Showing papers on "Stochastic process published in 2023"

Stochastic Mainshock–Aftershock Simulation and Its Applications in Dynamic Reliability of Structural Systems via DPIM

Abstract: A novel approach for nonlinear stochastic dynamic analysis is proposed and illustrated with nonlinear building structures subjected to mainshock–aftershock sequences. First, a stochastic seismic sequence model with stochastic parameters was established, and its generation method was derived based on the source–path–site mechanism. Then, the representative point sets of seismic parameters could be chosen based on generalized F-discrepancy, and the correlation between the mainshock and aftershock parameters could be determined by using Copula theory. Finally, the stochastic dynamic response was obtained by solving the probability density integral equation (PDIE). Furthermore, the first-passage dynamic reliability could be obtained by the direct probability integral method (DPIM) combined with the absorbing condition approach. This novel approach was used to obtain stochastic dynamic results for four structures subjected to stochastic seismic sequences, which were compared to those using Monte Carlo simulation (MCS) and probability density evolution method (PDEM) to demonstrate the proposed method’s correctness and efficiency. Additionally, the influence of aftershocks on nonlinear structures is explained from the perspective of probability for the first time.

Some integral inequalities for harmonical $ cr $-$ h $-Godunova-Levin stochastic processes

Abstract: An important part of optimization is the consideration of convex and non-convex functions. Furthermore, there is no denying the connection between the ideas of convexity and stochastic processes. Stochastic processes, often known as random processes, are groups of variables created at random and supported by mathematical indicators. Our study introduces a novel stochastic process for center-radius (cr) order based on harmonic h-Godunova-Levin ($ \mathcal{GL} $) in the setting of interval-valued functions ($ \mathcal{IVFS} $). With some interesting examples, we establish some variants of Hermite-Hadamard ($ \mathcal{H.H} $) types inequalities for generalized interval-valued harmonic cr-h-Godunova-Levin stochastic processes.

Nonequilibrium currents in stochastic field theories: A geometric insight

Abstract: We introduce a formalism to study nonequilibrium steady-state probability currents in stochastic field theories. We show that generalizing the exterior derivative to functional spaces allows identification of the subspaces in which the system undergoes local rotations. In turn, this allows prediction of the counterparts in the real, physical space of these abstract probability currents. The results are presented for the case of the Active Model B undergoing motility-induced phase separation, which is known to be out of equilibrium but whose steady-state currents have not yet been observed, as well as for the Kardar-Parisi-Zhang equation. We locate and measure these currents and show that they manifest in real space as propagating modes localized in regions with nonvanishing gradients of the fields.

Stochastic analysis of double-skin façades subjected to imprecise seismic excitation

Abstract: Abstract. Double-skin façades (DSFs), widely used in buildings to provide specific thermal efficiency, acoustic isolation, and weather resistance properties, have been recently used as passive control systems. The present study focuses on the stochastic analysis of a shear-type frame equipped with a DSF subjected to ground motion acceleration modelled as a zero-mean stationary Gaussian random process fully characterized by an imprecise power spectral density function i.e., with interval parameters. The influence of imprecision of the seismic excitation on the performance of the DSF is investigated by evaluating the bounds of the interval reliability function for a selected displacement process of the frame structure, in the framework of the classical first-passage problem.

Data‐Driven Stochastic Lie Transport Modeling of the 2D Euler Equations

Abstract: In this paper, we propose and assess several stochastic parametrizations for data-driven modelling of the two-dimensional Euler equations using coarse-grid SPDEs. The framework of Stochastic Advection by Lie Transport (SALT) [Cotter et al., 2019] is employed to define a stochastic forcing that is decomposed in terms of a deterministic basis (empirical orthogonal functions, EOFs) multiplied by temporal traces, here regarded as stochastic processes. The EOFs are obtained from a fine-grid data set and are defined in conjunction with corresponding deterministic time series. We construct stochastic processes that mimic properties of the measured time series. In particular, the processes are defined such that the underlying probability density functions (pdfs) or the estimated correlation time of the time series are retained. These stochastic models are compared to stochastic forcing based on Gaussian noise, which does not use any information of the time series. We perform uncertainty quantification tests and compare stochastic ensembles in terms of mean and spread. Reduced uncertainty is observed for the developed models. On short timescales, such as those used for data assimilation [Cotter et al., 2020], the stochastic models show a reduced ensemble mean error and a reduced spread. Particularly, using estimated pdfs yields stochastic ensembles which rarely fail to capture the reference solution on small time scales, whereas introducing correlation into the stochastic models improves the quality of the coarse-grid predictions with respect to Gaussian noise.

A closed-form solution to time-dependent reliability of aging structures described by gamma-based deterioration models

Abstract: Structural resistance deterioration is by nature a non-increasing stochastic process with autocorrelation on the temporal scale. The Gamma process is often used to describe the stochastic behavior of resistance deterioration. With Gamma-based deterioration models, the calculation of time-dependent reliability presents a serious challenge, and often Monte Carlo simulation is the only solution to this problem. This paper derives a closed-form solution for time-dependent reliability of aging structures, in which the resistance deterioration is described by a Gamma process and the applied load is modelled as a Gaussian process with a constant standard deviation. The accuracy of the proposed method is verified through a comparison with Monte Carlo simulation results, and its applicability is further illustrated in a time-dependent reliability analysis of an existing highway bridge whose vehicle load is modelled using weigh-in-motion (WIM) data. It is found that the measured vehicle load has a relatively small uncertainty, and the uncertainty associated with resistance deterioration is crucial to the reliability assessment because it dominates the overall uncertainty in the reliability calculation.

Analysis of a stochastic HIV model with cell-to-cell transmission and Ornstein–Uhlenbeck process

Abstract: In this paper, we establish and analyze a stochastic human immunodeficiency virus model with both virus-to-cell and cell-to-cell transmissions and Ornstein–Uhlenbeck process, in which we suppose that the virus-to-cell infection rate and the cell-to-cell infection rate satisfy the Ornstein–Uhlenbeck process. First, we validate that there exists a unique global solution to the stochastic model with any initial value. Then, we adopt a stochastic Lyapunov function technique to develop sufficient criteria for the existence of a stationary distribution of positive solutions to the stochastic system, which reflects the strong persistence of all CD4+ T cells and free viruses. In particular, under the same conditions as the existence of a stationary distribution, we obtain the specific form of the probability density around the quasi-chronic infection equilibrium of the stochastic system. Finally, numerical simulations are conducted to validate these analytical results. Our results suggest that the methods used in this paper can be applied to study other viral infection models in which the infected CD4+ T cells are divided into latently infected and actively infected subgroups.

Measuring Deviation from Stochasticity in Time-Series Using Autoencoder Based Time-Invariant Representation: Application to Black Hole Data

Abstract: We propose a novel approach to quantify "deviation from stochasticity" (DS) in a time-series. This is important to determine if the time-series is coming from a physical phenomenon or if it is noise. This approach utilizes time-invariant representation obtained using time- and frequency-domain analyses. Autoencoder based time-invariant features have been utilized to obtain multi-scale reconstruction as well as identification of prominent peaks in dissimilarity curves. We devise a DS measure based on the observation that a stochastic time-series exhibits similar behavior across multiple time scales. The values of DS are expected to be significantly small for stochastic time-series in comparison with those for non-stochastic time-series, leading to classification. As proof of concept, we illustrate this trend on synthetic data. Subsequently, the proposed methodology is applied on astronomical data which are 12 distinct temporal classes of time-series pertaining to the black hole GRS 1915 + 105, obtained from RXTE satellite. This dataset had been previously studied using correlation integration (CI) based approach to understand the underlying dynamics leading to time-series classification. Results obtained using the proposed methodology are compared with those obtained using CI. Concurrence is obtained for 11 temporal classes, while one is found to be non-concurrent. This could be attributed to the observation that the non-concurrence is due to that specific time-series exhibiting both stochastic and non-stochastic characteristics. Besides, these DS values can also be interpreted as quantification of signal-to-noise ratio (SNR) of a time-series.

Nonlinear stochastic dynamics of an array of coupled micromechanical oscillators

Abstract: The stochastic response of a multi‐degree‐of‐freedom nonlinear dynamical system is determined based on the recently developed Wiener path integral (WPI) technique. The system can be construed as a representative model of electrostatically coupled arrays of micromechanical oscillators, and relates to an experiment performed by Buks and Roukes. Compared to alternative modeling and solution treatments in the literature, the paper exhibits the following novelties. First, typically adopted linear, or higher‐order polynomial, approximations of the nonlinear electrostatic forces are circumvented. Second, for the first time, stochastic modeling is employed by considering a random excitation component representing the effect of diverse noise sources on the system dynamics. Third, the resulting high‐dimensional, nonlinear system of coupled stochastic differential equations governing the dynamics of the micromechanical array is solved based on the WPI technique for determining the response joint probability density function. Comparisons with pertinent Monte Carlo simulation data demonstrate a quite high degree of accuracy and computational efficiency exhibited by the WPI technique. Further, it is shown that the proposed model can capture, at least in a qualitative manner, the salient aspects of the frequency domain response of the associated experimental setup.

Uncertainty in the Flexural Behavior of Soft-Core Sandwich Beams

Abstract: The impact of uncertainty on the flexural response of sandwich beams is studied in this paper. The study focuses on the unique features typical to 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 ODEs use the 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 are 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 more 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.

Introduction to Stochastic Processes

Abstract: This chapter briefly discusses stochastic processes, including Markov processes, Poisson processes, renewal processes, quasi-renewal processes, and nonhomogeneous Poisson processes. The chapter also provides a short list of books for readers who are interested in advanced topics in stochastic processes.

Objectivity of classical quantum stochastic processes

Abstract: We investigate what can be concluded about the quantum system when the sequential quantum measurements of its observable -- a prominent example of the so-called quantum stochastic process -- fulfill the Kolmogorov consistency condition, and thus, appear to an observer as a sampling of classical trajectory. We identify a set of physical conditions imposed on the system dynamics, that when satisfied lead to the aforementioned trajectory interpretation of the measurement results. Then, we show that when another quantum system is coupled to the observable, the operator representing it can be replaced by an external noise. Crucially, the realizations of this surrogate (classical) stochastic process are following the same trajectories as those measured by the observer. Therefore, it can be said that the trajectory interpretation suggested by the Kolmogorov consistent measurements also applies in contexts other than sequential measurements.

Countable-state stochastic processes with c\`adl\`ag sample paths

Abstract: The Daniell-Kolmogorov Extension Theorem is a fundamental result in the theory of stochastic processes, as it allows one to construct a stochastic process with prescribed finite-dimensional distributions. However, it is well-known that the domain of the constructed probability measure - the product sigma-algebra in the set of all paths - is not sufficiently rich. This problem is usually dealt with through a modification of the stochastic process, essentially changing the sample paths so that they become c\`adl\`ag. Assuming a countable state space, we provide an alternative version of the Daniell-Kolmogorov Extension Theorem that does not suffer from this problem, in that the domain is sufficiently rich and we do not need a subsequent modification step: we assume a rather weak regularity condition on the finite-dimensional distributions, and directly obtain a probability measure on the product sigma-algebra in the set of all c\`adl\`ag paths.

Modelling Illiquid Stocks Using Quantum Stochastic Calculus: Asymptotic Methods

Abstract: This article investigates the Fokker-Planck equations that arise from the application of quantum stochastic calculus to the modelling of illiquid financial markets, using asymptotic methods. We present a power series solution for quantum stochastic processes with a non-zero conservation process. Whilst the series in question are in general divergent, we show they can be used to approximate solutions for longer time frames, and provide estimates for the relative error on the higher order terms.

Stochastic Dynamics on $$\mathbb {R}^d$$

Abstract: This chapter discusses the theory of Brownian motion (Wiener process) with drift in detail. It derives the stochastic differential equations of motion from a stochastic action principle in the Lagrangian formulation. It shows how the Feynman-Kac formula can be derived from the stochastic Hamilton-Jacobi equation. Moreover, it discusses some important differences between deterministic theories and stochastic theories.