105 research outputs found
Optimal Time Decay of Navier-Stokes Equations With Low Regularity Initial Data
In this paper, we study the optimal time decay rate of isentropic
Navier-Stokes equations under the low regularity assumptions about initial
data. In the previous works about optimal time decay rate, the initial data
need to be small in . Our work combined negative
Besov space estimates and the conventional energy estimates in Besov space
framework which is developed by R. Danchin. Though our methods, we can get
optimal time decay rate with initial data just small in and belong to some negative Besov space(need not to
be small). Finally, combining the recent results in \cite{zhang2014} with our
methods, we can only need the initial data to be small in homogeneous Besov
space to get the optimal time decay
rate in space .Comment: arXiv admin note: text overlap with arXiv:1410.794
Improved mutual coherence of some random overcomplete dictionaries for sparse repsentation
The letter presents a method for the reduction in the mutual coherence of an
overcomplete Gaussian or Bernoulli random matrix, which is fairly small due to
the lower bound given here on the probability of the event that the aforesaid
mutual coherence is less than any given number in (0, 1). The mutual coherence
of the matrix that belongs to a set which contains the two types of matrices
with high probability can be reduced by a similar method but a subset that has
Lebesgue measure zero. The numerical results are provided to illustrate the
reduction in the mutual coherence of an overcomplete Gaussian, Bernoulli or
uniform random dictionary. The effect on the third type is better than a former
result.Comment: The manuscript has been submitted to Linear Algebra and its
Applications with LAA-D-12-01061 in 201
Optimal Time Decay Rate for the Compressible Viscoelastic Equations in Critical Spaces
In this paper, we are concerned with the convergence rates of the global
strong solution to constant equilibrium state for the compressible viscoelastic
fluids in the whole space. We combine both analysis about Green's matrix method
and energy estimate method to get optimal time decay rate in critical Besov
space framework. Our result imply the optimal -time decay rate and only
need the initial data to be small in critical Besov space which have very low
regularity compared with traditional Sobolev space.Comment: 20 page
Bayesian approach to inverse problems for functions with variable index Besov prior
We adopt Bayesian approach to consider the inverse problem of estimate a
function from noisy observations. One important component of this approach is
the prior measure. Total variation prior has been proved with no discretization
invariant property, so Besov prior has been proposed recently. Different prior
measures usually connect to different regularization terms. Variable index TV,
variable index Besov regularization terms have been proposed in image analysis,
however, there are no such prior measure in Bayesian theory. So in this paper,
we propose a variable index Besov prior measure which is a Non-Guassian
measure. Based on the variable index Besov prior measure, we build the Bayesian
inverse theory. Then applying our theory to integer and fractional order
backward diffusion problems. Although there are many researches about
fractional order backward diffusion problems, we firstly apply Bayesian inverse
theory to this problem which provide an opportunity to quantify the
uncertainties for this problem.Comment: 31 pages. arXiv admin note: text overlap with arXiv:1302.6989 by
other author
Well-posedness for compressible MHD system with highly oscillating initial data
In this paper, we transform compressible MHD system written in Euler
coordinate to Lagrange coordinate in critical Besov space. Then we construct
unique local solutions for compressible MHD system. Our results improve the
range of Lebesgue exponent in Besov space from to with
stands for dimension. In addition, we give a lower bound for the maximal
existence time which is important for our construction of global solutions.
Based on the local solution, we obtain a unique global solution with high
oscillating initial velocity and density by using effective viscous flux and
Hoff's energy methods to explore the structure of compressible MHD system.Comment: 44 page
Explosive solutions of parabolic stochastic partial differential equations with Lvy noise
In this paper, we study the explosive solutions to a class of parbolic
stochastic semilinear differential equations driven by a L\acute{\mbox{e}}vy
type noise. The sufficient conditions are presented to guarantee the existence
of a unique positive solution of the stochastic partial differential equation
under investigation. Moreover, we show that the positive solutions will blow up
in finite time in mean -norm sense, provided that the initial data, the
nonlinear term and the multiplicative noise satisfies some conditions. Several
examples are presented to illustrated the theory. Finally, we establish a
global existence theorem based on a Lyapunov functional and prove that a
stochastic Allen-Cahn equation driven by L\acute{\mbox{e}}vy noise has a
global solution.Comment: arXiv admin note: text overlap with arXiv:1402.6365 by other author
On the Decay and Stability of Global Solutions to the 3D Inhomogeneous MHD system
In this paper, we investigative the large time decay and stability to any
given global smooth solutions of the D incompressible inhomogeneous MHD
systems. We proved that given a solution of (\ref{mhd_a}), the
velocity field and magnetic field decay to with an explicit rate, for
which coincide with incompressible inhomogeneous Navier-Stokes equations
\cite{zhangping}. In particular, we give the decay rate of higher order
derivatives of and which is useful to prove our main stability result.
For a large solutions of (\ref{mhd_a}) denoted by , we proved that a
small perturbation to the initial data still generates a unique global smooth
solution and the smooth solution keeps close to the reference solution . Due to the coupling between and , we used elliptic estimates to get
, which is
different to Navier-Stokes equations.Comment: arXiv admin note: text overlap with arXiv:1206.6144 by other author
When is P such that l_0-minimization Equals to l_p-minimization
In this paper, we present an analysis expression of p(A,b) such that the
unique solution to l_0-minimization also can be the unique solution to
l_p-minimization for any 0<p<p(A,b). Furthermore, the main contribution of this
paper isn't only the analysis expressed of such p^(A,b) but also its proof.
Finally, we display the results of two examples to confirm the validity of our
conclusionsComment: 16 pages, 3 figure
Non-convex Fraction Function Penalty: Sparse Signals Recovered from Quasi-linear Systems
The goal of compressed sensing is to reconstruct a sparse signal under a few
linear measurements far less than the dimension of the ambient space of the
signal. However, many real-life applications in physics and biomedical sciences
carry some strongly nonlinear structures, and the linear model is no longer
suitable. Compared with the compressed sensing under the linear circumstance,
this nonlinear compressed sensing is much more difficult, in fact also NP-hard,
combinatorial problem, because of the discrete and discontinuous nature of the
-norm and the nonlinearity. In order to get a convenience for sparse
signal recovery, we set most of the nonlinear models have a smooth quasi-linear
nature in this paper, and study a non-convex fraction function in
this quasi-linear compressed sensing. We propose an iterative fraction
thresholding algorithm to solve the regularization problem
for all . With the change of parameter , our algorithm could get a
promising result, which is one of the advantages for our algorithm compared
with other algorithms. Numerical experiments show that our method performs much
better compared with some state-of-art methods
Posterior contraction for empirical Bayesian approach to inverse problems under non-diagonal assumption
We investigate an empirical Bayesian nonparametric approach to a family of
linear inverse problems with Gaussian prior and Gaussian noise. We consider a
class of Gaussian prior probability measures with covariance operator indexed
by a hyperparameter that quantifies regularity. By introducing two auxiliary
problems, we construct an empirical Bayes method and prove that this method can
automatically select the hyperparameter. In addition, we show that this
adaptive Bayes procedure provides optimal contraction rates up to a slowly
varying term and an arbitrarily small constant, without knowledge about the
regularity index. Our method needs not the prior covariance, noise covariance
and forward operator have a common basis in their singular value decomposition,
enlarging the application range compared with the existing results.Comment: 24 pages; Accepted by Inverse Problems and Imagin
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