The mean first exit time and escape probability are utilized to quantify dynamical behaviors of stochastic differential equations with non-Gaussian α-stable type Lévy motions. An efficient and accurate numerical scheme is developed and validated for computing the mean exit time and escape probability from the governing differential-integral equation. An asymptotic solution for the mean exit time is given when the pure jump measure in the Lévy motion is small. From both the analytical and numerical results, it is observed that the mean exit time depends strongly on the domain size and the value of α in the α-stable Lévy jump measure. The mean exit time and escape probability could become discontinuous at the boundary of the domain, when the value of α is in (0, 1). Introduction.Random fluctuations in complex systems in engineering and science are often non-Gaussian [33,12,11]. For instance, it has been argued that diffusion by geophysical turbulence [29] corresponds, loosely speaking, to a series of "pauses," when the particle is trapped by a coherent structure, and "flights" or "jumps" or other extreme events, when the particle moves in the jet flow. Paleoclimatic data [13] also indicate such irregular processes.Lévy motions are thought to be appropriate models for non-Gaussian processes with jumps [26]. Recall that a Lévy motion L(t), or L t , is a stochastic process with stationary and independent increments. That is, for any s, t with 0 ≤ s < t, the distribution of L t − L s only depends on t − s, and for any 0 ≤ t 0 < t 1 < · · · < t n , L ti − L ti−1 , i = 1, . . . , n, are independent. Without loss of generality, we may assume that the sample paths of L t are almost surely right continuous with left limits.This generalizes the Brownian motion B(t), which satisfies all three conditions. But additionally, (i) almost all sample paths of the Brownian motion are continuous in time in the usual sense and (ii) Brownian motion's increments are Gaussian distributed.
The Fokker-Planck equations for stochastic dynamical systems, with non-Gaussian α−stable symmetric Lévy motions, have a nonlocal or fractional Laplacian term. This nonlocality is the manifestation of the effect of non-Gaussian fluctuations. Taking advantage of the Toeplitz matrix structure of the time-space discretization, a fast and accurate numerical algorithm is proposed to simulate the nonlocal Fokker-Planck equations, under either absorbing or natural conditions. The scheme is shown to satisfy a discrete maximum principle and to be convergent. It is validated against a known exact solution and the numerical solutions obtained by using other methods. The numerical results for two prototypical stochastic systems, the Ornstein-Uhlenbeck system and the double-well system are shown.This measure ν is the so called Lévy jump measure of the Lévy process L t . We also call (b, A, ν) the generating triplet [3,21].In this paper, we consider stochastic differential equations (SDEs) with a special class of Lévy processes, the α-stable symmetric Lévy motions. The corresponding Fokker-Planck equations (for the evolution of the probability density function of the solution) contain a nonlocal term, i.e., the fractional Laplacian term, which quantifies the non-Gaussian effect. It is hardly possible to have analytical solutions for these nonlocal Fokker-Planck equations even for simple systems. We thus consider numerical simulation for these nonlocal equations, either on a bounded domain or a unbounded domain. Due to the nonlocality, however,
We study the sum uncertainty relations based on variance and skew information for arbitrary finite N quantum mechanical observables. We derive new uncertainty inequalities which improve the exiting results about the related uncertainty relations. Detailed examples are provided to illustrate the advantages of our uncertainty inequalities.
In recent years, data-driven methods for discovering complex dynamical systems in various fields have attracted widespread attention. These methods make full use of data and have become powerful tools to study complex phenomena. In this work, we propose a framework for detecting dynamical behaviors, such as the maximum likelihood transition path, of stochastic dynamical systems from data. For a stochastic dynamical system, we use the Kramers–Moyal formula to link the sample path data with coefficients in the system, then use the extended sparse identification of nonlinear dynamics method to obtain these coefficients, and finally calculate the maximum likelihood transition path. With two examples of stochastic dynamical systems with additive or multiplicative Gaussian noise, we demonstrate the validity of our framework by reproducing the known dynamical system behavior.
Drag estimation and shape optimization of autonomous underwater vehicle (AUV) hulls are critical to energy utilization and endurance improvement. In the present work, a shape optimization platform composed of several commercial software packages is presented. Computational accuracy, efficiency and robustness were carefully considered and balanced. Comparisons between experiments and computational fluid dynamics (CFD) were conducted to prove that a two-dimensional (2D) unstructured mesh, a standard wall function and adaptive mesh refinement could greatly improve efficiency as well as guarantee accuracy. Details of the optimization platform were then introduced. A comparison of optimizers indicates that the multi-island genetic algorithm (MIGA) obtains a better hull shape than particle swarm optimization (PSO), despite being a little more time consuming. The optimized hull shape under general volume requirement could provide reference for AUV hull design. Specific requirements based on optimization testify of the platform's robustness.
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