On the Influence of Stochastic Moments in the Solution of the Neutron Point Kinetics Equation

On the Influence of Stochastic Moments in the Solution of the Neutron Point Kinetics EquationComment: 12 pages, 2 figure

Moment Lyapunov exponents of the stochastic parametrical Hill’s equation

AbstractThe Lyapunov exponent and moment Lyapunov exponents of Hill’s equation with frequency and damping coefficient fluctuated by white noise stochastic process are investigated. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises. The results are applied to the study of the almost-sure and the moment stability of the stationary solutions of the thin simply supported beam subjected to axial compressions and time-varying damping which are small intensity stochastic excitations

MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC STABILITY OF A THIN-WALLED BEAM SUBJECTED TO AXIAL LOADS AND END MOMENTS

In this paper, the Lyapunov exponent and moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation are investigated. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. The Lyapunov exponent and moment Lyapunov exponents are important characteristics for determining both the almost-sure and the moment stability of a stochastic dynamic system. As an example, we study the almost-sure and moment stability of a thin-walled beam subjected to stochastic axial load and stochastically fluctuating end moments.  The validity of the approximate results for moment Lyapunov exponents is checked by numerical Monte Carlo simulation method for this stochastic system

Explicit stabilized integration of stiff determinisitic or stochastic problems

Explicit stabilized methods for stiff ordinary differential equations have a long history. Proposed in the early 1960s and developed during 40 years for the integration of stiff ordinary differential equations, these methods have recently been extended to implicit-explicit or partitioned type methods for advection-diffusion-reaction problems, and to efficient explicit solvers for stiff mean-square stable stochastic problems. After a short review on the basic stabilized methods we discuss some recent developments

IMECE2003-44486 NUMERICAL DETERMINATION OF THE MOMENT LYAPUNOV EXPONENTS

Abstract Two numerical methods for the determination of the pth moment Lyapunov exponents of a two-dimensional system under bounded noise or real noise parametric excitation are presented. The first method is an analytical-numerical approach, in which the partial differential eigenvalue problems governing the moment Lyapunov exponents are established using the theory of stochastic dynamical systems. The eigenfunctions are expanded in double series to transform the partial differential eigenvalue problems to linear algebraic eigenvalue problems, which are then solved numerically. The second method is a Monte Carlo simulation approach. The numerical values obtained are compared with approximate analytical results with weak noise amplitudes

Efficient Computational Approaches for Treatment of Transformed Path Integrals

In this thesis, efficient computational approaches for treatment of transformed path integrals (TPIs) are proposed. The TPI-based method allows us to calculate the time evolution probability density functions (PDFs) using a short time propagator matrix that accounts for the transition probability in a transformed domain. A grid-based implementation of the TPI, in contrast to the conventional fixed-grid implementation of a path integral (PI), allows the propagation of the PDF to be performed on a dynamically adaptive grid parametrized by the mean and covariance of the PDF. TPI-based methods generate PDFs from all possible paths within the transformed space, and while these methods are found to be highly effective at capturing tail information in systems with large drifts, diffusions, and concentrations, they can become somewhat computationally expensive when applied to systems that must be represented by large numbers of data points. The purpose of this thesis is to develop computationally efficient TPI-based methods that largely preserve the accuracy and other desirable features of the original TPI method. The first proposed method, referred to as the bandlimited TPI (BL-TPI) method, takes advantage of the fact that the transition probability is often concentrated around a set of peaks, with one natural peak occurring for each source state. This allows us to consider sparse matrix representations of the transition probability matrix operator and consider a region of importance about the peak transition probability curve for consideration in PDF propagation while neglecting all values outside of this region. With the use of sparse matrix tools, the BL-TPI enables us to perform PDF propagation using far fewer operations than the standard implementation. In the second proposed method, a TPI implementation based on the Symmetric Fast Gauss Transform (SFGT) is proposed. This method utilizes a Taylor series expansion of the Gaussian kernel in the propagator matrix to reduce the convolution operation for the PDF to an infinite sum of moments. This allows us to perform calculations involving source and target terms separately, eliminating their convolution and in the process potentially reducing the associated computational complexity. In order to demonstrate the effectiveness of the proposed approaches, comparisons with the standard TPI implementation are performed for canonical problems in onedimensional and multi-dimensional state spaces. The results from the BL-TPI method appear promising and indicate that the method is applicable to a wide range of cases. In contrast, the effectiveness of the SFGT approach is found to be inherently conditional, and the computational cost of this method can exceed that of the standard TPI method in many cases

Fractional Stochastic Dynamics in Structural Stability Analysis

The objective of this thesis is to develop a novel methodology of fractional stochastic dynamics to study stochastic stability of viscoelastic systems under stochastic loadings. Numerous structures in civil engineering are driven by dynamic forces, such as seismic and wind loads, which can be described satisfactorily only by using probabilistic models, such as white noise processes, real noise processes, or bounded noise processes. Viscoelastic materials exhibit time-dependent stress relaxation and creep; it has been shown that fractional calculus provide a unique and powerful mathematical tool to model such a hereditary property. Investigation of stochastic stability of viscoelastic systems with fractional calculus frequently leads to a parametrized family of fractional stochastic differential equations of motion. Parametric excitation may cause parametric resonance or instability, which is more dangerous than ordinary resonance as it is characterized by exponential growth of the response amplitudes even in the presence of damping. The Lyapunov exponents and moment Lyapunov exponents provide not only the information about stability or instability of stochastic systems, but also how rapidly the response grows or diminishes with time. Lyapunov exponents characterizes sample stability or instability. However, this sample stability cannot assure the moment stability. Hence, to obtain a complete picture of the dynamic stability, it is important to study both the top Lyapunov exponent and the moment Lyapunov exponent. Unfortunately, it is very difficult to obtain the accurate values of theses two exponents. One has to resort to numerical and approximate approaches. The main contributions of this thesis are: (1) A new numerical simulation method is proposed to determine moment Lyapunov exponents of fractional stochastic systems, in which three steps are involved: discretization of fractional derivatives, numerical solution of the fractional equation, and an algorithm for calculating Lyapunov exponents from small data sets. (2) Higher-order stochastic averaging method is developed and applied to investigate stochastic stability of fractional viscoelastic single-degree-of-freedom structures under white noise, real noise, or bounded noise excitation. (3) For two-degree-of-freedom coupled non-gyroscopic and gyroscopic viscoelastic systems under random excitation, the Stratonovich equations of motion are set up, and then decoupled into four-dimensional Ito stochastic differential equations, by making use of the method of stochastic averaging for the non-viscoelastic terms and the method of Larionov for viscoelastic terms. An elegant scheme for formulating the eigenvalue problems is presented by using Khasminskii and Wedig’s mathematical transformations from the decoupled Ito equations. Moment Lyapunov exponents are approximately determined by solving the eigenvalue problems through Fourier series expansion. Stability boundaries, critical excitations, and stability index are obtained. The effects of various parameters on the stochastic stability of the system are discussed. Parametric resonances are studied in detail. Approximate analytical results are confirmed by numerical simulations.1 yea

Optimal filtering for Itô-stochastic continuous-time systems with multiple delayed measurements

This paper focuses on the problem of Kalman filtering for Itô stochastic continuous-time systems with multiple delayed measurements, for which very little work exist to date. For an Itô-stochastic system, its stochastic differential and integral have a significant place and are different from other stochastic systems owing to the Wiener or the Brownian process. In this paper, an Itô stochastic continuous-time system with multiple delayed measurements is first reduced to a system with delay free measurements by applying the stochastic analysis and calculus of stochastic variables. Next, the Itô differentials for the optimal filter and its error variance are derived. Finally, through an illustrative example, the performance of the designed optimal filter is verified. © 2011 AACC American Automatic Control Council

Optimal filtering for Itô-stochastic continuous-time systems with multiple delayed measurements

This technical note focuses on optimal filtering for Itô stochastic continuous-time systems with multiple delayed measurements. Stochastic analysis and calculus of stochastic variables are the main tools employed for the analysis and design. For an Itô-stochastic system, its stochastic differential and integral have a significant place and are different from that for other stochastic systems owing to the Wiener or the Brownian process. In this technical note, an Itô stochastic continuous-time system with multiple delayed measurements is first reduced to an equivalent system with delay free measurements by solving stochastic equation via the non-singularity of the transition matrix instead of reorganizing the innovation. Then, based on the delay-free measurements the optimal filter is derived through calculation of the conditional expectation. It is should be stressed that the optimal filter follows directly from the manipulation of the performance. Finally, a short interest rate model in mathematical finance is chosen to demonstrate the design of the optimal filter via the approach proposed in the technical note. © 1963-2012 IEEE