422 research outputs found
Maximum principles for a time-space fractional diffusion equation
In this paper, we focus on maximum principles of a time-space fractional
diffusion equation. Maximum principles for classical solution and weak solution
are all obtained by using properties of the time fractional derivative operator
and the fractional Laplace operator. We deduce maximum principles for a full
fractional diffusion equation, other than time-fractional and spatial-integer
order diffusion equations.Comment: 6 page
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
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
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
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
A Hybrid Solution to improve Iteration Efficiency in the Distributed Learning
Currently, many machine learning algorithms contain lots of iterations. When
it comes to existing large-scale distributed systems, some slave nodes may
break down or have lower efficiency. Therefore traditional machine learning
algorithm may fail because of the instability of distributed system.We presents
a hybrid approach which not only own a high fault-tolerant but also achieve a
balance of performance and efficiency.For each iteration, the result of slow
machines will be abandoned. Then, we discuss the relationship between accuracy
and abandon rate. Next we debate the convergence speed of this process.
Finally, our experiments demonstrate our idea can dramatically reduce
calculation time and be used in many platforms.Comment: Unfinished job, with completed mathematical proof while experiments
are in process now. We plan to submit this paper to icml201
Studies on an inverse source problem for a space-time fractional diffusion equation by constructing a strong maximum principle
In this paper, we focus on a space-time fractional diffusion equation with
the generalized Caputo's fractional derivative operator and a general space
nonlocal operator (with the fractional Laplace operator as a special case). A
weak Harnack's inequality has been established by using a special test function
and some properties of the space nonlocal operator. Based on the weak Harnack's
inequality, a strong maximum principle has been obtained which is an important
characterization of fractional parabolic equations. With these tools, we
establish a uniqueness result for an inverse source problem on the
determination of the temporal component of the inhomogeneous term.Comment: 30 pages. arXiv admin note: text overlap with arXiv:1009.4852 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
- β¦