First Passage Time Densities in Resonate-and-Fire Models

Motivated by the dynamics of resonant neurons we discuss the properties of the first passage time (FPT) densities for nonmarkovian differentiable random processes. We start from an exact expression for the FPT density in terms of an infinite series of integrals over joint densities of level crossings, and consider different approximations based on truncation or on approximate summation of this series. Thus, the first few terms of the series give good approximations for the FPT density on short times. For rapidly decaying correlations the decoupling approximations perform well in the whole time domain. As an example we consider resonate-and-fire neurons representing stochastic underdamped or moderately damped harmonic oscillators driven by white Gaussian or by Ornstein-Uhlenbeck noise. We show, that approximations reproduce all qualitatively different structures of the FPT densities: from monomodal to multimodal densities with decaying peaks. The approximations work for the systems of whatever dimension and are especially effective for the processes with narrow spectral density, exactly when markovian approximations fail.Comment: 11 pages, 8 figure

A Method for the Combination of Stochastic Time Varying Load Effects

The problem of evaluating the probability that a structure becomes unsafe under a combination of loads, over a given time period, is addressed. The loads and load effects are modeled as either pulse (static problem) processes with random occurrence time, intensity and a specified shape or intermittent continuous (dynamic problem) processes which are zero mean Gaussian processes superimposed 'on a pulse process. The load coincidence method is extended to problems with both nonlinear limit states and dynamic responses, including the case of correlated dynamic responses. The technique of linearization of a nonlinear limit state commonly used in a time-invariant problem is investigated for timevarying combination problems, with emphasis on selecting the linearization point. Results are compared with other methods, namely the method based on upcrossing rate, simpler combination rules such as Square Root of Sum of Squares and Turkstra's rule. Correlated effects among dynamic loads are examined to see how results differ from correlated static loads and to demonstrate which types of load dependencies are most important, i.e., affect' the exceedance probabilities the most. Application of the load coincidence method to code development is briefly discussed.National Science Foundation Grants CME 79-18053 and CEE 82-0759

Reliability Analysis of Systems Subject to First-Passage Failure

An obvious goal of reliability analysis is the avoidance of system failure. However, it is generally recognized that it is often not feasible to design a practical or useful system for which failure is impossible. Thus it is necessary to use techniques that estimate the likelihood of failure based on modeling the uncertainty about such items as the demands on and capacities of various elements in the system. This usually involves the use of probability theory, and a design is considered acceptable if it has a sufficiently small probability of failure. This report contains findings of analyses of systems subject to first-passage failure

Probabilistic engineering analysis and design under time-dependent uncertainty

Time-dependent uncertainties, such as time-variant stochastic loadings and random deterioration of material properties, are inherent in engineering applications. Not considering these uncertainties in the design process may result in catastrophic failures after the designed products are put into operation. Although significant progress has been made in probabilistic engineering design, quantifying and mitigating the effects of time-dependent uncertainty is still challenging. This dissertation aims to help build high reliability into products under time-dependent uncertainty by addressing two research issues. The first one is to efficiently and accurately predict the time-dependent reliability while the second one is to effectively design the time-dependent reliability into the product. For the first research issue, new time-dependent reliability analysis methodologies are developed, including the joint upcrossing rate method, the surrogate model method, the global efficient optimization, and the random field approach. For the second research issue, a time-dependent sequential optimization and reliability analysis method is proposed. The developed approaches are applied to the reliability analysis of designing a hydrokinetic turbine blade subjected to stochastic river flow loading. Extension of the proposed methods to the reliability analysis with mixture of random and interval variables is also a contribution of this dissertation. The engineering examples tested in in this work demonstrate that the proposed time-dependent reliability methods can improve the efficiency and accuracy more than 100% and that high reliability can be successfully built into products with the proposed method. The research results can benefit a wide range of areas, such as life cycle cost optimization and decision making --Abstract, page iv

Counting of level crossings for inertial random processes: Generalization of the Rice formula

We address the counting of level crossings for inertial stochastic processes. We review Rice's approach to the problem and generalize the classical Rice formula to include all Gaussian processes in their most general form. We apply the results to some second-order (i.e., inertial) processes of physical interest, such as Brownian motion, random acceleration and noisy harmonic oscillators. For all models we obtain the exact crossing intensities and discuss their long- and short-time dependence. We illustrate these results with numerical simulations.Comment: 29 pages, 16 figures. Minor changes in title and some figure caption

Excursion set approach to the clustering of dark matter haloes in Lagrangian space

We present a stochastic approach to the spatial clustering of dark matter haloes in Lagrangian space. Our formalism is based on a local formulation of the `excursion set' approach by Bond et al., which automatically accounts for the `cloud-in-cloud' problem in the identification of bound systems. Our method allows to calculate correlation functions of haloes in Lagrangian space using either a multi-dimensional Fokker-Planck equation with suitable boundary conditions or an array of Langevin equations with spatially correlated random forces. We compare the results of our method with theoretical predictions for the halo auto-correlation function considered in the literature and find good agreement with the results recently obtained within a treatment of halo clustering in terms of `counting fields' by Catelan et al.. The possible effect of spatial correlations on numerical simulations of halo merger trees is finally discussed.Comment: LaTeX, 19 pages, 3 figures. Submitted to MNRA

A Lagrangian Dynamical Theory for the Mass Function of Cosmic Structures: II Statistics

The statistical tools needed to obtain a mass function from realistic collapse time estimates are presented. Collapse dynamics has been dealt with in paper I of this series by means of the powerful Lagrangian perturbation theory and the simple ellipsoidal collapse model. The basic quantity considered here is the inverse collapse time F; it is a non-linear functional of the initial potential, with a non-Gaussian distribution. In the case of sharp k-space smoothing, it is demonstrated that the fraction of collapsed mass can be determined by extending to the F process the diffusion formalism introduced by Bond et al. (1991). The problem is then reduced to a random walk with a moving absorbing barrier, and numerically solved; an accurate analytical fit, valid for small and moderate resolutions, is found. For Gaussian smoothing, the F trajectories are strongly correlated in resolution. In this case, an approximation proposed by Peacock & Heavens (1990) can be used to determine the mass functions. Gaussian smoothing is preferred, as it optimizes the performances of dynamical predictions and stabilizes the F trajectories. The relation between resolution and mass is treated at a heuristic level, and the consequences of this approximation are discussed. The resulting mass functions, compared to the classical Press & Schechter (1974) one, are shifted toward large masses (confirming the findings of Monaco 1995), and tend to give more intermediate-mass objects at the expense of small-mass objects. However, the small-mass part of the mass function, which depends on uncertain dynamics and is likely to be affected by uncertainties in the resolution--mass relation, is not considered a robust prediction of this theory.Comment: 18 pages, Latex, uses mn.sty and psfig, 11 postscript figures . Revised version; important changes in the demonstration of the Markov nature of the F process. MNRAS, in pres

Extremes and First Passage Times of Correlated Fractional Brownian Motions

Let {X-i (t), t >= 0}, i = 1, 2 be two standard fractional Brownian motions being jointly Gaussian with constant cross-correlation. In this paper, we derive the exact asymptotics of the joint survival function P {sups(is an element of)[(0,1]) X-1(s) > u, sup(t is an element of)[(0,1]) X-2(t) > u} as u ->infinity. A novel finding of this contribution is the exponential approximation of the joint conditional first passage times of X-1, X-2. As a by-product, we obtain generalizations of the Borell-TIS inequality and the Piterbarg inequality for 2-dimensional Gaussian random fields. Keywords Borell-TIS inequality; Extremes; First passage times; Fractional Brownian motion; Gaussian random fields; Piterbarg inequality