3,375 research outputs found
Decay of a model system of radiating gas
This paper is concerned with optimal time-decay estimates of solutions of the
Cauchy problem to a model system of the radiating gas in .
Compared to Liu and Kawashima (2011) \cite{Liu1} and Wang and Wang (2009)
\cite{Wang}, without smallness assumption of initial perturbation in
-norm, we study large time behavior of small amplitude classical solutions
to the Cauchy problem. The optimal -norm time-decay rates of the solutions
in with are obtained by applying the Fourier
splitting method introduced in Schonbek (1980) \cite{Schonbek1} with a slight
modification and an energy method. Furthermore, basing on a refined pure energy
method introduced in Guo and Wang \cite{Guo} (2011), we give optimal
- decay estimates of the derivatives of solutions when
initial perturbation is bounded in -norm with some .Comment: 15 page
Improved Kernels and Algorithms for Claw and Diamond Free Edge Deletion Based on Refined Observations
In the {claw, diamond}-free edge deletion problem, we are given a graph
and an integer , the question is whether there are at most edges whose
deletion results in a graph without claws and diamonds as induced graphs. Based
on some refined observations, we propose a kernel of vertices and
edges, significantly improving the previous kernel of
vertices and edges. In addition, we derive an -time
algorithm for the {claw, diamond}-free edge deletion problem.Comment: 13 page
Partitioned AVF methods
The classic second-order average vector field (AVF) method can exactly
preserve the energy for Hamiltonian ordinary differential equations and partial
differential equations. However, the AVF method inevitably leads to
fully-implicit nonlinear algebraic equations for general nonlinear systems. To
address this drawback and maintain the desired energy-preserving property, a
first-order partitioned AVF method is proposed which first divides the
variables into groups and then applies the AVF method step by step. In
conjunction with its adjoint method we present the partitioned AVF composition
method and plus method respectively to improve its accuracy to second order.
Concrete schemes for two classic model equations are constructed with
semi-implicit, linear-implicit properties that make considerable lower cost
than the original AVF method. Furthermore, additional conservative property can
be generated besides the conventional energy preservation for specific
problems. Numerical verification of these schemes further conforms our results.Comment: 23 pages, 20 figure
Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions
This paper presents two kinds of strategies to construct structure-preserving
algorithms with homogeneous Neumann boundary conditions for the sine-Gordon
equation, while most existing structure-preserving algorithms are only valid
for zero or periodic boundary conditions. The first strategy is based on the
conventional second-order central difference quotient but with a cell-centered
grid, while the other is established on the regular grid but incorporated with
summation by parts (SBP) operators. Both the methodologies can provide
conservative semi-discretizations with different forms of Hamiltonian
structures and the discrete energy. However, utilizing the existing SBP
formulas, schemes obtained by the second strategy can directly achieve
higher-order accuracy while it is not obvious for schemes based on the
cell-centered grid to make accuracy improved easily. Further combining the
symplectic Runge-Kutta method and the scalar auxiliary variable (SAV) approach,
we construct symplectic integrators and linearly implicit energy-preserving
schemes for the two-dimensional sine-Gordon equation, respectively. Extensive
numerical experiments demonstrate their effectiveness with the homogeneous
Neumann boundary conditions.Comment: 23 pages, 47 figure
A Novel Sixth Order Energy-Conserved Method for Three-Dimensional Time-Domain Maxwell's Equations
In this paper, a novel sixth order energy-conserved method is proposed for
solving the three-dimensional time-domain Maxwell's equations. The new scheme
preserves five discrete energy conservation laws, three momentum conservation
laws, symplectic conservation law as well as two divergence-free properties and
is proved to be unconditionally stable, non-dissipative. An optimal error
estimate is established based on the energy method, which shows that the
proposed method is of sixth order accuracy in time and spectral accuracy in
space in discrete -norm. The constant in the error estimate is proved to
be only . Furthermore, the numerical dispersion relation is analyzed in
detail and a fast solver is presented to solve the resulting discrete linear
equations efficiently. Numerical results are addressed to verify our
theoretical analysis.Comment: 36 page
Detect or Track: Towards Cost-Effective Video Object Detection/Tracking
State-of-the-art object detectors and trackers are developing fast. Trackers
are in general more efficient than detectors but bear the risk of drifting. A
question is hence raised -- how to improve the accuracy of video object
detection/tracking by utilizing the existing detectors and trackers within a
given time budget? A baseline is frame skipping -- detecting every N-th frames
and tracking for the frames in between. This baseline, however, is suboptimal
since the detection frequency should depend on the tracking quality. To this
end, we propose a scheduler network, which determines to detect or track at a
certain frame, as a generalization of Siamese trackers. Although being
light-weight and simple in structure, the scheduler network is more effective
than the frame skipping baselines and flow-based approaches, as validated on
ImageNet VID dataset in video object detection/tracking.Comment: Accepted to AAAI 201
Higher Order Elastic Instabilities of Metals: From Atom to Continuum Level
Strain-based theory on elastic instabilities is being widely employed for
studying onset of plasticity, phase transition or melting in crystals. And size
effects, observed in nano-materials or solids under dynamic loadings, needs to
account for contributions from strain gradient. However, the strain gradient
based higher order elastic theories on the elastic instabilities are not well
established to enable one to predict high order instabilities of solids
directly at atom level. In present work, a general continuum theory for higher
order elastic instabilities is established and justified by developing an
equivalent description at atom level. Our results show that mechanical
instability of solids, triggered by either strain or strain gradient, is
determined by a simple stability condition consisting of strain or strain
gradient related elastic constants. With the atom-level description of the
higher order elasticity, the strain-gradient elastic constants could be
directly obtained by a molecular statics procedure and then serve as inputs of
the stability condition. In this way, mechanical instabilities of three metals,
i.e., copper, aluminum and iron, are predicted. Alternatively, ramp compression
technique by nonequilibrium molecular dynamics (NEMD) simulations is employed
to study the higher order instabilities of the three metals. The predicted
critical strains at onset of instabilities agree well with the results from the
NEMD simulations for all the metals. Since the only inputs for the established
higher order elastic theory are the same as atomic simulations, i.e., atomic
potentials and structures of solids, the established theory is completely
equivalent to empirical-potential based atomic simulations methods, at least,
for crystals.Comment: 30 pages, 4 figure
A novel linearized and momentum-preserving Fourier pseudo-spectral scheme for the Rosenau-Korteweg de Vries equation
In this paper, we design a novel linearized and momentum-preserving Fourier
pseudo-spectral scheme to solve the Rosenau-Korteweg de Vries equation. With
the aid of a new semi-norm equivalence between the Fourier pseudo-spectral
method and the finite difference method, a prior bound of the numerical
solution in discrete -norm is obtained from the discrete momentum
conservation law. Subsequently, based on the energy method and the bound of the
numerical solution, we show that, without any restriction on the mesh ratio,
the scheme is convergent with order in discrete
-norm, where is the number of collocation points used in the
spectral method and is the time step. Numerical results are addressed to
confirm our theoretical analysis.Comment: 24 pages.arXiv admin note: text overlap with arXiv:1808.0685
Deeply-Fused Nets
In this paper, we present a novel deep learning approach, deeply-fused nets.
The central idea of our approach is deep fusion, i.e., combine the intermediate
representations of base networks, where the fused output serves as the input of
the remaining part of each base network, and perform such combinations deeply
over several intermediate representations. The resulting deeply fused net
enjoys several benefits. First, it is able to learn multi-scale representations
as it enjoys the benefits of more base networks, which could form the same
fused network, other than the initial group of base networks. Second, in our
suggested fused net formed by one deep and one shallow base networks, the flows
of the information from the earlier intermediate layer of the deep base network
to the output and from the input to the later intermediate layer of the deep
base network are both improved. Last, the deep and shallow base networks are
jointly learnt and can benefit from each other. More interestingly, the
essential depth of a fused net composed from a deep base network and a shallow
base network is reduced because the fused net could be composed from a less
deep base network, and thus training the fused net is less difficult than
training the initial deep base network. Empirical results demonstrate that our
approach achieves superior performance over two closely-related methods, ResNet
and Highway, and competitive performance compared to the state-of-the-arts
Stimulated brillouin scattering in slow light waveguides
We develop a general method of calculating Stimulated Brillouin Scattering
(SBS) gain coefficient in axially periodic waveguides. Applying this method to
a silicon periodic waveguide suspended in air, we demonstrate that SBS
nonlinearity can be dramatically enhanced at the brillouin zone boundary where
the decreased group velocity of light magnifies photon-phonon interaction. In
addition, we show that the symmetry plane perpendicular to the propagation axis
plays an important role in both forward and backward SBS processes. In forward
SBS, only elastic modes which are even about this plane are excitable. In
backward SBS, the SBS gain coefficients of elastic modes approach to either
infinity or constants, depending on their symmetry about this plane at
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