Numerical Studies of Three-dimensional Stochastic Darcy’s Equation and Stochastic Advection-Diffusion-Dispersion Equation

Solute transport in randomly heterogeneous porous media is commonly described by stochastic flow and advection-dispersion equations with a random hydraulic conductivity field. The statistical distribution of conductivity of engineered and naturally occurring porous material can vary, depending on its origin. We describe solutions of a three-dimensional stochastic advection-dispersion equation using a probabilistic collocation method (PCM) on sparse grids for several distributions of hydraulic conductivity. Three random distributions of log hydraulic conductivity are considered: uniform, Gaussian, and truncated Gaussian (beta). Log hydraulic conductivity is represented by a Karhunen-Loève (K-L) decomposition as a second-order random process with an exponential covariance function. The convergence of PCM has been demonstrated. It appears that the accuracy in both the mean and the standard deviation of PCM solutions can be improved by using the Jacobi-chaos representing the truncated Gaussian distribution rather than the Hermite-chaos for the Gaussian distribution. The effect of type of distribution and parameters such as the variance and correlation length of log hydraulic conductivity and dispersion coefficient on leading moments of the advection velocity and solute concentration was investigated.     

Stochastic Simulation of Ligand–Receptor Interaction

We have developed an algorithm for the stochastic simulation of ligand–receptor interactions based on 10⁴–10⁵fictitious binding sites. Reversible receptor binding was simulated by alternate random selection of sites, the first selection resulting in “occupation” if the selected site was “free,” the second selection resulting in “dissociation” if the selected site was “occupied.” We show that the mathematical formalism of mass action kinetics is predicted on purely statistical grounds. The model was extended by the introduction of two further selections, simulating a conformational change in the ligand–receptor complex (“receptor isomerization model”). All random selections were gauged separately by “probability barriers,” taking the place of macroscopic kinetic rate constants. Simulation of gradual increases and gradual decreases of the fraction of occupied fictitious binding sites in the receptor isomerization model, using various combinations of “rate constants,” resulted in biexponential time dependencies, in agreement with predictions from the integrated rate equations. Stochastic simulation of molecular processes is a powerful and versatile technique, providing the researcher with a means of studying mechanisms of increasing complexity.     

Stochastic modeling and analysis of road–tramway intersections

In the last decades, the socio-demographic evolution of the population has substantially changed mobility demand, posing new challenges in minimizing urban congestion and reducing environmental impact. In this scenario, understanding how different modes of transport can efficiently share (partially or totally) a common infrastructure is crucial for urban development. To this aim, we present a stochastic model-based analysis of critical intersections shared by tram traffic and private traffic, combining a microscopic model of the former with a macroscopic model of the latter. Advanced simulation tools are typically used for such kind of analyses, by playing various traffic scenarios. However, simulation is not an exhaustive approach, and some critical, possibly rare, event may be ignored. For this reason, our aim is instead to adopt suitable analytical solution techniques and tools that can support instead a complete, exhaustive analysis, so being able to take into account rare events as well. Transient analysis of the overall traffic model using the method of stochastic state classes is adopted to support the evaluation of relevant performance measures, namely the probability of traffic congestion over time and the average number of private vehicles in the queue over time. A sensitivity analysis is performed with respect to multiple parameters, notably including the arrival rate of private vehicles, the frequency of tram rides, and the time needed to recover from traffic congestion.

     

Stochastic Distribution Control of Singular Systems: Output PDF Shaping

This paper presents a new algorithm designed to control the shape of the output probability density function (PDF) of singular systems subjected to non-Gaussian input. The aim is to select a control input uk such that the output PDF is made as close as possible to a given PDF. Based on the B-spline neural network approximation of the output PDF, the control algorithm is formulated by extending the developed PDF control strategies of non-singular systems to singular systems. It has been shown that under certain conditions the stability of the closed-loop system can be guaranteed. Simulation examples are given to show the effectiveness of the proposed control algorithm.     

Stochastic analysis of real-time systems under preemptive priority-driven scheduling

Exact stochastic analysis of most real-time systems under preemptive priority-driven scheduling is unaffordable in current practice. Even assuming a periodic and independent task model, the exact calculation of the response time distribution of tasks is not possible except for simple task sets. Furthermore, in practice, tasks introduce complexities such as release jitter, blocking in shared resources, etc., which cannot be handled by the periodic independent task set model. In order to solve these problems, exact analysis must be abandoned for an approximated analysis. However, in the real-time field, approximations must not be optimistic, i.e. the deadline miss ratios predicted by the approximated analysis must be greater than or equal to the exact ones. In order to achieve this goal, the concept of pessimism needs to be mathematically defined in the stochastic context, and the pessimistic properties of the analysis carefully derived. This paper provides a mathematical framework for reasoning about stochastic pessimism, and obtaining mathematical properties of the analysis and its approximations. This framework allows us to prove the safety of several proposed approximations and extensions. We analyze and solve some practical problems in the implementation of the stochastic analysis, such as the problem of the finite precision arithmetic or the truncation of the probability functions. In addition, we extend the basic model in several ways, such as the inclusion of shared resources, release jitter or non-preemptive sections.

Stochastic stability of cubature predictive filterCubature 预测滤波的随机稳定性分析

In this paper, the cubature predictive filter (CPF) is derived based on a third-degree spherical-radial cubature rule. It provides a set of cubature-points scaling linearly with the state-vector dimension, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear predictive filter (PF). In order to facilitate the new method, the algorithm CPF is given firstly. Then, the theoretical analyses demonstrate that the estimated accuracy of the model error and system for the proposed CPF is higher than that of the traditional PF. Moreover, the authors analyze the stochastic boundedness and the error behavior of CPF for general nonlinear systems in a stochastic framework. In particular, the theoretical results present that the estimation error remains bounded and the covariance keeps stable if the system’s initial estimation error, disturbing noise terms as well as the model error are small enough, which is the core part of the CPF theory. All of the results have been demonstrated by numerical simulations for a nonlinear example system.     

A Priori Open Loop Optimal Control of Continuous Time Stochastic Systems†

This paper is presented, not so much as an end in itself concerning the solution of a definite engineering problem, but as an exploratory step toward the solution of the problem of optimal control of continuous time stochastic non-linear systems when only noisy observations of the state are available. In this paper, the problem of determining the optimal open loop control, when no observations at all are available, is treated by dynamic programming. The main results of the paper are to obtain the functional equation of dynamic programming and to present a quasi-linearization type of algorithm for its solution. The author's intention was to illustrate by these results that infinite dimensional function space is the most natural setting for stochastic non-linear problems. The results obtained give some insight into what can be expected in the more general case of noisy observations. In the Appendices an argument is presented to justify the proposed algorithm and an example is given for which an exact solution to the functional equation of dynamic programming can be obtained.     

Stochastic learning of multi-instance dictionary for earth mover’s distance-based histogram comparison

Dictionary plays an important role in multi-instance data representation. It maps bags of instances to histograms. Earth mover’s distance (EMD) is the most effective histogram distance metric for the application of multi-instance retrieval. However, up to now, there is no existing multi-instance dictionary learning methods designed for EMD-based histogram comparison. To fill this gap, we develop the first EMD-optimal dictionary learning method using stochastic optimization method. In the stochastic learning framework, we have one triplet of bags, including one basic bag, one positive bag, and one negative bag. These bags are mapped to histograms using a multi-instance dictionary. We argue that the EMD between the basic histogram and the positive histogram should be smaller than that between the basic histogram and the negative histogram. Base on this condition, we design a hinge loss. By minimizing this hinge loss and some regularization terms of the dictionary, we update the dictionary instances. The experiments over multi-instance retrieval applications shows its effectiveness when compared to other dictionary learning methods over the problems of medical image retrieval and natural language relation classification.

     

Stochastic elastic–plastic finite elements

A computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic–plastic materials. Uncertain elastic–plastic material properties are modeled as random fields, which appear as the coefficient term in the governing partial differential equation of mechanics. A spectral stochastic elastic–plastic finite element method with Fokker–Planck–Kolmogorov equation based probabilistic constitutive integrator is proposed for solution of this non-linear (elastic–plastic) partial differential equation with stochastic coefficient. To this end, the random field material properties are discretized, in both spatial and stochastic dimension, into finite numbers of independent basic random variables, using Karhunen–Loève expansion. Those random variables are then propagated through the elastic–plastic constitutive rate equation using Fokker–Planck–Kolmogorov equation approach, to obtain the evolutionary material properties, as the material plastifies. The unknown displacement (solution) random field is then assembled - using polynomial chaos - as a function of known basic random variables and unknown deterministic coefficients, which are obtained by minimizing the error of finite representation, by Galerkin technique.     

Linear Systems with Stochastic Coefficients. II†

The paper is an extension of a previous one (Ariaratnam and Graefe 1965), and presents the analysis of general linear systems governed by n first-Order state equations whose coefficients are subjected to either of two forms of random variation, namely Gaussian white or incremental Brownian type of process. The differences between the two cases are pointed out. The Fokker-Planck equation is set up for both cases and the equations governing the response moments are obtained. An example is given which incorporates both types of random variation of the coefficients.     

Stochastic resonance–a nonlinear control theory interpretation

Stochastic resonance (SR) is an effect that has been known (Benzi, R., Sutera, A., and Vulpiani, A. (1981 Chapeau-Blondeau, F. 1997. Input-Output Gains for Signal in Noise in Stochastic Resonance. Physics Letters A, 232: 41–48.  [Google Scholar]), ‘The Mechanism of Stochastic Resonance’, Journal of Physics, A14, L453–L457) for almost three decades and has been extensively studied in biology, statistics, signal processing and in numerous other eclectic areas (Wiesenfeld, K., and Moss, F. (1995 Xu, B, Jiang, Z-P, Wu, X and Repperger, DW. 2009. Investigation of Two-dimensional Parameter-Induced Stochastic Resonance and Applications in Nonlinear Image Processing. Journal of Physics A–Mathematical and Theoretical, 42: 145207, 1-45207, 9 [Google Scholar]), ‘Stochastic Resonance and the Benefits of Noise: From Ice Ages to Crayfish and Squids’, Nature, 373, 33–36). Herein, a nonlinear control theory analysis is conducted on how to better understand the class of systems that may exhibit the SR effect. Using nonlinear control theory methods, equilibrium points are manipulated to create the SR response (similar to shaping dynamical response in a phase plane). From this approach, a means of synthesising and designing the appropriate class of nonlinear systems is introduced. New types of nonlinear dynamics that demonstrate the SR effects are discovered, which may have utility in control theory as well as in many diverse applications. A numerical simulation illustrates some powerful attributes of these systems.     

Stochastic control in non‐ homogeneous markov systems

We study the problem of maintainability of the structure in a non‐homogeneous Markov ystem (NHMS) by input control. The set of maintainable structures is given as the convex hull of n‐points in Rn where n is the number of states of the NHMS. We also study the problem of attainability in a NHMS by input control and an algorithm is provided for the optimal policy in this respect. Finally, we provide applications of the present results to a large British firm     

Further Results on Input-to-State Stability of Stochastic Cohen–Grossberg BAM Neural Networks with Probabilistic Time-Varying Delays

Neural Processing Letters - In this article, the problem of stochastic Cohen–Grossberg Bidirectional Associative Memory (CGBAM) neural networks with probabilistic time-varying delay is...     

Existence,Uniqueness and Stability of Mild Solutions to a Stochastic Nonlocal Delayed Reaction–Diffusion Equation

Neural Processing Letters - The aim of this paper is to investigate the existence, uniqueness and stability of mild solutions to a stochastic delayed reaction–diffusion equation with spatial...     

Stochastic bottom–up fixation prediction and saccade generation

In this article, a novel technique for fixation prediction and saccade generation will be introduced. The proposed model simulates saccadic eye movement to incorporate the underlying eye movement mechanism into saliency estimation. To this end, a simple salience measure is introduced. Afterwards, we derive a system model for saccade generation and apply it in a stochastic filtering framework. The proposed model will dynamically make a saccade toward the next predicted fixation and produces saliency maps. Evaluation of the proposed model is carried out in terms of saccade generation performance and saliency estimation. Saccade generation evaluation reveals that the proposed model outperforms inhibition of return. Also, experiments signify integration of eye movement mechanism into saliency estimation boosts the results. Finally, comparison with several saliency models shows the proposed model performs aptly.     

Stochastic comb-pilot design for NC-OFDM based cognitive radio system＇

NC-OFDM is a transmission mode of cognitive radio systems. The paper models and analyzes NC-OFDM stochastic process, then designs stochastic comb-pilot pattern and discusses its characteristics in detail. At the end of the paper simulation results show that channel estimation performance by Stochastic comb-pilots is better than those of conventional schemes, and prove feasibilities of the stochastic comb-pilot strategies in NC-OFDM system.     

Predictive Compensation for Stochastic Time Delay in Network Control Systems

This paper proposes a predictive compensation strategy to reduce the detrimental effect of stochastic time delays induced by communication networks on control performance. Values of a manipulated variable at the present sampling instant and future time instants can be determined by performing a receding horizon optimal procedure only once. When the present value of the manipulated variable does not arrive at a smart actuator, its predictive one is imposed to the corresponding process Switching of a manipulated variable between its true present value and the predictive one usually results in unsmooth operation of a control system. This paper shows: 1) for a steady process, as long as its input is sufficiently smooth, the smoothness of its output can be guaranteed; 2) a manipulated variable can be switched smoothly by filtering the manipulated variable just using a simple low-pass filter. Thus the control performance can be improved. Finally, the effectiveness of the proposed method is demonstrated by simulation study.