Solutions of the First-Passage Problem by Importance Sampling

An efficient simulation method for reliability analysis of linear dynamical systems using simple additive rules of probability

A single-loop reliability sensitivity analysis strategy for time-dependent rare events with both random variables and stochastic processes

Efficient High-Dimensional Importance Sampling in Mixture Frameworks

Importance sampling methods have been developed with the aim of reducing the computational costs inherent in Monte Carlo methods. This study proposes a new algorithm called the adaptive kernel method which combines and modifies some of the concepts from adaptive sampling and the simple kernel method to evaluate the structural reliability of time variant problems. The essence of the resulting algorithm is to select an appropriate starting point from which the importance sampling density can be generated efficiently. Numerical results show that the method is unbiased and substantially increases the efficiency over other methods.

The research presented in this thesis focuses on analytical approximations for the time-variant first-passage failure probability (FPFP) of linear elastic models of structural systems subjected to stochastic excitations. The FPFP is defined as the probability that a response quantity of an engineering system subjected to a dynamic stochastic loading outcrosses a specified threshold within a given exposure time. The FPFP is an important and useful quantity for many structural engineering applications. The classical first-passage reliability problem is studied for linear elastic single-degree-of-freedom (SDOF) and multi-degrees-of-freedom (MDOF) systems subjected to stationary and nonstationary Gaussian excitations. The absolute and relative accuracy of several analytical approximations available in the literature (i.e., the Poisson’s (P), classical Vanmarcke’s (cVM), and modified Vanmarcke’s (mVM) approximations) are studied through an extensive parametric study for SDOF systems. In addition, a new analytical approximation for the FPFP of linear SDOF systems is developed. The new proposed approximation is verified by comparing its analytical estimates of the failure probability with the corresponding results obtained using existing analytical approximations and the importance sampling using elementary events (ISEE) method for a wide range of oscillator properties and different types of input excitations. It is found that the newly developed analytical approximation provides estimates of the time-variant FPFP of SDOF systems that are significantly more accurate than the estimates obtained using the P, cVM, and mVM approximations. Real-valued and complex-valued modal analysis with modal truncation is used to study the time-variant FPFP of MDOF subjected to stationary and nonstationary stochastic excitations. The absolute and relative accuracy of the P, cVM, and mVM approximations as well as of the newly developed approximation are studied for a select number of case studies. It is found that the newly proposed analytical approximation cannot be directly extended to the computation of the FPFP of MDOF systems, while the mVM approximation appears to be more accurate than the other existing analytical approximations.

The classical first-passage reliability problem for linear elastic single-degree-of-freedom (SDOF) oscillators subjected to stationary and nonstationary Gaussian excitations is explored. Several analytical approximations are available in the literature for this problem: the Poisson, classical Vanmarcke, and modified Vanmarcke approximations. These analytical approximations are widely used because of their simplicity and their lower computational cost compared with simulation techniques. However, little is known about their accuracy in estimating the time-variant first-passage failure probability (FPFP) for varying oscillator properties, failure thresholds, and types of loading. In this paper, a new analytical approximation of the FPFP for linear SDOF systems is proposed by modifying the classical Vanmarcke hazard function. This new approximation is verified by comparing its failure probability estimates with the results obtained using existing analytical approximations and the importance sampling ...

In this paper, we present a variance minimization (VM) procedure for rare event simulation in tandem queueing networks. We prove that the VM method can produce a zero variance. The VM method is suitable to compute optimal importance sampling (IS) parameters for small scale tandem networks. For large scale tandem networks we propose a sub-optimal IS (SOIS) method, which projects the optimal biased transition probabilities of the corresponding small scale system into those of a large scale system. In other words, we establish an efficient IS method for a large scale system by zooming into a small scale system and then projecting our findings into the large scale system. The numerical results show that our SOIS method can produce accurate results with very short CPU time, while many other methods often require much longer.

In this paper a torque estimation model for induction machines is studied. The model requires several machine parameters as input. Since the input parameters are deduced from measurements burdened with inaccuracy, it is necessary to take into account uncertainty in the input of the model. A first order and second order reliability method are performed to evaluate the probability of failure of the system under the uncertain inputs. The system failure is defined through the limit state function and occurs when the calculated value of the torque differs significantly from the rated torque at rated conditions. Monte Carlo simulations are performed for validation and importance sampling is discussed to reduce the Monte Carlo simulation error. The probability of failure is also derived from the torque probability density function by means of a stochastic response surface method. The analysis includes current and speed measurement uncertainties as well.

An investigation was performed to study the effect of uncertainty in applied loads, helix angles, and failure stresses on the probability of failure of composite cylinders. Also, the effect of the factor of safety on the probability of failure of the cylinder was studied. The probability of failure for a composite cylinder subject to axisymmetric loading was obtained using the First Order Reliability method (FORM). The failure of the cylinder was studied based on first ply failure. Ply failure was predicted using the non-smooth Hashin criterion. Hashin criterion is based on writing a different interaction equation for separate failure modes. Four failure modes are considered associated with matrix or fiber failure, in either tension or compression. The accuracy of FORM as applied to this problem was verified by comparing the FORM results to those obtained by Monte-Carlo Simulation method with importance sampling. i x µ = mean values of the design parameters i x

Reliability analysis of laminated composite structures using finite elements and neural networks

An important sampling scheme to determine the time-variant reliability of structures that are subjected to Poisson processes is presented in this paper. The suggested approach combines the first order reliability method with the standard importance sampling techniques. It becomes therefore feasible to determine extremely low probabilities of failure with a moderate number of samples. As numerical examples, a fatigue crack growth problem and an oscillator subjected to exponentially distributed random noise, are investigated and the computational efficiency of the proposed method is demonstrated in this paper.

Risk analysis using stochastic reliability methods applied to two cases of deterministic water quality models

The importance of design robustness and reliability is a well-established notion and practice in today’s industry. Manufacturing companies strive to achieve Six-Sigma quality measures. While Virtual Prototyping is a key-factor in accelerating the product development process while reducing development costs; it has not contributed in the quest for improved product reliability and robustness performance since it is based on deterministic approaches. This paper provides a systematic approach to design for sixsigma simultaneously addressing variability and uncertainty present in real life on most design parameters. The first step is the ability to flexibly integrate and federate best-of-class virtual prototyping simulation software programs -most based on Finite Element Modeling techniques- thus, capturing existing engineering processes. These engineering processes provide the deterministic calculation engines for assessing design performance outputs of interest, such as -but not limited to- stress, structural impact, eigenfrequencies, forced response functions, fatigue life. Second is the assessment of design parameter variability on the design performance outputs. Analytical methods for the efficient calculation of dynamic response variability examined in this work include reliability-based methods such as First and Second Order Reliability methods, the Importance Sampling Monte Carlo (ISMC) method and the Transformed Importance Latin Hypercube Sampling (TILHS) method. They have been assessed for their ability to accurately predict the failure probability in a reasonable amount of time. The latter criterion is paramount for successful application to large industrial FE models, where the computation time is the main bottleneck. The accuracy has been examined with the comparison of the aforementioned methods with classical and advanced Monte-Carlo methodologies. Uncertainty is treated with the use of Transformation Method based on Fuzzy arithmetic concepts. Fuzzy arithmetic is an attractive mathematical concept to solve problems with uncertain parameters. The Transformation Method (TM) follows a fuzzy approach based on the concept of α-cuts, i.e. discretization of the parameter’s magnitude into a number of intervals. Uncertainty is then assessed by solving the problem in a systematic way: for each combination of the values attained by the uncertain parameters a deterministic computation of the response is performed. In this paper, the TM is demonstrated on an airplane FE model, to assess the effect of material stiffness uncertainty on the FRF from wing tip to pilot seat. A second demonstration example on this model aims to ensure a certain frequency difference between two free-free modes that may result in flutter at certain air speed; several reliability methods are applied for this purpose, namely Monte Carlo simulations, FORM and Importance Sampling.

The marginal likelihood has become an important tool for model selection in Bayesian analysis because it can be used to rank the models. We discuss the marginal likelihood for Poisson regression models that are potentially useful in small area estimation. Computation in these models is intensive and it requires an implementation of Markov chain Monte Carlo (MCMC) methods. Using importance sampling and multivariate density estimation, we demonstrate a computation of the marginal likelihood through an output analysis from an MCMC sampler.

Modified efficiency importance sampling (MEIS), an importance sampling technique which can be used to yield a significant saving of run time for the assessment of bit error rate (BER) in digital communication systems using computer simulation, is described. The concept of equivalent noise source is introduced to avoid the effect of system memory on the efficiency of importance sampling, and a nonGaussian probability density function (p.d.f.) is selected as the biased equivalent noise to further improve run time. MEIS has been used successfully in the simulation of M-ary quadrature amplitude modulation (MQAM) systems. The MEIS is shown to be effective in the simulation of systems with large memory length, and in mildly nonlinear systems. The experimental results show that MEIS is a great improvement over Monte Carlo (MC) methods and that is even more efficient than the efficient importance sampling (EIS) technique. >