Simulation of non-Gaussian/non-stationary stochastic processes: beyond second-order orthogonality

The theory of stochastic processes and their generations are indispensable to characterize wind fluctuations, ocean waves, and earthquake excitations among other quantities in engineering. To computationally analyze and simulate these stochastic systems, practical realization of samples of stochastic processes is essential. The object of this thesis is to introduce new state-of-the-art methodologies for the generation of stochastic processes with non-Gaussianity/non-stationarity possessing higher-order properties than the second-order orthogonality. A new type of Iterative Translation Approximation Method (ITAM) using the Karhunen-Loève expansion was developed for simulating non-Gaussian and non-stationary processes utilizing translation process theory. The proposed methodology enhances the accuracy of simulated processes in matching a prescribed autocorrelation, maintains the computational efficiency, and resolves limitations caused by utilizing evolutionary power spectra for non-stationary processes. A new generalized stochastic expansion, the bispectral representation method (BSRM), expanded from the traditional spectral representation method is introduced to simulate skewed nonlinear stochastic processes. With new orthogonal increments to satisfy the conditions of the Cramér spectral representation up to third order orthogonality, the BSRM generates samples that match both the power spectrum and bispectrum of the process by modeling complex nonlinear wave interactions. A model of phase angle distributions to characterize phase coupling in higher-order stochastic processes is presented. Relationships between the trigonometric moments of circular distributions of phase differences and higher-order cumulant spectra are derived. The prescribed properties are shown to accurately model quadratic and cubic phase couplings in simple stochastic processes and can easily be extended to general n-wave couplings. Lastly, as applications of the prescribed methods, wind pressure and turbulent wind velocity time histories are generated with SRM, ITAM, and BSRM and applied to two different nonlinear dynamic structural systems. For structures having material and geometrical nonlinearities, performance of an elastic perfectly-plastic structure and the buffeting response of a long-span bridge with coupled aerodynamic forces are examined. The structures are investigated to observe the effect of higher-order properties of the excitations on the response when compared to conventional second-order Gaussian and non-Gaussian excitations

Modeling and Simulation of Random Processes and Fields in Civil Engineering and Engineering Mechanics

This thesis covers several topics within computational modeling and simulation of problems arising in Civil Engineering and Applied Mechanics. There are two distinct parts. Part 1 covers work in modeling and analyzing heterogeneous materials using the eXtended Finite Element Method (XFEM) with arbitrarily shaped inclusions. A novel enrichment function, which can model arbitrarily shaped inclusions within the framework of XFEM, is proposed. The internal boundary of an arbitrarily shaped inclusion is first discretized, and a numerical enrichment function is constructed "on the fly" using spline interpolation. This thesis considers a piecewise cubic spline which is constructed from seven localized discrete boundary points. The enrichment function is then determined by solving numerically a nonlinear equation which determines the distance from any point to the spline curve. Parametric convergence studies are carried out to show the accuracy of this approach, compared to pointwise and linear segmentation of points, for the construction of the enrichment function in the case of simple inclusions and arbitrarily shaped inclusions in linear elasticity. Moreover, the viability of this approach is illustrated on a Neo-Hookean hyperelastic material with a hole undergoing large deformation. In this case, the enrichment is able to adapt to the deformation and effectively capture the correct response without remeshing. Part 2 then moves on to research work in simulation of random processes and fields. Novel algorithms for simulating random processes and fields such as earthquakes, wind fields, and properties of functionally graded materials are discussed. Specifically, a methodology is presented to determine the Evolutionary Spectrum (ES) for non-stationary processes from a prescribed or measured non-stationary Auto-Correlation Function (ACF). Previously, the existence of such an inversion was unknown, let alone possible to compute or estimate. The classic integral expression suggested by Priestley, providing the ACF from the ES, is not invertible in a unique way so that the ES could be determined from a given ACF. However, the benefits of an efficient inversion from ACF to ES are vast. Consider for example various problems involving simulation of non-stationary processes or non-homogeneous fields, including non-stationary seismic ground motions as well as non-homogeneous material properties such as those of functionally graded materials. In such cases, it is sometimes more convenient to estimate the ACF from measured data, rather than the ES. However, efficient simulation depends on knowing the ES. Even more important, simulation of non-Gaussian and non-stationary processes depends on this inversion, when following a spectral representation based approach. This work first examines the existence and uniqueness of such an inversion from the ACF to the ES under a set of special conditions and assumptions (since such an inversion is clearly not unique in the most general form). It then moves on to efficient methodologies of computing the inverse, including some established optimization techniques, as well as proposing a novel methodology. Its application within the framework of translation models for simulation of non-Gaussian, non-stationary processes is developed and discussed. Numerical examples are provided demonstrating the capabilities of the methodology. Additionally in Part 2, a methodology is presented for efficient and accurate simulation of wind velocities along long span structures at a virtually infinite number of points. Currently, the standard approach is to model wind velocities as a multivariate stochastic process, characterized by a Cross-Spectral Density Matrix (CSDM). In other words, the wind velocities are modeled as discrete components of a vector process. To simulate sample functions of the vector process, the Spectral Representation Method (SRM) is used. The SRM involves a Cholesky decomposition of the CSDM. However, it is a well known issue that as the length of the structure, and consequently the size of the vector process, increases, this Cholesky decomposition breaks down (from the numerical point of view). To avoid this issue, current research efforts in the literature center around approximate techniques to simplify the decomposition. Alternatively, this thesis proposes the use of the frequency-wavenumber (F-K) spectrum to model the wind velocities as a stochastic "wave," continuous in both space and time. This allows the wind velocities to be modeled at a virtually infinite number of points along the length of the structure. In this work, the relationship between the CSDM and the F-K spectrum is first examined, as well as simulation techniques for both. The F-K spectrum for wind velocities is then derived. Numerical examples are then carried out demonstrating that the simulated wave samples exhibit the desired spectral and coherence characteristics. The efficiency of this method, specifically through the use of the Fast Fourier Transform, is demonstrated

MCMC inference for Markov Jump Processes via the Linear Noise Approximation

Bayesian analysis for Markov jump processes is a non-trivial and challenging problem. Although exact inference is theoretically possible, it is computationally demanding thus its applicability is limited to a small class of problems. In this paper we describe the application of Riemann manifold MCMC methods using an approximation to the likelihood of the Markov jump process which is valid when the system modelled is near its thermodynamic limit. The proposed approach is both statistically and computationally efficient while the convergence rate and mixing of the chains allows for fast MCMC inference. The methodology is evaluated using numerical simulations on two problems from chemical kinetics and one from systems biology

A brief history of long memory: Hurst, Mandelbrot and the road to ARFIMA

Long memory plays an important role in many fields by determining the behaviour and predictability of systems; for instance, climate, hydrology, finance, networks and DNA sequencing. In particular, it is important to test if a process is exhibiting long memory since that impacts the accuracy and confidence with which one may predict future events on the basis of a small amount of historical data. A major force in the development and study of long memory was the late Benoit B. Mandelbrot. Here we discuss the original motivation of the development of long memory and Mandelbrot's influence on this fascinating field. We will also elucidate the sometimes contrasting approaches to long memory in different scientific communitiesComment: 40 page

Limits and dynamics of stochastic neuronal networks with random heterogeneous delays

Realistic networks display heterogeneous transmission delays. We analyze here the limits of large stochastic multi-populations networks with stochastic coupling and random interconnection delays. We show that depending on the nature of the delays distributions, a quenched or averaged propagation of chaos takes place in these networks, and that the network equations converge towards a delayed McKean-Vlasov equation with distributed delays. Our approach is mostly fitted to neuroscience applications. We instantiate in particular a classical neuronal model, the Wilson and Cowan system, and show that the obtained limit equations have Gaussian solutions whose mean and standard deviation satisfy a closed set of coupled delay differential equations in which the distribution of delays and the noise levels appear as parameters. This allows to uncover precisely the effects of noise, delays and coupling on the dynamics of such heterogeneous networks, in particular their role in the emergence of synchronized oscillations. We show in several examples that not only the averaged delay, but also the dispersion, govern the dynamics of such networks.Comment: Corrected misprint (useless stopping time) in proof of Lemma 1 and clarified a regularity hypothesis (remark 1

Understanding how kurtosis is transferred from input acceleration to stress response and it's influence on fatigue life

High cycle fatigue of metals typically occurs through long term exposure to time varying loads which, although modest in amplitude, give rise to microscopic cracks that can ultimately propagate to failure. The fatigue life of a component is primarily dependent on the stress amplitude response at critical failure locations. For most vibration tests, it is common to assume a Gaussian distribution of both the input acceleration and stress response. In real life, however, it is common to experience non-Gaussian acceleration input, and this can cause the response to be non-Gaussian. Examples of non-Gaussian loads include road irregularities such as potholes in the automotive world or turbulent boundary layer pressure fluctuations for the aerospace sector or more generally wind, wave or high amplitude acoustic loads. The paper first reviews some of the methods used to generate non-Gaussian excitation signals with a given power spectral density and kurtosis. The kurtosis of the response is examined once the signal is passed through a linear time invariant system. Finally an algorithm is presented that determines the output kurtosis based upon the input kurtosis, the input power spectral density and the frequency response function of the system. The algorithm is validated using numerical simulations. Direct applications of these results include improved fatigue life estimations and a method to accelerate shaker tests by generating high kurtosis, non-Gaussian drive signals

On the Nature and Shape of Tubulin Trails: Implications on Microtubule Self-Organization

Microtubules, major elements of the cell skeleton are, most of the time, well organized in vivo, but they can also show self-organizing behaviors in time and/or space in purified solutions in vitro. Theoretical studies and models based on the concepts of collective dynamics in complex systems, reaction-diffusion processes and emergent phenomena were proposed to explain some of these behaviors. In the particular case of microtubule spatial self-organization, it has been advanced that microtubules could behave like ants, self-organizing by 'talking to each other' by way of hypothetic (because never observed) concentrated chemical trails of tubulin that are expected to be released by their disassembling ends. Deterministic models based on this idea yielded indeed like-looking spatio-temporal self-organizing behaviors. Nevertheless the question remains of whether microscopic tubulin trails produced by individual or bundles of several microtubules are intense enough to allow microtubule self-organization at a macroscopic level. In the present work, by simulating the diffusion of tubulin in microtubule solutions at the microscopic scale, we measure the shape and intensity of tubulin trails and discuss about the assumption of microtubule self-organization due to the production of chemical trails by disassembling microtubules. We show that the tubulin trails produced by individual microtubules or small microtubule arrays are very weak and not elongated even at very high reactive rates. Although the variations of concentration due to such trails are not significant compared to natural fluctuations of the concentration of tubuline in the chemical environment, the study shows that heterogeneities of biochemical composition can form due to microtubule disassembly. They could become significant when produced by numerous microtubule ends located in the same place. Their possible formation could play a role in certain conditions of reaction. In particular, it gives a mesoscopic basis to explain the collective dynamics observed in excitable microtubule solutions showing the propagation of concentration waves of microtubules at the millimeter scale, although we doubt that individual microtubules or bundles can behave like molecular ants

Simulation of non-Gaussian wind field as a 3rd3^{rd}-order stochastic wave

This paper presents a methodology for the simulation of non-Gaussian wind field as a stochastic wave using the 3rd-order Spectral Representation Method. Traditionally, the wind field is modeled as a stochastic vector process at discrete locations in space. But the simulation of vector process is well-known to be computationally challenging and numerically unstable when modeling wind at a large number of discrete points in space. Recently, stochastic waves have been used to model the field as a continuous process indexed both in time and space. We extend the classical Spectral Representation Method for simulation of Gaussian stochastic waves to a third-order representation modeling asymmetrically skewed non-Gaussian stochastic waves from a prescribed power spectrum and bispectrum. We present an efficient implementation using the fast Fourier transform, which reduces the computational time dramatically. We then apply the method for simulation of a non-Gaussian wind velocity field along a long-span bridge.Comment: 22 pages, 4 figure, 1 tabl