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Concept# Monte Carlo method

Summary

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.
In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases).
Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business

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João Miguel De Oliveira Durães Alves Martins

Safety assessments of road bridges to braking events combine the braking force, acting along the longitudinal axis of the deck, with a vertical load that accounts for the vertical component of the traffic action. In modern design standards the vertical load models result from probabilistic calibration procedures targeting predefined return periods. On the contrary, the braking force was derived from a deterministic characterization of the vehicle configurations and of the braking process. Therefore, the return period of the braking force is unclear and may not be consistent with that of the vertical load model. Significant deviations from the target return period might lead to either uneconomical decisions, e.g. uncalled-for retrofitting interventions, or to inaccurate structural safety verifications. This thesis presents an original stochastic model to compute site-specific values of the braking force as a function of the return period. The developed stochastic model takes into account the length of the bridge deck and its dynamic properties for vibrations in the longitudinal direction, as well as different sources of randomness related to braking events, all of which comply with real-world measurements, including: - vehicle configurations, resorting to a time-history of crossing vehicles; - driver response times, randomly generated from probability distributions defined in the scope of this project; - deceleration profiles of the vehicles, resampled from catalogues of realistic deceleration profiles. The stochastic model uses Monte Carlo simulation of braking events and computes the maximum of the dynamic response of the bridge to each event. The computed maxima are collected in an empirical distribution function of the braking force. In the end, the model returns the quantile of this distribution that is suitable for safety assessments. This value of braking force is specific to the bridge given properties, to the traffic characteristics, and to the target return period. An additional novelty of this research work is the estimation of a rate of occurrence on motorways of braking events per vehicle-distance travelled. This parameter enables the estimation of the period of time covered by the simulations of braking events as a function of traffic flow and of the total number of braking events simulated. This step is fundamental to determine the value of the braking force that has a given return period. The braking forces returned by the stochastic model show significant dependence on the bridge length, the natural vibration period of the deck in the longitudinal direction, and the number of directions of traffic on the deck. On the contrary, damping ratio, traffic on the fast-lane or on weekends, and an augmentation of traffic in 20% show no substantial influence on the braking force. Moreover, the two motorway locations considered as sources of traffic data, Denges and Monte Ceneri, both in Switzerland, yielded braking forces with similar magnitudes, despite the significant differences in traffic characteristics. Finally, the results compiled served to calibrate an updated braking force that depends explicitly on the parameters found relevant, as well as on the return period so that it can be adopted by different standards even if they enforce different safety targets. This updated expression evidences that the braking forces of current codes tend to be conservative and, hence, can be improved based on the findings of this project.

xtreme value analysis is concerned with the modelling of extreme events such as floods and heatwaves, which can have large impacts. Statistical modelling can be useful to better assess risks even if, due to scarcity of measurements, there is inherently very large residual uncertainty in any analysis. Driven by the increase in environmental databases, spatial modelling of extremes has expanded rapidly in the last decade. This thesis presents contributions to such analysis.
The first chapter is about likelihood-based inference in the univariate setting and investigates the use of bias-correction and higher-order asymptotic methods for extremes, highlighting through examples and illustrations the unique challenge posed by data scarcity. We focus on parametric modelling of extreme values, which relies on limiting distributional results and for which, as a result, uncertainty quantification is complicated. We find that, in certain cases, small-sample asymptotic methods can give improved inference by reducing the error rate of confidence intervals. Two data illustrations, linked to assessment of the frequency of extreme rainfall episodes in Venezuela and the analysis of survival of supercentenarians, illustrate the methods developed.
In the second chapter, we review the major methods for the analysis of spatial extremes models. We highlight the similarities and provide a thorough literature review along with novel simulation algorithms. The methods described therein are made available through a statistical software package.
The last chapter focuses on estimation for a Bayesian hierarchical model derived from a multivariate generalized Pareto process. We review approaches for the estimation of censored components in models derived from (log)-elliptical distributions, paying particular attention to the estimation of a high-dimensional Gaussian distribution function via Monte Carlo methods. The impacts of model misspecification and of censoring are explored through extensive simulations and we conclude with a case study of rainfall extremes in Eastern Switzerland.