2,446 research outputs found
High order algorithm for the time-tempered fractional Feynman-Kac equation
We provide and analyze the high order algorithms for the model describing the
functional distributions of particles performing anomalous motion with
power-law jump length and tempered power-law waiting time. The model is derived
in [Wu, Deng, and Barkai, Phys. Rev. E., 84 (2016), 032151], being called the
time-tempered fractional Feynman-Kac equation. The key step of designing the
algorithms is to discretize the time tempered fractional substantial
derivative, being defined as where
and , , and is a real number. The
designed schemes are unconditionally stable and have the global truncation
error , being theoretically proved and numerically
verified in {\em complex} space. Moreover, some simulations for the
distributions of the first passage time are performed, and the second order
convergence is also obtained for solving the `physical' equation (without
artificial source term).Comment: 21 pages, 4 figure
WSLD operators II: the new fourth order difference approximations for space Riemann-Liouville derivative
High order discretization schemes play more important role in fractional
operators than classical ones. This is because usually for classical
derivatives the stencil for high order discretization schemes is wider than low
order ones; but for fractional operators the stencils for high order schemes
and low order ones are the same. Then using high order schemes to solve
fractional equations leads to almost the same computational cost with first
order schemes but the accuracy is greatly improved. Using the fractional linear
multistep methods, Lubich obtains the -th order ()
approximations of the -th derivative () or integral
(\alpha \in(1,2)$ for time dependent problem. By
weighting and shifting Lubich's 2nd order discretization scheme, in [Chen &
Deng, arXiv:1304.7425] we derive a series of effective high order
discretizations for space fractional derivative, called WSLD opeartors there.
As the sequel of the previous work, we further provide new high order schemes
for space fractional derivatives by weighting and shifting Lubich's 3rd and 4th
order discretizations. In particular, we prove that the obtained 4th order
approximations are effective for space fractional derivatives. And the
corresponding schemes are used to solve the space fractional diffusion equation
with variable coefficients.Comment: 22 pages, 2 figure
Convergence proof for the multigrid method of the nonlocal model
Recently, nonlocal models attract the wide interests of scientist. They
mainly come from two applied scientific fields: peridyanmics and anomalous
diffusion. Even though the matrices of the algebraic equation corresponding the
nonlocal models are usually Toeplitz (denote a0 as the principal diagonal
element, a1 as the trailing diagonal element, etc). There are still some
differences for the models in these two fields. For the model of anomalous
diffusion, a0/a1 is uniformly bounded; most of the time, a0/a1 of the model for
peridyanmics is unbounded as the stepsize h tends to zero. Based on the uniform
boundedness of a0/a1, the convergence of the two-grid method is well
established [Chan, Chang, and Sun, SIAM J. Sci. Comput., 19 (1998), pp.
516--529; Pang and Sun, J. Comput. Phys., 231 (2012), pp. 693--703; Chen, Wang,
Cheng, and Deng, BIT, 54 (2014), pp. 623--647]. This paper provides the
detailed proof of the convergence of the two-grid method for the nonlocal model
of peridynamics. Some special cases of the full multigrid and the V-cycle
multigrid are also discussed. The numerical experiments are performed to verify
the convergence.Comment: 21 page
Efficient numerical algorithms for three-dimensional fractional partial differential equations
This paper detailedly discusses the locally one-dimensional numerical methods
for efficiently solving the three-dimensional fractional partial differential
equations, including fractional advection diffusion equation and Riesz
fractional diffusion equation. The second order finite difference scheme is
used to discretize the space fractional derivative and the Crank-Nicolson
procedure to the time derivative. We theoretically prove and numerically verify
that the presented numerical methods are unconditionally stable and second
order convergent in both space and time directions. In particular, for the
Riesz fractional diffusion equation, the idea of reducing the splitting error
is used to further improve the algorithm, and the unconditional stability and
convergency are also strictly proved and numerically verified for the improved
scheme.Comment: 21 page
Discretized fractional substantial calculus
This paper discusses the properties and the numerical discretizations of the
fractional substantial integral and the fractional substantial derivative where ,
can be a constant or a function without related to , say
; and is the smallest integer that exceeds . The Fourier
transform method and fractional linear multistep method are used to analyze the
properties or derive the discretized schemes. And the convergences of the
presented discretized schemes with the global truncation error
are theoretically proved and numerically
verified.Comment: 20 page
Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion
This paper further discusses the tempered fractional Brownian motion, its
ergodicity, and the derivation of the corresponding Fokker-Planck equation.
Then we introduce the generalized Langevin equation with the tempered
fractional Gaussian noise for a free particle, called tempered fractional
Langevin equation (tfLe). While the tempered fractional Brownian motion
displays localization diffusion for the long time limit and for the short time
its mean squared displacement has the asymptotic form , we show that
the asymptotic form of the mean squared displacement of the tfLe transits from
(ballistic diffusion for short time) to , and then to
(again ballistic diffusion for long time). On the other hand, the overdamped
tfLe has the transition of the diffusion type from to
(ballistic diffusion). The tfLe with harmonic potential is also considered.Comment: 19 pages, 9 figure
Langevin dynamics for L\'evy walk with memory
Memory effects, sometimes, can not be neglected. In the framework of
continuous time random walk, memory effect is modeled by the correlated waiting
times. In this paper, we derive the two-point probability distribution of the
stochastic process with correlated increments as well as the one of its inverse
process, and present the Langevin description of L\'evy walk with memory, i.e.,
correlated waiting times. Based on the built Langevin picture, the properties
of aging and nonstationary are discussed. The Langevin system exhibits
sub-ballistic superdiffusion if the friction force is involved, while it
displays super-ballistic diffusion or hyperdiffusion if there is no friction.
It is discovered that the correlation of waiting times suppresses the diffusion
behavior whether there is friction or not, and the stronger the correlation of
waiting times becomes, the slower the diffusion is. In particular, the
correlation function, correlation coefficient, ergodicity, and scaling property
of the corresponding stochastic process are also investigated.Comment: 11 pages, 4 figure
Numerical algorithms for the forward and backward fractional Feynman-Kac equations
The Feynman-Kac equations are a type of partial differential equations
describing the distribution of functionals of diffusive motion. The probability
density function (PDF) of Brownian functionals satisfies the Feynman-Kac
formula, being a Schr\"{o}dinger equation in imaginary time. The functionals of
no-Brownian motion, or anomalous diffusion, follow the fractional Feynman-Kac
equation [J. Stat. Phys. 141, 1071-1092, 2010], where the fractional
substantial derivative is involved. Based on recently developed discretized
schemes for fractional substantial derivatives [arXiv:1310.3086], this paper
focuses on providing algorithms for numerically solving the forward and
backward fractional Feynman-Kac equations; since the fractional substantial
derivative is non-local time-space coupled operator, new challenges are
introduced comparing with the general fractional derivative. Two ways (finite
difference and finite element) of discretizing the space derivative are
considered. For the backward fractional Feynman-Kac equation, the numerical
stability and convergence of the algorithms with first order accuracy are
theoretically discussed; and the optimal estimates are obtained. For all the
provided schemes, including the first order and high order ones, of both
forward and backward Feynman-Kac equations, extensive numerical experiments are
performed to show their effectiveness.Comment: 27 pages, 8 figure
An equalised global graphical model-based approach for multi-camera object tracking
Non-overlapping multi-camera visual object tracking typically consists of two
steps: single camera object tracking and inter-camera object tracking. Most of
tracking methods focus on single camera object tracking, which happens in the
same scene, while for real surveillance scenes, inter-camera object tracking is
needed and single camera tracking methods can not work effectively. In this
paper, we try to improve the overall multi-camera object tracking performance
by a global graph model with an improved similarity metric. Our method treats
the similarities of single camera tracking and inter-camera tracking
differently and obtains the optimization in a global graph model. The results
show that our method can work better even in the condition of poor single
camera object tracking.Comment: 13 pages, 17 figure
L\'{e}vy-walk-like Langevin dynamics
Continuous time random walks and Langevin equations are two classes of
stochastic models for describing the dynamics of particles in the natural
world. While some of the processes can be conveniently characterized by both of
them, more often one model has significant advantages (or has to be used)
compared with the other one. In this paper, we consider the weakly damped
Langevin system coupled with a new subordinator|-dependent subordinator
with . We pay attention to the diffusion behaviour of the
stochastic process described by this coupled Langevin system, and find the
super-ballistic diffusion phenomena for the system with an unconfined potential
on velocity but sub-ballistic superdiffusion phenomenon with a confined
potential, which is like L\'{e}vy walk for long times. One can further note
that the two-point distribution of inverse subordinator affects mean square
displacement of this coupled weakly damped Langevin system in essential.Comment: 24 pages, 4 figure
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