Siu-Kui Au

A New Adaptive Importance Sampling Scheme for Reliability Calculations

An adaptive importance sampling methodology is proposed to compute the multidimensional integrals encountered in reliability analysis. It is based on a Markov simulation algorithm due to Metropolis et al. (Metropolis, Rosenbluth, Rosenbluth and Teller, Equations of state calculatons by fast computing machines. Journal of Chemical Physics, 1953;21(6): 1087-1092). In the proposed methodology, samples are simulated as the states of a Markov chain and are distributed asymptotically according to the optimal importance sampling density. A kernel sampling density is then constructed from these samples which is used as the sampling density in an importance sampling simulation. The Markov chain samples populate the region of higher probability density in the failure region and so the kernel sampling density approximates the optimal importance sampling density for a large variety of shapes of the failure region. This adaptive feature is insensitive to the probability level to be estimated. A variety of numerical examples demonstrates the accuracy, efficiency and robustness of the methodology.

Important sampling in high dimensions

This paper draws attention to a fundamental problem that occurs in applying importance sampling to ‘high-dimensional’ reliability problems, i.e., those with a large number of uncertain parameters. This question of applicability carries an important bearing on the potential use of importance sampling for solving dynamic first-excursion problems and static reliability problems for structures with a large number of uncertain structural model parameters. The conditions under which importance sampling is applicable in high dimensions are investigated, where the focus is put on the common case of standard Gaussian uncertain parameters. It is found that importance sampling densities using design points are applicable if the covariance matrix associated with each design point does not deviate significantly from the identity matrix. The study also suggests that importance sampling densities using random pre-samples are generally not applicable in high dimensions.

Entropy-Based Optimal Sensor Location for Structural Model Updating

A statistical methodology is presented for optimally locating the sensors in a structure for the purpose of extracting from the measured data the most information about the parameters of the model used to represent structural behavior. The methodology can be used in model updating and in damage detection and localization applications. It properly handles the unavoidable uncertainties in the measured data as well as the model uncertainties. The optimality criterion for the sensor locations is based on information entropy, which is a unique measure of the uncertainty in the model parameters. The uncertainty in these parameters is computed by a Bayesian statistical methodology, and then the entropy measure is minimized over the set of possible sensor configurations using a genetic algorithm. The information entropy measure is also extended to handle large uncertainties expected in the pretest nominal model of a structure. In experimental design, the proposed entropy-based measure of uncertainty is also well-suited for making quantitative evaluations and comparisons of the quality of the parameter estimates that can be achieved using sensor configurations with different numbers of sensors in each configuration. Simplified models for a shear building and a truss structure are used to illustrate the methodology.

Fast Bayesian FFT method for ambient modal identification with separated modes

Previously a Bayesian theory for modal identification using the fast Fourier transform (FFT) of ambient data was formulated. That method provides a rigorous way for obtaining modal properties as well as their uncertainties by operating in the frequency domain. This allows a natural partition of information according to frequencies so that well-separated modes can be identified independently. Determining the posterior most probable modal parameters and their covariance matrix, however, requires solving a numerical optimization problem. The dimension of this problem grows with the number of measured channels; and its objective function involves the inverse of an ill-conditioned matrix, which makes the approach impractical for realistic applications. This paper analyzes the mathematical structure of the problem and develops efficient methods for computations, focusing on well-separated modes. A method is developed that allows fast computation of the posterior most probable values and covariance matrix. The analysis reveals a scientific definition of signal-to-noise ratio that governs the behavior of the solution in a characteristic manner. Asymptotic behavior of the modal identification problem is investigated for high signal-to-noise ratios. The proposed method is applied to modal identification of two field buildings. Using the proposed algorithm, Bayesian modal identification can now be performed in a few seconds even for a moderate to large number of measurement channels.

Monitoring Structural Health Using a Probabilistic Measure

A Bayesian probabilistic methodology for structural health monitoring is presented. The method uses a sequence of identified modal parameter data sets to continually compute the probability of damage. In this approach, a high likelihood of a reduction in model stiffness at a location is taken as a proxy for damage at the corresponding structural location. The concept extends the idea of using as indicators of damage the changes in model parameters identified using a linear finite-element model and modal parameter data sets from the structure in undamaged and possibly damaged states. This extension is needed because of uncertainties in the updated model parameters that in practice obscure health assessment. These uncertainties arise due to effects such as variation in the identified modal parameters in the absence of damage, as well as unavoidable model error. The method is illustrated by simulating on-line monitoring, wherein specified modal parameters are identified on a regular basis and the probability of damage for each substructure is continually updated. Examples are given for abrupt onset of damage and progressive deterioration.

Reliability of uncertain dynamical systems with multiple design points

Asymptotic approximations and importance sampling methods are presented for evaluating a class of probability integrals with multiple design points that may arise in the calculation of the reliability of uncertain dynamical systems. An approximation based on asymptotics is used as a first step to provide a computationally efficient estimate of the probability integral. The importance sampling method utilizes information of the integrand at the design points to substantially accelerate the convergence of available importance sampling methods that use information from one design point only. Implementation issues related to the choice of importance sampling density and sample generation for reducing the variance of the estimate are addressed. The computational efficiency and improved accuracy of the proposed methods is demonstrated by investigating the reliability of structures equipped with a tuned mass damper for which multiple design points are shown to contribute significantly to the value of the reliability integral.

Effect of Seismic Risk on Lifetime Property Value

We examine seismic risk from the commercial real estate investors viewpoint. We present a methodology to estimate the uncertain net asset value (NAV) of an investment opportunity considering market risk and seismic risk. For seismic risk, we employ a performance-based earthquake engineering methodology called assembly-based vulnerability (ABV). For market risk, we use evidence of volatility of return on investment in the United States. We find that uncertainty in NAV can be significant compared with investors’ risk tolerance, making it appropriate to adopt a decision-analysis approach to the investment decision, in which one optimizes certainty equivalent, CE, as opposed to NAV. Uncertainty in market value appears greatly to exceed uncertainty in earthquake repair costs. Consequently, CE is sensitive to the mean value of earthquake repair costs but not to its variance. Thus, to a real estate investor, seismic risk matters only in the mean, at least for the demonstration buildings examined here.

A Bayesian probabilistic approach to structural health monitoring

Some general issues associated with online structural health monitoring are discussed. In order to address the problem of determining the existence and location of damage in the presence of uncertainties, a global model-based structural health monitoring method which utilizes Bayesian probabilistic inference is developed. The results of tests using simulated data are described.

Analyzing the sensitivity of WRF's single-layer urban canopy model to parameter uncertainty using advanced Monte Carlo simulation

Single-layer physically based urban canopy models (UCM) have gained popularity for modeling urban‐ atmosphere interactions, especially the energy transport component. For a UCM to capture the physics of conductive, radiative, and turbulent advective transport of energy, it is important to provide it with an accurate parameter space, including both mesoscale meteorological forcing and microscale surface inputs. While field measurement of all input parameters to a UCM is rarely possible, understanding the model sensitivity to individual parameters is essential todeterminethe relative importanceof parameteruncertainty for model performance. In this paper, an advanced Monte Carlo approach—namely, subset simulation—is used to quantifythe impact of the uncertaintyof surface input parameters on the output of an offline modified version of the Weather Research and Forecasting (WRF)-UCM. On the basis of the conditional sampling technique, the importance of surface parameters is determined in terms of their impact on critical model responses. It is found that model outputs (both critical energy fluxes and surface temperatures) are highly sensitive to uncertainties in urban geometry, whereas variations in emissivities and building interior temperatures are relatively insignificant. In addition, the sensitivity of the model to input surface parameters is also shown to be very weakly dependent on meteorological parameters. The statistical quantification of the model’s sensitivity to input parameters has practical implications, such as surface parameter calibrations in UCM and guidance for urban heat island mitigation strategies.

Fast Bayesian Ambient Modal Identification Incorporating Multiple Setups

AbstractIn full-scale ambient vibration tests, many situations exist where it is required to obtain a detailed mode shape of a structure with a limited number of sensors. A common feasible strategy is to perform multiple setups with each one covering a different part of the structure while sharing some reference degrees of freedom (DOF) in common. Methods exist that assemble the mode shapes identified in individual setups to form a global one covering all measured DOF. This paper presents a fast Bayesian method for modal identification capable of incorporating the fast Fourier transform information in different setups consistent with probability logic. The method allows the global mode shape to be determined, taking into account the quality of data in different setups. A fast iterative algorithm is developed that allows practical implementation even for a large number of DOF. The method is illustrated with synthetic and field test data. Challenges of the mode shape assembly problem arising in field applic...