On the Joint Probability Density Function of Non-Linear Rolling Motion

Random response of a Duffing oscillator by the stochastic central difference method

Effects of time step in stochastic central difference method

This work focuses on modeling of film porosity in thin film deposition using stochastic differential equations. A deposition process is modeled via kinetic Monte Carlo (kMC) simulation on a triangular lattice. The microscopic process events involve atom adsorption and migration and the film growth allows for vacancies and overhangs to develop inside the film. Appropriate definitions of film site occupancy ratio (SOR), i.e., fraction of film sites occupied by particles over total number of film sites, and its fluctuation are introduced to describe film porosity. Deterministic and stochastic ordinary differential equation (ODE) models are also derived to describe the time evolution of film SOR and its fluctuation. The coefficients of the ODE models are estimated on the basis of data obtained from the kMC simulator of the deposition process using least-square methods. Simulation results demonstrate the applicability and effectiveness of the proposed film porosity modeling methods in the context of the deposition process under consideration.

Parameter and uncertainty estimation in stochastic differential equation models with multi-rate data and nonstationary disturbances

Efficient Estimation of Interrupted Energy with Time-Varying Load Models for Distribution Systems Planning Studies

For the safety problem of flight anti-collision in route area and terminal area, a flight conflict detection method is proposed based on Monte Carlo method. In the method, the ellipsoidal aircraft protection area model is selected and the position of aircraft is predicted through the stochastic differential equation, and then Monte Carlo method is used to calculate the probability of flight conflict in terminal area. The experiments show that the proposed method can obtain a more accurate overall flight conflict probability, thus verifying its feasibility in the application of conflict detection studies.

Modeling real-time balancing power demands in wind power systems using stochastic differential equations

Method of Iterated Integrals for Solution of Stochastic Integrals with Applications to Monte Carlo Simulation of Stochastic Differential Equations.

In the analysis of metastable diffusion processes, Transition Path Theory (TPT) provides a way to quantify the probability of observing a given transition between two disjoint metastable subsets of state space. However, many TPT-based methods for diffusion processes compute the primary objects from TPT, such as the committor and probability current, by solving partial differential equations. The computational performance of these methods is limited by the need for mesh-based computations, the need to estimate the coefficients of the stochastic differential equation that defines the diffusion process, and the use of Markovian processes to approximate the diffusion process. We propose a Monte Carlo method for approximating the primary objects from TPT from sample trajectory data of the diffusion process, without estimating drift or diffusion coefficients. We discretise the state space of the diffusion process using Voronoi tessellations and construct a non-Markovian jump process on the dual Delaunay graph. For the jump process, we define committors, probability currents, and streamlines, and use these to define piecewise constant approximations of the corresponding objects from TPT for diffusion processes. Rigorous error bounds and convergence theorems establish the validity of our approach. A comparison of our method with TPT for Markov chains (Metzner et al., Multiscale Model Simul. 2009) on a triple-well 2D potential provides proof of principle.

Fractional Brownian Motions, Fractional Noises and Applications

In this paper we present a family of balanced stochastic Runge-Kutta (BSRK) methods for stiff systems of stochastic differential equations (SDEs). This class of methods will have general framework for balancing the stochastic terms in arbitrary SRK methods of stochastic strong order p to get some balanced derivative-free methods of arbitrary stochastic order p. We also investigate the mean-square stability (MS-stability) properties for SDEs with stiffness in their drift and diffusion parts for some selected BSRK methods. Using such new methods, contrary to most existing derivative based balanced methods, in addition to having superior stability properties, one can get some significant reduction in computational cost. Numerical examples are presented to demonstrate the effectiveness of these methods for the pathwise approximation of solution of Ito SDEs.

The impact of uncertainty on the flexural response of sandwich beams is studied in this paper. The study focuses on the unique features of typical soft-core sandwich beams, including local effects and stresses concentrations. Those are critical for the integrity and performance of the structural system and, to the best knowledge of the authors, were never considered from a stochastic viewpoint. Several structural features of the beam are considered, separately, as uncertain, and the stochastic characteristics of the distributions of the displacements and stresses are investigated. To capture the unique local effects, the extended high-order sandwich panel theory is adopted. Numerical solutions of the stochastic ordinary differential equations use the finite element (FE) method. The parametric uncertainty and its spatial distribution are modeled by random fields. The stochastic analysis uses both the Karhunen–Loeve polynomial-chaos and the perturbation-based stochastic finite element methods. The two approaches are compared and comprehensively validated against Monte Carlo simulations. The numerical results quantify the impact of uncertainty on the response, but show significant disagreements between the two stochastic approaches, and the perturbation-based stochastic finite element method is chosen as the most suitable one. The effects of the coefficient of variation of the uncertain input parameter and its correlation length are also studied. Highly amplified levels of uncertainty are revealed by the stochastic analysis near points of local stress concentration. These localized yet significant uncertainties, reported here for the first time, shed new light on the design, analysis, and safety of such sandwich beams.

A complexity study of Monte Carlo methods for Fredholm integral equations is carried out. We analyze the problem of computing a functional μ ( y ) \mu (y) , where y is the solution of a Fredholm integral equation \[ y ( s ) = ∫ I m k ( s , t ) y ( t ) d t + f ( s ) , s ∈ I m y(s) = \int _{{I^m}} {k(s,t)y(t)\;dt + f(s),\quad s \in {I^m}} \] , on the m -dimensional unit cube I m {I^m} , where the kernel k and right-hand side f are given r times differentiable functions. We permit stochastic numerical methods which can make use of function evaluations of k and f only. All Monte Carlo methods known to the authors for solving the above problem are of the order n − 1 / 2 {n^{ - 1/2}} , while the optimal deterministic methods yield rate n − r / ( 2 m ) {n^{ - r/(2m)}} , thus taking into account the given smoothness of the data. Here, n denotes the (average) number of function evaluations performed. The optimal algorithm we present combines deterministic and stochastic methods in an optimal way. It can be seen that both rates—the standard Monte Carlo rate for general continuous data and the deterministic rate for r -smooth data—multiply. This provides the smallest error that stochastic methods of given computational cost can achieve.

Preview control of random response of a half-car vehicle model traversing rough road

European anchovies and round sardinella play a crucial role, both ecological and commercial, in the Mediterranean Sea. In this paper, we investigate the distribution of their larval stages by analyzing a dataset collected over time (1998–2016) and spaced along the area of the Strait of Sicily. Environmental factors are also integrated. We employ a hierarchical spatio-temporal Bayesian model and approximate the spatial field by a Gaussian Markov Random Field to reduce the computation effort using the Stochastic Partial Differential Equation method. Furthermore, the Integrated Nested Laplace Approximation is used for the posterior distributions of model parameters. Moreover, we propose an index that enables the temporal evaluation of species abundance by using an abundance aggregation within a spatially confined area. This index is derived through Monte Carlo sampling from the approximate posterior distribution of the fitted models. Model results suggest a strong relationship between sea currents’ directions and the distribution of larval European anchovies. For round sardinella, the analysis indicates increased sensitivity to warmer ocean conditions. The index suggests no clear overall trend over the years.