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 $${^S\!}D_t^{\gamma,\widetilde{\lambda}} G(x,p,t)\!=\!D_t^{\gamma,\widetilde{\lambda}} G(x,p,t)\!-\!\lambda^\gamma G(x,p,t) ~{\rm with}~\widetilde{\lambda}=\lambda+ pU(x),\, p=\rho+J\eta,\, J=\sqrt{-1},$$ where $$D_t^{\gamma,\widetilde{\lambda}} G(x,p,t) =\frac{1}{\Gamma(1-\gamma)} \left[\frac{\partial}{\partial t}+\widetilde{\lambda} \right] \int_{0}^t{\left(t-z\right)^{-\gamma}}e^{-\widetilde{\lambda}\cdot(t-z)}{G(x,p,z)}dz,$$ and $\lambda \ge 0$, $00$, and $\eta$ is a real number. The designed schemes are unconditionally stable and have the global truncation error $\mathcal{O}(\tau^2+h^2)$, 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 $\nu$-th order ($\nu\leq 6$) approximations of the $\alpha$-th derivative ($\alpha>0$) or integral ($\alpha<0) [Lubich, SIAM J. Math. Anal., 17, 704-719, 1986], because of the stability issue the obtained scheme can not be directly applied to the space fractional operator with$\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 $$I_s^\nu f(x)=\frac{1}{\Gamma(\nu)} \int_{a}^x{\left(x-\tau\right)^{\nu-1}}e^{-\sigma(x-\tau)}{f(\tau)}d\tau,\nu>0,$$ and the fractional substantial derivative $$D_s^\mu f(x)=D_s^m[I_s^\nu f(x)], \nu=m-\mu,$$ where $D_s=\frac{\partial}{\partial x}+\sigma=D+\sigma$, $\sigma$ can be a constant or a function without related to $x$, say $\sigma(y)$; and $m$ is the smallest integer that exceeds $\mu$. 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 $\mathcal{O}(h^p)$$(p=1,2,3,4,5)$ 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 $t^{2H}$, we show that the asymptotic form of the mean squared displacement of the tfLe transits from $t^2$ (ballistic diffusion for short time) to $t^{2-2H}$, and then to $t^2$ (again ballistic diffusion for long time). On the other hand, the overdamped tfLe has the transition of the diffusion type from $t^{2-2H}$ to $t^2$ (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|$\alpha$-dependent subordinator with $1<\alpha<2$. 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