A new approach for multistep numerical methods in several frequencies for perturbed oscillators

This article presents a new multi-step numerical method based on φ-function series and designed to integrate forced oscillators with precision. The new algorithm retains the good properties of the MDF_pPC methods while presenting the advantages of greater precision and that of integrating the non-perturbed problem without any discretization error. In addition, this new method permits a single formulation to be obtained from the MDF_pPC schemes independently of the parity of the number of steps, which facilitates the design of a computational algorithm thus permitting improved implementation in a computer.The construction of a new method for accurately integrating the homogenous problem is necessary if a method is sought which would be comparable to the methods based on Scheifele G-function series, very often used when problems of satellite orbital dynamics need to be resolved without discretization error.Greater precision compared to the MDF_pPC methods and other known integrators is demonstrated by overcoming stiff and highly oscillatory problems with the new method and comparing approximations obtained with those calculated by means of other integrators.     

Stability of weak numerical schemes for stochastic differential equations

This paper considers numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise. For different explicit and implicit schemes, the regions of stability are also examined.     

Exact and fast numerical algorithms for the stochastic wave equation

On the basis of integral representations we propose fast numerical methods to solve the Cauchy problem for the stochastic wave equation without boundaries and with Dirichlet boundary conditions. The algorithms are exact in a probabilistic sense.     

Simulation of stochastic processes using graphics hardware

Graphics Processing Units (GPUs) were originally designed to manipulate images, but due to their intrinsic parallel nature, they turned into a powerful tool for scientific applications. In this article, we evaluated GPU performance in an implementation of a traditional stochastic simulation – the correlated Brownian motion. This movement can be described by the Generalized Langevin Equation (GLE), which is a stochastic integro-differential equation, with applications in many areas like anomalous diffusion, transport in porous media, noise analysis, quantum dynamics, among many others. Our results show the power inherent in GPU programming when compared to traditional CPUs (Intel): we observed acceleration values up to sixty times by using a NVIDIA GPU in place of a single-core Intel CPU.     

On stochastic Riccati equations for the stochastic LQR problem

This paper studies an approximation of stochastic Riccati equations for stochastic LQR problems some of which may be even with indefinite control weight costs.     

A numerical solution of the stochastic discrete algebraic Riccati equation

This article proposes two algorithms for solving a stochastic discrete algebraic Riccati equation which arises in a stochastic optimal control problem for a discrete-time system. Our algorithms are generalized versions of Hewer’s algorithm. Algorithm I has quadratic convergence, but needs to solve a sequence of extended Lyapunov equations. On the other hand, Algorithm II only needs solutions of standard Lyapunov equations which can be solved easily, but it has a linear convergence. By a numerical example, we shall show that Algorithm I is superior to Algorithm II in cases of large dimensions. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

On detectability of stochastic systems

We discuss notions of detectability for stochastic linear control systems of Itô type. A natural concept of detectability requires a non-zero output, if the state process is unstable. We show that this property can equivalently be characterized by a generalized version of the Hautus-test. The proof is based on spectral theory for positive operators.     

One-step explicit methods for the numerical integration of perturbed oscillators

In this paper, we present a class of one-step explicit zero-dissipative nonlinear methods for the numerical integration of perturbed oscillators, which have second algebraic order and high phase-lag order. For multi-dimensional problems, we give the vector form of the methods with the aid of a special vector operation. Some numerical results are reported to illustrate the efficiency of our methods.     

On the infinite time solution to state-constrained stochastic optimal control problems

A method is presented for solving the infinite time Hamilton-Jacobi-Bellman (HJB) equation for certain state-constrained stochastic problems. The HJB equation is reformulated as an eigenvalue problem, such that the principal eigenvalue corresponds to the expected cost per unit time, and the corresponding eigenfunction gives the value function (up to an additive constant) for the optimal control policy. The eigenvalue problem is linear and hence there are fast numerical methods available for finding the solution.     

On the numerical properties of the schur approach for solving the matrix riccati equation

An a posteriori estimate of the relative error in the solution of the algebraic matrix Riccati equation, found by the Schur approach, is derived. It is shown that in some cases the Schur approach is numerically unstable.     

On the global nonlinear stochastic dynamical behavior of a class of exothermic CSTRs

The problem of assessing the short and long time effects of stochastic fluctuations on the global-nonlinear dynamics of a class of closed-loop continuous exothermic reactors with temperature control and mono or bistable isothermal dynamics is addressed. The consideration of the problem within a Fokker–Planck (FP) stochastic framework yields: (i) the characterization of the global-nonlinear stochastic dynamics, and (ii) the connection between the deterministic and stochastic modeling approaches. The evolution of the state probability density function (PDF) is explained as the result of a complex interplay between deterministic dynamical features, initial PDF shape, and noise intensity. The correspondence between stationary PDF mono (or bi) modality and deterministic mono (or bi) stability is established, and the stochastic settling time is put in perspective with the deterministic, noise-diffusion, and escape times. The conditions for the occurrence of a retarded response, with respect to deterministic and noise-diffusion times, are identified. The proposed approach: (i) is illustrated with representative case class example, and (ii) constitutes an inductive step towards the development of a general-purpose stochastic modeling approach in chemical process systems engineering.     

On stable and explicit numerical methods for the advection–diffusion equation

In this paper two stable and explicit numerical methods to integrate the one-dimensional (1D) advection–diffusion equation are presented. These schemes are stable by design and follow the main general concept behind the semi-Lagrangian method by constructing a virtual grid where the explicit method becomes stable. It is shown that the new schemes compare well with analytic solutions and are often more accurate than implicit schemes. In particular, the diffusion-only case is explored in some detail. The error produced by the stable and explicit method is a function of the ratio between the standard deviation σ₀ of the initial Gaussian state and the characteristic virtual grid distance ΔS. Larger values of this ratio lead to very accurate results when compared to implicit methods, while lower values lead to less accuracy. It is shown that the σ₀/ΔS ratio is also significant in the advection–diffusion problem: it determines the maximum error generated by new methods, obtained with a certain combination of the advection and diffusion values. In addition, the error becomes smaller when the problem becomes more advective or more diffusive.     

On the controllability of a class of nonlinear stochastic systems

We study the controllability properties of the class of stochastic differential systems characterized by a linear controlled diffusion perturbed by a smooth, bounded, uniformly Lipschitz nonlinearity. We obtain conditions that guarantee the weak and strong controllability of the system. Also, given any open set in the state space we construct a control, depending only on the Lipschitz constant and the infinity-norm of the nonlinear perturbation, such that the hitting time of the set has a finite expectation with respect to all initial conditions.     

On the applicability of stochastic volatility models

The applicability of the stochastic volatility (SV) model and the SV model with jumps for US. Treasury Bill yields data is investigated. The transformation of the continuous time models into regression models is considered and their error terms are examined. The applicability of the continuous time models to the real data is assessed by comparing some atypical properties of such error terms with an application to the real data and the generated data from the models. The empirical results indicate that the SV model and the SV model with jumps are not applicable to modeling the daily/weekly released US T-Bill secondary market yields data. Some trends and correlation structure are detected to exist in the error terms of the transformed regression models for the daily/weekly released US T-Bill yields data, while the error terms of the continuous time models are supposed to be uncorrelated. These results suggest that alternative models are needed to model such T-Bill yields data.     

Estimation of the variances of TCP/RED using stochastic differential equation

By using the stochastic differential equation (SDE), a model is built to describe the system behavior, especially the queue length behavior, of TCP/RED (transmission control protocol with random early detection) on a single bottleneck network. Then, not only the static equilibrium point of the system is calculated, but also its variance is estimated. After calculating the variance of the queue length in the router, the authors show how such random oscillation affects the maximum connection number that the link admits while keeping low dropping probability, and how this relation is influenced by other network parameters. This model can be applied to the case where only homogeneous TCP flows exist, as well as where TCP and UDP (user data protocol) flows coexist. The simulations by NS‐2 show that our deductive method is effective. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Non-asymptotic confidence bounds for the optimal value of a stochastic program

We discuss a general approach to building non-asymptotic confidence bounds for Stochastic Optimization problems. Our principal contribution is the observation that a Sample Average Approximation of a problem supplies upper and lower bounds for the optimal value of the problem which are essentially better than the quality of the corresponding optimal solutions. At the same time, such bounds are more reliable than ‘standard’ confidence bounds obtained through the asymptotic approach. We also discuss bounding the optimal value of MinMax Stochastic Optimization and stochastically constrained problems. We conclude with a simulation study illustrating the numerical behaviour of the proposed bounds.     

Analysis on the stability of numerical schemes for a class of stochastic partial differential systems

The high-order accurate compact finite difference scheme which belongs to the finite difference methods is constructed to solve the system of partial differential equations with random noise. The error analysis and stability analysis are given and then the numerical simulation is executed. The simulation results verify the theoretical analysis results and have the faster computation speed and higher accuracy.     

On the exponential stability in mean square of neutral stochastic functional differential equations

In [1 and 2], some efforts have been devoted to the investigation of exponential stability in mean square of neutral stochastic functional differential equations. However, the results derived there are either difficult to demonstrate in a straightforward way for practical situations or somewhat too restricted to be applied to general neutral stochastic functional differential equations, for instance, nonautonomous cases. In this paper, we shall establish some results which are more effective and relatively easy to verify to obtain the required stability.     

On the quasi-decentralized estimation and control of linear stochastic systems

It is shown that the quasi-decentralized estimation and control of linear stochastic systems have extremely simple and computationally very efficient solutions in the domain of nonoptimal local filters. These solutions reduce both off-line and on-line computation requirements. The optimal global estimate is obtained simply by the addition of nonoptimal local estimates.