Q-tensor, is established on a bounded domain
in the case of homogeneous Dirichlet boundary conditions. The classical
Beris--Edwards model is enriched by including a dependence of the fluid
viscosity on the Q-tensor. The proof is based on a linearization of the system
and Banach's fixed-point theorem.Comment: 31 page
An algorithm to find the spectral radius of nonnegative tensors and its convergence analysis
In this paper we propose an iterative algorithm to find out the spectral
radius of nonnegative tensors. This algorithm is an extension of the smoothing
method for finding the largest eigenvalue of a nonnegative matrix \cite{s14}.
For nonnegative irreducible tensors, we establish the converges of the
algorithm. Finally we report some numerical results and conclude this paper
with some remarks
Deep Approximately Orthogonal Nonnegative Matrix Factorization for Clustering
Nonnegative Matrix Factorization (NMF) is a widely used technique for data
representation. Inspired by the expressive power of deep learning, several NMF
variants equipped with deep architectures have been proposed. However, these
methods mostly use the only nonnegativity while ignoring task-specific features
of data. In this paper, we propose a novel deep approximately orthogonal
nonnegative matrix factorization method where both nonnegativity and
orthogonality are imposed with the aim to perform a hierarchical clustering by
using different level of abstractions of data. Experiment on two face image
datasets showed that the proposed method achieved better clustering performance
than other deep matrix factorization methods and state-of-the-art single layer
NMF variants.Comment: 5 page
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