arXiv subjects

Weihua Deng

Publications and source records attributed to Weihua Deng.

At least 73 records · Page 4Linked to original sources

arXiv2018

Tempered fractional Langevin-Brownian motion with inverse $β$-stable subordinator

Time-changed stochastic processes have attracted great attention and wide interests due to their extensive applications, especially in financial time series, biology and physics. This paper pays attention to a special stochastic process, tempered fractional Langevin motion, which is non-Markovian and undergoes ballistic diffusion for long times. The corresponding time-changed Langevin system with inverse $β$-stable subordinator is discussed in detail, including its diffusion type, moments, Klein-Kramers equation, and the correlation structure. Interestingly, this subordination could result in both subdiffusion and superdiffusion, depending on the value of $β$. The difference between the subordinated tempered fractional Langevin equation and the subordinated Langevin equation with external biasing force is studied for a deeper understanding of subordinator. The time-changed tempered fractional Brownian motion by inverse $β$-stable subordinator is also considered, as well as the correlation structure of its increments. Some properties of the statistical quantities of the time-changed process are discussed, displaying striking differences compared with the original process.

cond-mat.stat-mech↗

arXiv2018

Feynman-Kac equation revisited

Functionals of particles' paths have diverse applications in physics, mathematics, hydrology, economics, and other fields. Under the framework of continuous time random walk (CTRW), the governing equations for the probability density functions (PDFs) of the functionals, including the ones of the paths of stochastic processes of normal diffusion, anomalous diffusion, and even the diffusion with reaction, have been derived. Sometimes, the stochastic processes in physics and chemistry are naturally described by Langevin equations. The Langevin picture has the advantages in studying the dynamics with an external force field and analyzing the effect of noise resulting from a fluctuating environment. We derive the governing equations of the PDFs of the functionals of paths of Langevin system with both space and time dependent force field and arbitrary multiplicative noise; and the backward version is proposed for the system with arbitrary additive noise or multiplicative Gaussian white noise together with a force field. For the newly built equations, their applications of solving the PDFs of the occupation time and area under the trajectory curve are provided, and the results are confirmed by simulations.

cond-mat.stat-mech↗

arXiv2018

Nonlocal Diffusion Operators for Normal and Anomalous Dynamics

The Laplacian $Δ$ is the infinitesimal generator of isotropic Brownian motion, being the limit process of normal diffusion, while the fractional Laplacian $Δ^{β/2}$ serves as the infinitesimal generator of the limit process of isotropic Lévy process. Taking limit, in some sense, means that the operators can approximate the physical process well after sufficient long time. We introduce the nonlocal operators (being effective from the starting time), which describe the general processes undergoing normal diffusion. For anomalous diffusion, we extend to the anisotropic fractional Laplacian $Δ_m^{β/2}$ and the tempered one $Δ_m^{β/2,λ}$ in $\mathbb{R}^n$. Their definitions are proved to be equivalent to an alternative one in Fourier space. Based on these new nonlocal diffusion operators, we further derive the deterministic governing equations of some interesting statistical observables of the very general jump processes with multiple internal states. Finally, we consider the associated initial and boundary value problems and prove their well-posedness of the Galerkin weak formulation in $\mathbb{R}^n$. To obtain the coercivity, we claim that the probability density function $m(Y)$ should be nondegenerate.

math.AP↗

arXiv2018

A reduced finite element formulation for space fractional partial differential equation

Applying proper orthogonal decomposition to a usual finite element (FE) formulation for space fractional partial differential equation, we get a reduced FE model, which greatly reduces the complexity of computation. Then, the stability analysis and error estimate for the reduced model are presented. Finally, we verify the effectiveness of the algorithm by numerical experiments.

math.NA↗

arXiv2018

Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian

Tempered fractional Laplacian is the generator of the tempered isotropic Lévy process [W.H. Deng, B.Y. Li, W.Y. Tian, and P.W. Zhang, Multiscale Model. Simul., 16(1), 125-149, 2018]. This paper provides the finite difference discretization for the two dimensional tempered fractional Laplacian $(Δ+λ)^{\fracβ{2}}$. Then we use it to solve the tempered fractional Poisson equation with Dirichlet boundary conditions and derive the error estimates. Numerical experiments verify the convergence rates and effectiveness of the schemes.

math.NA↗

arXiv2018

Resonant behavior of the generalized Langevin system with tempered Mittag-Leffler memory kernel

The generalized Langevin equation describes anomalous dynamics. Noise is not only the origin of uncertainty but also plays a positive role in helping to detect signal with information, termed stochastic resonance (SR). This paper analyzes the anomalous resonant behaviors of the generalized Langevin system with a multiplicative dichotomous noise and an internal tempered Mittag-Leffler noise. For the system with fluctuating harmonic potential, we obtain the exact expressions of several SR, such as, the first moment, the amplitude and the autocorrelation function for the output signal as well as the signal-noise ratio. We analyze the influence of the tempering parameter and memory exponent on the bona fide SR and the general SR. Moreover, it is detected that the critical memory exponent changes regularly with the increase of tempering parameter. Almost all the theoretical results are validated by numerical simulations.

cond-mat.stat-mech↗

arXiv2017

Levy walk with multiple internal states

Levy walk is a fundamental model with applications ranging from quantum physics to paths of animal foraging. Taking animal foraging as an example, a natural idea that comes to one's mind is to introduce the multiple internal states for dealing with the dependence of the PDF of waiting time on the energy of the animal and richness of the food at a particular location, etc; the framework can also be used to model the moving trajectories of smart animals without returning to the directions or locations which they come from immediately. After building the Levy walk model with multiple internal states and deriving the governing equation of the distribution of the positions of the particles, some applications are discussed with specific transition matrices. The type of diffusion for non-immediately-repeating Lévy walk is uncovered, and the distribution and average of first passage time are numerically simulated.

cond-mat.stat-mech↗

arXiv2017

A Riesz basis Galerkin method for the tempered fractional Laplacian

The fractional Laplacian $Δ^{β/2}$ is the generator of $β$-stable Lévy process, which is the scaling limit of the Lévy fight. Due to the divergence of the second moment of the jump length of the Lévy fight it is not appropriate as a physical model in many practical applications. However, using a parameter $λ$ to exponentially temper the isotropic power law measure of the jump length leads to the tempered Lévy fight, which has finite second moment. For short time the tempered Lévy fight exhibits the dynamics of Lévy fight while after sufficiently long time it turns to normal diffusion. The generator of tempered $β$-stable Lévy process is the tempered fractional Laplacian $(Δ+λ)^{β/2}$ [W.H. Deng, B.Y. Li, W.Y. Tian, and P.W. Zhang, Multiscale Model. Simul., in press, 2017]. In the current work, we present new computational methods for the tempered fractional Laplacian equation, including the cases with the homogeneous and nonhomogeneous generalized Dirichlet type boundary conditions. We prove the well-posedness of the Galerkin weak formulation and provide convergence analysis of the single scaling B-spline and multiscale Riesz bases finite element methods. We propose a technique for efficiently generating the entries of the dense stiffness matrix and for solving the resulting algebraic equation by preconditioning. We also present several numerical experiments to verify the theoretical results.

math.NA↗

arXiv2017

Aging Feynman-Kac Equation

Aging, the process of growing old or maturing, is one of the most widely seen natural phenomena in the world. For the stochastic processes, sometimes the influence of aging can not be ignored. For example, in this paper, by analyzing the functional distribution of the trajectories of aging particles performing anomalous diffusion, we reveal that for the fraction of the occupation time $T_+/t$ of strong aging particles, $\langle (T^+(t)^2)\rangle=\frac{1}{2}t^2$ with coefficient $\frac{1}{2}$, having no relation with the aging time $t_a$ and $α$ and being completely different from the case of weak (none) aging. In fact, we first build the models governing the corresponding functional distributions, i.e., the aging forward and backward Feynman-Kac equations; the above result is one of the applications of the models. Another application of the models is to solve the asymptotic behaviors of the distribution of the first passage time, $g(t_a,t)$. The striking discovery is that for weakly aging systems, $g(t_a,t)\sim t_a^{\fracα{2}}t^{-1-\fracα{2}}$, while for strongly aging systems, $g(t_a,t)$ behaves as $ t_a^{α-1}t^{-α}$.

physics.chem-ph↗

arXiv2017

Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations

This paper focuses on the adaptive discontinuous Galerkin (DG) methods for the tempered fractional (convection) diffusion equations. The DG schemes with interior penalty for the diffusion term and numerical flux for the convection term are used to solve the equations, and the detailed stability and convergence analyses are provided. Based on the derived posteriori error estimates, the local error indicator is designed. The theoretical results and the effectiveness of the adaptive DG methods are respectively verified and displayed by the extensive numerical experiments. The strategy of designing adaptive schemes presented in this paper works for the general PDEs with fractional operators.

math.NA↗

arXiv2017

Uniform convergence of V-cycle multigrid algorithms for two-dimensional fractional Feynman-Kac equation

In this paper we derive new uniform convergence estimates for the V-cycle MGM applied to symmetric positive definite Toeplitz block tridiagonal matrices, by also discussing few connections with previous results. More concretely, the contributions of this paper are as follows: (1) It tackles the Toeplitz systems directly for the elliptic PDEs. (2) Simple (traditional) restriction operator and prolongation operator are employed in order to handle general Toeplitz systems at each level of the recursion. Such a technique is then applied to systems of algebraic equations generated by the difference scheme of the two-dimensional fractional Feynman-Kac equation, which describes the joint probability density function of non-Brownian motion. In particular, we consider the two coarsening strategies, i.e., doubling the mesh size (geometric MGM) and Galerkin approach (algebraic MGM), which lead to the distinct coarsening stiffness matrices in the general case: however, several numerical experiments show that the two algorithms produce almost the same error behaviour.

math.NA↗

arXiv2017

Feynman-Kac Equations for Reaction and Diffusion Processes

This paper provides a theoretical framework of deriving the forward and backward Feynman-Kac equations for the distribution of functionals of the path of a particle undergoing both diffusion and chemical reaction. Very general forms of the equations are obtained. Once given the diffusion type and reaction rate, a specific forward or backward Feynman-Kac equation can be obtained. The listed in the paper include the ones for normal/anomalous diffusions and reactions with linear/nonlinear rates. Using the derived equations, we also study the occupation time in half-space, the first passage time to a fixed boundary, and the occupation time in half-space with absorbing or reflecting boundary conditions.

cond-mat.stat-mech↗

arXiv2017

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.

cond-mat.stat-mech↗

arXiv2017

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 $${^S\!}D_t^{γ,\widetildeλ} G(x,p,t)\!=\!D_t^{γ,\widetildeλ} G(x,p,t)\!-\!λ^γG(x,p,t) ~{\rm with}~\widetildeλ=λ+ pU(x),\, p=ρ+Jη,\, J=\sqrt{-1},$$ where $$D_t^{γ,\widetildeλ} G(x,p,t) =\frac{1}{Γ(1-γ)} \left[\frac{\partial}{\partial t}+\widetildeλ \right] \int_{0}^t{\left(t-z\right)^{-γ}}e^{-\widetildeλ\cdot(t-z)}{G(x,p,z)}dz,$$ and $λ\ge 0$, $0<γ<1$, $ρ>0$, and $η$ is a real number. The designed schemes are unconditionally stable and have the global truncation error $\mathcal{O}(τ^2+h^2)$, 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).

math.NA↗

arXiv2017

Fractional compound Poisson processes with multiple internal states

For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and functional distributions of the trajectories of particles; in particular, the equations governing the functional distribution of internal states are also obtained. The dynamics of the stochastic processes are analyzed and the applications, calculating the distribution of the first passage time and the distribution of the fraction of the occupation time, of the equations are given.

math.ST↗

arXiv2017

Boundary problems for the fractional and tempered fractional operators

For characterizing the Brownian motion in a bounded domain: $Ω$, it is well-known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; on the contrary, for the Lévy flights or tempered Lévy flights in a bounded domain, it involves the information of a solution in the complementary set of $Ω$, i.e., $\mathbb{R}^n\backslash Ω$, with the potential reason that paths of the corresponding stochastic process are discontinuous. Guided by probability intuitions and the stochastic perspectives of anomalous diffusion, we show the reasonable ways, ensuring the clear physical meaning and well-posedness of the partial differential equations (PDEs), of specifying `boundary' conditions for space fractional PDEs modeling the anomalous diffusion. Some properties of the operators are discussed, and the well-posednesses of the PDEs with generalized boundary conditions are proved.

math.AP↗

arXiv2016

Mean exit time and escape probability for the anomalous processes with the tempered power-law waiting times

The mean first exit (passage) time characterizes the average time of a stochastic process never leaving a fixed region in the state space, while the escape probability describes the likelihood of a transition from one region to another for a stochastic system driven by discontinuous (with jumps) Lévy motion. This paper discusses the two deterministic quantities, mean first exit time and escape probability, for the anomalous processes having the tempered Lévy stable waiting times with the tempering index $λ>0$ and the stability index $0<α\le 1$; as for the distribution of jump lengths or the type of the noises driving the system, two cases are considered, i.e., Gaussian white noise and non-Gaussian (tempered) $β$-stable ($0<β<2$) Lévy noise. Firstly, we derive the nonlocal elliptic partial differential equations (PDEs) governing the mean first exit time and escape probability. Based on the derived PDEs, it is observed that the mean first exit time depends strongly on the domain size and the values of $α$, $β$ and $λ$; when $λ$ is close to zero, the mean first exit time tends to $\infty$. In particular, we also find an interesting result that the escape probability of a particle with (tempered) power-law jumping length distribution has no relation with the distribution of waiting times for the model considered in this paper. For the solutions of the derived PDEs, the boundary layer phenomena are observed, which inspires the motivation for developing the boundary layer theory for nonlocal PDEs.

math.PR↗

arXiv2016

Applications of Wavelet Bases to The Numerical Solutions of Fractional PDEs

For describing the probability distribution of the positions and times of particles performing anomalous motion, fractional PDEs are derived from the continuous time random walk models with waiting time distribution having divergent first order moment and/or jump length distribution which has divergent second order moment. It can be noted that the fractional PDEs are essentially dealing with the multiscale issues. Generally the regularity of the solutions for fractional PDEs is weak at the areas close to boundary and initial time. This paper focuses on developing the applications of wavelet bases to numerically solving fractional PDEs and digging out the potential benefits of wavelet methods comparing with other numerical methods, especially in the aspects of realizing preconditioning, adaptivity, and keeping the Toeplitz structure. More specifically, the contributions of this paper are as follows: 1. the techniques of efficiently generating stiffness matrix with computational cost $\mathcal{O}(2^J)$ are provided for first, second, and any order bases; 2. theoretically and numerically discuss the effective preconditioner for time-independent equation and multigrid method for time-dependent equation, respectively; 3. the wavelet adaptivity is detailedly discussed and numerically applied to solving the time-dependent (independent) equations. In fact, having reliable, simple, and local regularity indicators is the striking benefit of the wavelet in adaptively solving fractional PDEs (it seems hard to give a local posteriori error estimate for the adaptive finite element method because of the global property of the operator).

math.NA↗