Probabilistic approaches in civil engineering: generation of random fields and structural identification with genetic algorithms

The inherent stochastic character of most of the physical quantities involved in engineering models has led to an always increasing interest for probabilistic analysis. Many approaches to stochastic analysis have been proposed. However, it is widely acknowledged that the only universal method available to solve accurately any kind of stochastic mechanics problem is Monte Carlo Simulation. One of the key parts in the implementation of this technique is the accurate and efficient generation of samples of the random processes and fields involved in the problem at hand. In the present thesis an original method for the simulation of homogeneous, multi-dimensional, multi-variate, non-Gaussian random fields is proposed. The algorithm has proved to be very accurate in matching both the target spectrum and the marginal probability. The computational efficiency and robustness are very good too, even when dealing with strongly non-Gaussian distributions. What is more, the resulting samples posses all the relevant, welldefined and desired properties of “translation fields”, including crossing rates and distributions of extremes. The topic of the second part of the thesis lies in the field of non-destructive parametric structural identification. Its objective is to evaluate the mechanical characteristics of constituent bars in existing truss structures, using static loads and strain measurements. In the cases of missing data and of damages that interest only a small portion of the bar, Genetic Algorithm have proved to be an effective tool to solve the problem.

EoR in alternative cosmological scenarios

The aim of this Thesis is to investigate the possibility that the observations related to the epoch of reionization can probe not only the evolution of the IGM state, but also the cosmological background in which this process occurs. In fact, the history of the IGM ionization is indeed affected by the evolution of the sources of ionizing photons that, under the assumption of a structure formation paradigm determined by the hierarchic growth of the matter uctuations, results strongly dependent on the characteristics of the background universe. For the purpose of our investigation, we have analysed the reionization history in innovative cosmological frameworks, still in agreement with the recent observational tests related to the SNIa and the CMB probes, comparing our results with the reionization scenario predicted by the commonly used LCDM cosmology. In particular, in this Thesis we have considered two different alternative universes. The first one is a at universe dominated at late epochs by a dynamic dark energy component, characterized by an equation of state evolving in time. The second cosmological framework we have assumed is a LCDM characterized by a primordial overdensity field having a non-Gaussian probability distribution. The reionization scenario have been investigated, in this Thesis, through semi-analytic approaches based on the hierarichic growth of the matter uctuations and on suitable assumptions concerning the ionization and the recombination of the IGM. We make predictions for the evolution and the distribution of the HII regions, and for the global features of reionization, that can be constrained by future observations. Finally, we brie y discuss the possible future prospects of this Thesis work.

L'efficienza degli indici azionari: un'analisi empirica

La studio dell’efficienza di un indice azionario ha accresciuto la propria importanza nell’industria dell’asset management a seguito della diffusione dell’utilizzo di benchmark e investimenti indicizzati. Il presente lavoro valuta il livello di efficienza dei principali indici del mercato azionario statunitense, dell’Area Euro e italiano. Lo studio empirico ricorre a quattro misure di efficienza: il GRS, un test small-sample multivariato fondato sul CAPM; il test large sample di Wald, implementato tramite una simulazione bootstrap; il test GMM, che è stato applicato in una cornice non-gaussiana attraverso una simulazione block bootstrap; la misura di efficienza relativa di Kandel e Stambaugh. I risultati empirici forniscono una prova evidente della superiore efficienza degli indici equiponderati. Questa conclusione è interpretata sulla base della letteratura scientifica esistente, analizzando le diverse cause di ordine teorico ed empirico che sono state proposte.

Rainfall spatial predictions: a two-part model and its assessment

Spatial prediction of hourly rainfall via radar calibration is addressed. The change of support problem (COSP), arising when the spatial supports of different data sources do not coincide, is faced in a non-Gaussian setting; in fact, hourly rainfall in Emilia-Romagna region, in Italy, is characterized by abundance of zero values and right-skeweness of the distribution of positive amounts. Rain gauge direct measurements on sparsely distributed locations and hourly cumulated radar grids are provided by the ARPA-SIMC Emilia-Romagna. We propose a three-stage Bayesian hierarchical model for radar calibration, exploiting rain gauges as reference measure. Rain probability and amounts are modeled via linear relationships with radar in the log scale; spatial correlated Gaussian effects capture the residual information. We employ a probit link for rainfall probability and Gamma distribution for rainfall positive amounts; the two steps are joined via a two-part semicontinuous model. Three model specifications differently addressing COSP are presented; in particular, a stochastic weighting of all radar pixels, driven by a latent Gaussian process defined on the grid, is employed. Estimation is performed via MCMC procedures implemented in C, linked to R software. Communication and evaluation of probabilistic, point and interval predictions is investigated. A non-randomized PIT histogram is proposed for correctly assessing calibration and coverage of two-part semicontinuous models. Predictions obtained with the different model specifications are evaluated via graphical tools (Reliability Plot, Sharpness Histogram, PIT Histogram, Brier Score Plot and Quantile Decomposition Plot), proper scoring rules (Brier Score, Continuous Rank Probability Score) and consistent scoring functions (Root Mean Square Error and Mean Absolute Error addressing the predictive mean and median, respectively). Calibration is reached and the inclusion of neighbouring information slightly improves predictions. All specifications outperform a benchmark model with incorrelated effects, confirming the relevance of spatial correlation for modeling rainfall probability and accumulation.

Advanced Mission Synthesis Algorithms for Vibration-based Accelerated Life-testing

In the field of vibration qualification testing, with the popular Random Control mode of shakers, the specimen is excited by random vibrations typically set in the form of a Power Spectral Density (PSD). The corresponding signals are stationary and Gaussian, i.e. featuring a normal distribution. Conversely, real-life excitations are frequently non-Gaussian, exhibiting high peaks and/or burst signals and/or deterministic harmonic components. The so-called kurtosis is a parameter often used to statistically describe the occurrence and significance of high peak values in a random process. Since the similarity between test input profiles and real-life excitations is fundamental for qualification test reliability, some methods of kurtosis-control can be implemented to synthesize realistic (non-Gaussian) input signals. Durability tests are performed to check the resistance of a component to vibration-based fatigue damage. A procedure to synthesize test excitations which starts from measured data and preserves both the damage potential and the characteristics of the reference signals is desirable. The Fatigue Damage Spectrum (FDS) is generally used to quantify the fatigue damage potential associated with the excitation. The signal synthesized for accelerated durability tests (i.e. with a limited duration) must feature the same FDS as the reference vibration computed for the component’s expected lifetime. Current standard procedures are efficient in synthesizing signals in the form of a PSD, but prove inaccurate if reference data are non-Gaussian. This work presents novel algorithms for the synthesis of accelerated durability test profiles with prescribed FDS and a non-Gaussian distribution. An experimental campaign is conducted to validate the algorithms, by testing their accuracy, robustness, and practical effectiveness. Moreover, an original procedure is proposed for the estimation of the fatigue damage potential, aiming to minimize the computational time. The research is thus supposed to improve both the effectiveness and the efficiency of excitation profile synthesis for accelerated durability tests.

Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.