First-passage Probability Research Articles

Molecular search with conformational change: One-dimensional discrete-state stochastic model.

Molecular search phenomena are observed in a variety of chemical and biological systems. During the search, the participating particles frequently move in complex inhomogeneous environments with random transitions between different dynamic modes. To understand the mechanisms of molecular search with alternating dynamics, we investigate the search dynamics with stochastic transitions between two conformations in a one-dimensional discrete-state stochastic model. It is explicitly analyzed using the first-passage time probability method to obtain a full dynamic description of the search process. A general dynamic phase diagram is developed. It is found that there are several dynamic regimes in the molecular search with conformational transitions, and they are determined by the relative values of the relevant length scales in the system. Theoretical predictions are fully supported by Monte Carlo computer simulations.

Reliability function determination of nonlinear oscillators under evolutionary stochastic excitation via a Galerkin projection technique

An approximate semi-analytical technique is developed for determining the response first-passage time probability density function of a class of lightly damped nonlinear oscillators subject to evolutionary stochastic excitation. Specifically, relying on a Markovian approximation of the response energy envelope, and on a stochastic averaging treatment, yields a backward Kolmogorov equation governing the evolution in time of the oscillator reliability function. Next, the backward Kolmogorov equation is solved by employing an appropriate orthogonal basis in conjunction with a Galerkin projection scheme. It is noted that the technique can account for arbitrary evolutionary excitation forms, even of the non-separable type. The special case of an undamped oscillator, for which relevant analytical results exist in the literature, is also included and studied in detail. Further, Markovian approximation of the potential energy envelope is considered as well. In comparison with the conventional amplitude-based energy envelope formulation, the intermediate step of linearizing the nonlinear stiffness element is circumvented, thus reducing the overall approximation degree of the technique. An additional significant advantage of the potential energy envelope formulation relates to the fact that its degree of accuracy appears rather insensitive to the nonlinearity magnitude (at least in the considered examples). Pertinent Monte Carlo simulation data are included in the numerical examples as well for assessing the accuracy of the technique.

System-reliability-based design and topology optimization of structures under constraints on first-passage probability

For the purpose of reliability assessment of a structure subject to stochastic excitations, the probability of the occurrence of at least one failure event over a time interval, i.e. the first-passage probability, often needs to be evaluated. In this paper, a new method is proposed to incorporate constraints on the first-passage probability into reliability-based optimization of structural design or topology. For efficient evaluations of first-passage probability during the optimization, the failure event is described as a series system event consisting of instantaneous failure events defined at discrete time points. The probability of the series system event is then computed by use of a system reliability analysis method termed as the sequential compounding method. The adjoint sensitivity formulation is derived for calculating the parameter sensitivity of the first-passage probability to facilitate the use of efficient gradient-based optimization algorithms. The proposed method is successfully demonstrated by numerical examples of a space truss and building structures subjected to stochastic earthquake ground motions.

SDE models with exponential drift and diffusion for approximating fatigue crack growth dynamics

A scalar stochastic differential equation (SDE) is studied for modeling fatigue crack growth dynamics. The drift and diffusion coefficients in the SDE model are exponential functions of the number of fatigue cycles. Exponential coefficients are shown to be reasonable based on experimental, physical, and theoretical arguments. The SDE model is examined for two fatigue crack growth datasets. For each dataset, the four parameters in the SDE model are fit using observed crack lengths for two different low values for the number of cycles. Then, using the SDE model with the fitted parameters, probability distributions for several higher values of the number of cycles and several first-passage probability distributions are calculated and compared to the observed distributions. The comparisons indicate that the proposed SDE model provides a good approximation for fatigue crack growth.

A note on first passage probabilities of a Lévy process reflected at a general barrier

A NOTE ON FIRST PASSAGE PROBABILITIES OF A LÉVY PROCESS REFLECTED AT A GENERAL BARRIERIn this paper we analyze a Lévy process reflected at a general possibly random barrier. For this process we prove the Central Limit Theorem for the first passage time. We also give the finite-time first passage probability asymptotics.

Reliability Based Structural Topology Optimization Considering Non-stationary Stochastic Excitations

This paper studies the optimum structural design considering non-stationary stochastic excitations. The topology optimization of the lateral bracing system of frame structures is conducted and the first-passage probability of a displacement response is minimized under the material volume constraint. The concept of Solid Isotropic Material with Penalization (SIMP) model is employed for describing the material distribution. An efficient optimization algorithm based on explicit time-domain method is developed. Numerical examples of frame structures subjected to non-stationary seismic excitations are investigated to demonstrate the effectiveness of the proposed approach.

Influence of the random walk finite step on the first-passage probability

A well known connection between first-passage probability of random walk and distribution of electrical potential described by Laplace equation is studied. We simulate random walk in the plane numerically as a discrete time process with fixed step length. We measure first-passage probability to touch the absorbing sphere of radius R in 2D. We found a regular deviation of the first-passage probability from the exact function, which we attribute to the finiteness of the random walk step.

Approximate survival probability determination of hysteretic systems with fractional derivative elements

A Galerkin scheme-based approach is developed for determining the survival probability and first-passage probability of a randomly excited hysteretic systems endowed with fractional derivative elements. Specifically, by employing a combination of statistical linearization and of stochastic averaging, the amplitude of the system response is modeled as one-dimensional Markovian Process. In this manner the corresponding backward Kolmogorov equation which governs the evolution of the survival probability of the system is determined. An approximate solution of this equation is sought by employing a Galerkin scheme in which a convenient set of confluent hypergeometric functions is used as an orthogonal basis. This set is well documented in the literature, as it is related to the solution of the first-passage problem of a randomly excited linear oscillator with integer-order derivatives. Applications to oscillators with bilinear, Preisach, and Bouc–Wen hysteretic models are presented. Comparisons with pertinent Monte Carlo simulations data demonstrate the efficiency and reliability of the proposed semi-analytical approach.

Unified path integral approach to theories of diffusion-influenced reactions.

Building on mathematical similarities between quantum mechanics and theories of diffusion-influenced reactions, we develop a general approach for computational modeling of diffusion-influenced reactions that is capable of capturing not only the classical Smoluchowski picture but also alternative theories, as is here exemplified by a volume reactivity model. In particular, we prove the path decomposition expansion of various Green's functions describing the irreversible and reversible reaction of an isolated pair of molecules. To this end, we exploit a connection between boundary value and interaction potential problems with δ- and δ^{'}-function perturbation. We employ a known path-integral-based summation of a perturbation series to derive a number of exact identities relating propagators and survival probabilities satisfying different boundary conditions in a unified and systematic manner. Furthermore, we show how the path decomposition expansion represents the propagator as a product of three factors in the Laplace domain that correspond to quantities figuring prominently in stochastic spatially resolved simulation algorithms. This analysis will thus be useful for the interpretation of current and the design of future algorithms. Finally, we discuss the relation between the general approach and the theory of Brownian functionals and calculate the mean residence time for the case of irreversible and reversible reactions.

Stochastic nonlinear vibration and reliability of orthotropic membrane structure under impact load

Orthotropic membrane structure is widely applied in construction buildings, mechanical engineering, electronic meters, space and aeronautics, etc. During their serving period, membrane structure is prone to vibrate stochastically and seriously under stochastic dynamic loads, which may lead to structural failure. For this purpose, this paper investigates the stochastic dynamic response and reliability analysis of membrane structure under impact load obeying Gaussian distribution. The equation of stochastic motion of membrane structure is established by Von Karman's large deformation theory. The results of stochastic dynamic response are obtained with perturbation method solving the equation. Then, reliability parameters of extreme value of dynamic response are calculated by Moment method based on first-passage probabilities of level crossing. Furthermore, the theoretical model proposed is validated by experimental study using Monte Carlo method. The effects of parameters including impact velocity, pretension force and radius on structural reliability are discussed in addition. The model proposed herein provides some theoretical basis for the stochastic vibration control and dynamic design of orthotropic membrane structure based on reliability theory.

First-passage probability of the deflection of a cable-stayed bridge under long-term site-specific traffic loading

Long-span bridges suffer from higher traffic loads and the simultaneous presence of multiple vehicles, which in conjunction with the steady traffic growth may pose a threat to the bridge safety. This study presents a methodology for first-passage probability evaluation of long-span bridges subject to stochastic heavy traffic loading. Initially, the stochastic heavy traffic loading was simulated based on long-term weigh-in-motion measurements of a highway bridge in China. A computational framework was presented integrating Rice’s level-crossing theory and the first-passage criterion. The effectiveness of the computational framework was demonstrated through a case study of a cable-stayed bridge. Numerical results show that the upper tail fitting of the up-crossing rate is an appropriate description of probability characteristics of the extreme traffic load effects of long-span bridges. The average daily truck traffic growth increases the probability of exceedance due to an intensive heavy traffic flow and results in a higher first-passage probability, but this increased trend is weakening as the continuous increase of the traffic volume. Since the sustained growth of gross vehicle weight has a constant impact on the probability of failure, setting a reasonable threshold overload ratio is an effective scheme as a traffic management to ensure the bridge serviceability.

Analytical Solutions for an Escape Problem in a Disc with an Arbitrary Distribution of Exit Holes Along Its Boundary

We analytically construct solutions for the mean first-passage time and splitting probabilities for the escape problem of a particle moving with continuous Brownian motion in a confining planar disc with an arbitrary distribution (i.e., of any number, size and spacing) of exit holes/absorbing sections along its boundary. The governing equations for these quantities are Poisson’s equation with a (non-zero) constant forcing term and Laplace’s equation, respectively, and both are subject to a mixture of homogeneous Neumann and Dirichlet boundary conditions. Our solutions are expressed as explicit closed formulae written in terms of a parameterising variable via a conformal map, using special transcendental functions that are defined in terms of an associated Schottky group. They are derived by exploiting recent results for a related problem of fluid mechanics that describes a unidirectional flow over “no-slip/no-shear” surfaces, as well as results from potential theory, all of which were themselves derived using the same theory of Schottky groups. They are exact up to the determination of a finite set of mapping parameters, which is performed numerically. Their evaluation also requires the numerical inversion of the parameterising conformal map. Computations for a series of illustrative examples are also presented.

An efficient estimation of probability of first-passage in a linear system

Reliability analysis of a structure under random vibratory loads involves estimation of the probability of the response exceeding a limit. The classical, brute force approach to such analysis is the Monte Carlo method. However, due to its slow convergence rate, it is often impractical for large-scale engineering structures. In many engineering applications, such as offshore platforms under wave loads, the excitation is represented by Power Spectral Density (PSD) functions. Random time histories of the excitation are generated using a linear combination of sinusoids that are consistent with the PSD of input load. This paper proposes a method that reduces the computational cost of MCs of a linear system with a separable performance function; that is, a function that can be decomposed into parts and calculated independently. The method generates sinusoidal functions of the excitation, finds the system response to each sinusoid, and stores the responses in a database. Then it samples with replacement the sinusoids of the response from the database, finds the system response to the superposition of these sinusoids and checks for failure. This procedure yields a very large number of values of the failure indicator function even from a database with a modest number of sinusoids because it uses sampling with replacement. The efficiency of the proposed approach is demonstrated by estimating the probability of the first excursion in a ten-bar truss model. In this example, the method predicts the probability of failure using less than 0.2 % of the calculated values of the failure indicator function than the standard MCs.

Equivalent linearization method using Gaussian mixture (GM-ELM) for nonlinear random vibration analysis

A new equivalent linearization method is developed for nonlinear random vibration analysis. The method employs a Gaussian mixture distribution model to approximate the probabilistic distribution of a nonlinear system response. The parameters of the Gaussian mixture model are estimated by an optimization algorithm which requires a few rounds of dynamic analysis of the nonlinear system. Due to properties of the Gaussian mixture distribution model, the proposed Gaussian mixture based equivalent linearization method (GM-ELM) can decompose the non-Gaussian response of a nonlinear system into multiple Gaussian responses of linear single-degree-of-freedom oscillators. Using a probabilistic combination technique, the linear system of GM-ELM can provide the response probability distribution equal to the Gaussian mixture estimation of the nonlinear response distribution. Using the linear system of GM-ELM in conjunction with linear random vibration theories, response statistics such as the mean up-crossing rate and first-passage probability of the nonlinear system can be conveniently computed. In order to facilitate applications of GM-ELM in earthquake engineering practice, a response spectrum formula is also proposed to compute the mean peak response of the nonlinear system by using the elastic response spectra representing the peak responses of the linear single-degree-of-freedom oscillators. Finally, two numerical examples are presented to illustrate and test GM-ELM. The analysis results obtained from GM-ELM are compared with those obtained from the conventional ELM and Monte-Carlo simulation. The supporting source code and data are available for download at https://github.com/ziqidwang/GitHub-GM-ELM-code.git.

Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements

In this paper, an approximate semi-analytical approach is developed for determining the first-passage probability of randomly excited linear and lightly nonlinear oscillators endowed with fractional derivative elements. The amplitude of the system response is modeled as one-dimensional Markovian process by employing a combination of the stochastic averaging and the statistical linearization techniques. This leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. Next, an approximate solution of this equation is sought by resorting to a Galerkin scheme. Specifically, a convenient set of confluent hypergeometric functions, related to the corresponding linear oscillator with integer-order derivatives, is used as orthogonal basis for this scheme. Applications to the standard viscous linear and to nonlinear (Van der Pol and Duffing) oscillators are presented. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed approximate analytical solution.

A very efficient approach to compute the first-passage probability density function in a time-changed Brownian model: Applications in finance

Abstract We propose a numerical method to compute the first-passage probability density function in a time-changed Brownian model. In particular, we derive an integral representation of such a density function in which the integrand functions must be obtained solving a system of Volterra equations of the first kind. In addition, we develop an ad-hoc numerical procedure to regularize and solve this system of integral equations. The proposed method is tested on three application problems of interest in mathematical finance, namely the calculation of the survival probability of an indebted firm, the pricing of a single-knock-out put option and the pricing of a double-knock-out put option. The results obtained reveal that the novel approach is extremely accurate and fast, and performs significantly better than the finite difference method.

Real-time reliability assessment based on acceleration monitoring for bridge

Bridge health monitoring (BHM) has become increasingly significant in the life-cycle of the structure such as maintenance, repair and rehabilitation. It is necessary to use BHM information efficiently to assess the working conditions of the bridge. The main objective of this study is to develop an effective method and establish a framework for the real-time reliability assessment based on BHM acceleration information. The first-passage probability and its further development have been proposed to assess the reliability probability. The first-passage probability shows the probability of that a scalar process exceeds a designated threshold during a given time interval. The advantage of the proposed method is the assessment of the real-time reliability probability based on the monitoring information during an assessment reference period. Furthermore, the velocity data and displacement data are calculated from the acceleration monitoring data using the relationships between their power spectral density (PSD) functions. The real-time reliability assessment of Donghai Bridge, which is the first large scale cross-sea bridge in China, demonstrates that the proposed method is efficient and effective.

First-passage problem for nonlinear systems under Lévy white noise through path integral method

In this paper, the first-passage problem for nonlinear systems driven by \(\alpha \)-stable Levy white noises is considered. The path integral solution (PIS) is adopted for determining the reliability function and first-passage time probability density function of nonlinear oscillators. Specifically, based on the properties of \(\alpha \)-stable random variables and processes, PIS is extended to deal with Levy white noises with any value of the stability index \(\alpha \). Application to linear and nonlinear systems considering different values of \(\alpha \) is reported. Comparisons with pertinent Monte Carlo simulation data demonstrate the accuracy of the results.

Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Lévy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Lévy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Lévy processes, which includes Brownian motion with drift, Kou's double-exponential model, and hyperexponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Lévy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient.

Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Lévy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Lévy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Lévy processes, which includes Brownian motion with drift, Kou's double-exponential model, and hyperexponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Lévy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient.