Bayesian Markov-chain-Monte-Carlo inversion of time-lapse cross hole ground-penetrating radar data to characterize the vadose zone at the Arrenaes field site, Denmark

The ground-penetrating radar (GPR) geophysical method has the potential to provide valuable information on the hydraulic properties of the vadose zone because of its strong sensitivity to soil water content. In particular, recent evidence has suggested that the stochastic inversion of crosshole GPR traveltime data can allow for a significant reduction in uncertainty regarding subsurface van Genuchten-Mualem (VGM) parameters. Much of the previous work on the stochastic estimation of VGM parameters from crosshole GPR data has considered the case of steady-state infiltration conditions, which represent only a small fraction of practically relevant scenarios. We explored in detail the dynamic infiltration case, specifically examining to what extent time-lapse crosshole GPR traveltimes, measured during a forced infiltration experiment at the Arreneas field site in Denmark, could help to quantify VGM parameters and their uncertainties in a layered medium, as well as the corresponding soil hydraulic properties. We used a Bayesian Markov-chain-Monte-Carlo inversion approach. We first explored the advantages and limitations of this approach with regard to a realistic synthetic example before applying it to field measurements. In our analysis, we also considered different degrees of prior information. Our findings indicate that the stochastic inversion of the time-lapse GPR data does indeed allow for a substantial refinement in the inferred posterior VGM parameter distributions compared with the corresponding priors, which in turn significantly improves knowledge of soil hydraulic properties. Overall, the results obtained clearly demonstrate the value of the information contained in time-lapse GPR data for characterizing vadose zone dynamics.

Networks of noisy oscillators with correlated degree and frequency dispersion

We investigate how correlations between the diversity of the connectivity of networks and the dynamics at their nodes affect the macroscopic behavior. In particular, we study the synchronization transition of coupled stochastic phase oscillators that represent the node dynamics. Crucially in our work, the variability in the number of connections of the nodes is correlated with the width of the frequency distribution of the oscillators. By numerical simulations on Erdös-Rényi networks, where the frequencies of the oscillators are Gaussian distributed, we make the counterintuitive observation that an increase in the strength of the correlation is accompanied by an increase in the critical coupling strength for the onset of synchronization. We further observe that the critical coupling can solely depend on the average number of connections or even completely lose its dependence on the network connectivity. Only beyond this state, a weighted mean-field approximation breaks down. If noise is present, the correlations have to be stronger to yield similar observations.

MCMC Analysis for Optimization of Stochastic Models

In any decision making under uncertainties, the goal is mostly to minimize the expected cost. The minimization of cost under uncertainties is usually done by optimization. For simple models, the optimization can easily be done using deterministic methods.However, many models practically contain some complex and varying parameters that can not easily be taken into account using usual deterministic methods of optimization. Thus, it is very important to look for other methods that can be used to get insight into such models. MCMC method is one of the practical methods that can be used for optimization of stochastic models under uncertainty. This method is based on simulation that provides a general methodology which can be applied in nonlinear and non-Gaussian state models. MCMC method is very important for practical applications because it is a uni ed estimation procedure which simultaneously estimates both parameters and state variables. MCMC computes the distribution of the state variables and parameters of the given data measurements. MCMC method is faster in terms of computing time when compared to other optimization methods. This thesis discusses the use of Markov chain Monte Carlo (MCMC) methods for optimization of Stochastic models under uncertainties .The thesis begins with a short discussion about Bayesian Inference, MCMC and Stochastic optimization methods. Then an example is given of how MCMC can be applied for maximizing production at a minimum cost in a chemical reaction process. It is observed that this method performs better in optimizing the given cost function with a very high certainty.

Estimation of State Space Models and Stochastic Volatility

Ma thèse est composée de trois chapitres reliés à l'estimation des modèles espace-état et volatilité stochastique. Dans le première article, nous développons une procédure de lissage de l'état, avec efficacité computationnelle, dans un modèle espace-état linéaire et gaussien. Nous montrons comment exploiter la structure particulière des modèles espace-état pour tirer les états latents efficacement. Nous analysons l'efficacité computationnelle des méthodes basées sur le filtre de Kalman, l'algorithme facteur de Cholesky et notre nouvelle méthode utilisant le compte d'opérations et d'expériences de calcul. Nous montrons que pour de nombreux cas importants, notre méthode est plus efficace. Les gains sont particulièrement grands pour les cas où la dimension des variables observées est grande ou dans les cas où il faut faire des tirages répétés des états pour les mêmes valeurs de paramètres. Comme application, on considère un modèle multivarié de Poisson avec le temps des intensités variables, lequel est utilisé pour analyser le compte de données des transactions sur les marchés financières. Dans le deuxième chapitre, nous proposons une nouvelle technique pour analyser des modèles multivariés à volatilité stochastique. La méthode proposée est basée sur le tirage efficace de la volatilité de son densité conditionnelle sachant les paramètres et les données. Notre méthodologie s'applique aux modèles avec plusieurs types de dépendance dans la coupe transversale. Nous pouvons modeler des matrices de corrélation conditionnelles variant dans le temps en incorporant des facteurs dans l'équation de rendements, où les facteurs sont des processus de volatilité stochastique indépendants. Nous pouvons incorporer des copules pour permettre la dépendance conditionnelle des rendements sachant la volatilité, permettant avoir différent lois marginaux de Student avec des degrés de liberté spécifiques pour capturer l'hétérogénéité des rendements. On tire la volatilité comme un bloc dans la dimension du temps et un à la fois dans la dimension de la coupe transversale. Nous appliquons la méthode introduite par McCausland (2012) pour obtenir une bonne approximation de la distribution conditionnelle à posteriori de la volatilité d'un rendement sachant les volatilités d'autres rendements, les paramètres et les corrélations dynamiques. Le modèle est évalué en utilisant des données réelles pour dix taux de change. Nous rapportons des résultats pour des modèles univariés de volatilité stochastique et deux modèles multivariés. Dans le troisième chapitre, nous évaluons l'information contribuée par des variations de volatilite réalisée à l'évaluation et prévision de la volatilité quand des prix sont mesurés avec et sans erreur. Nous utilisons de modèles de volatilité stochastique. Nous considérons le point de vue d'un investisseur pour qui la volatilité est une variable latent inconnu et la volatilité réalisée est une quantité d'échantillon qui contient des informations sur lui. Nous employons des méthodes bayésiennes de Monte Carlo par chaîne de Markov pour estimer les modèles, qui permettent la formulation, non seulement des densités a posteriori de la volatilité, mais aussi les densités prédictives de la volatilité future. Nous comparons les prévisions de volatilité et les taux de succès des prévisions qui emploient et n'emploient pas l'information contenue dans la volatilité réalisée. Cette approche se distingue de celles existantes dans la littérature empirique en ce sens que ces dernières se limitent le plus souvent à documenter la capacité de la volatilité réalisée à se prévoir à elle-même. Nous présentons des applications empiriques en utilisant les rendements journaliers des indices et de taux de change. Les différents modèles concurrents sont appliqués à la seconde moitié de 2008, une période marquante dans la récente crise financière.

Essays on numerically efficient inference in nonlinear and non-Gaussian state space models, and commodity market analysis.

The first two articles build procedures to simulate vector of univariate states and estimate parameters in nonlinear and non Gaussian state space models. We propose state space speci fications that offer more flexibility in modeling dynamic relationship with latent variables. Our procedures are extension of the HESSIAN method of McCausland[2012]. Thus, they use approximation of the posterior density of the vector of states that allow to : simulate directly from the state vector posterior distribution, to simulate the states vector in one bloc and jointly with the vector of parameters, and to not allow data augmentation. These properties allow to build posterior simulators with very high relative numerical efficiency. Generic, they open a new path in nonlinear and non Gaussian state space analysis with limited contribution of the modeler. The third article is an essay in commodity market analysis. Private firms coexist with farmers' cooperatives in commodity markets in subsaharan african countries. The private firms have the biggest market share while some theoretical models predict they disappearance once confronted to farmers cooperatives. Elsewhere, some empirical studies and observations link cooperative incidence in a region with interpersonal trust, and thus to farmers trust toward cooperatives. We propose a model that sustain these empirical facts. A model where the cooperative reputation is a leading factor determining the market equilibrium of a price competition between a cooperative and a private firm

Studies on gaussian effective potentials in some models

In classical field theory, the ordinary potential V is an energy density for that state in which the field assumes the value ¢. In quantum field theory, the effective potential is the expectation value of the energy density for which the expectation value of the field is ¢o. As a result, if V has several local minima, it is only the absolute minimum that corresponds to the true ground state of the theory. Perturbation theory remains to this day the main analytical tool in the study of Quantum Field Theory. However, since perturbation theory is unable to uncover the whole rich structure of Quantum Field Theory, it is desirable to have some method which, on one hand, must go beyond both perturbation theory and classical approximation in the points where these fail, and at that time, be sufficiently simple that analytical calculations could be performed in its framework During the last decade a nonperturbative variational method called Gaussian effective potential, has been discussed widely together with several applications. This concept was described as a means of formalizing our intuitive understanding of zero-point fluctuation effects in quantum mechanics in a way that carries over directly to field theory.

Analysis of Stochastic Volatility Sequences Generated by Product Autoregressive Models

The classical methods of analysing time series by Box-Jenkins approach assume that the observed series uctuates around changing levels with constant variance. That is, the time series is assumed to be of homoscedastic nature. However, the nancial time series exhibits the presence of heteroscedasticity in the sense that, it possesses non-constant conditional variance given the past observations. So, the analysis of nancial time series, requires the modelling of such variances, which may depend on some time dependent factors or its own past values. This lead to introduction of several classes of models to study the behaviour of nancial time series. See Taylor (1986), Tsay (2005), Rachev et al. (2007). The class of models, used to describe the evolution of conditional variances is referred to as stochastic volatility modelsThe stochastic models available to analyse the conditional variances, are based on either normal or log-normal distributions. One of the objectives of the present study is to explore the possibility of employing some non-Gaussian distributions to model the volatility sequences and then study the behaviour of the resulting return series. This lead us to work on the related problem of statistical inference, which is the main contribution of the thesis

Organic field-effect transistors and phototransistors based on amorphous materials

A comparison between the charge transport properties in low molecular amorphous thin films of spiro-linked compound and their corresponding parent compound has been demonstrated. The field-effect transistor method is used for extracting physical parameters such as field-effect mobility of charge carriers, ON/OFF ratios, and stability. In addition, phototransistors have been fabricated and demonstrated for the first time by using organic materials. In this case, asymmetrically spiro-linked compounds are used as active materials. The active materials used in this study can be divided into three classes, namely Spiro-linked compounds (symmetrically spiro-linked compounds), the corresponding parent-compounds, and photosensitive spiro-linked compounds (asymmetrically spiro-linked com-pounds). Some of symmetrically spiro-linked compounds used in this study were 2,2',7,7'-Tetrakis-(di-phenylamino)-9,9'-spirobifluorene (Spiro-TAD),2,2',7,7'-Tetrakis-(N,N'-di-p-methylphenylamino)-9,9'-spirobifluorene (Spiro-TTB), 2,2',7,7'-Tetra-(m-tolyl-phenylamino)-9,9'-spirobifluorene (Spiro-TPD), and 2,2Ž,7,7Ž-Tetra-(N-phenyl-1-naphtylamine)-9,9Ž-spirobifluorene (Spiro alpha-NPB). Related parent compounds of the symmetrically spiro-linked compound used in this study were N,N,N',N'-Tetraphenylbenzidine (TAD), N,N,N',N'-Tetrakis(4-methylphenyl)benzidine (TTB), N,N'-Bis(3-methylphenyl)-(1,1'-biphenyl)-4,4'-diamine (TPD), and N,N'-Diphenyl-N,N'-bis(1-naphthyl)-1,1'-biphenyl-4,4'-diamine (alpha-NPB). The photosensitive asymmetrically spiro-linked compounds used in this study were 2,7-bis-(N,N'-diphenylamino)-2',7'-bis(biphenyl-4-yl)-9,9'-spirobifluorene (Spiro-DPSP), and 2,7-bis-(N,N'-diphenylamino)-2',7'-bis(spirobifluorene-2-yl)-9,9'-spirobifluorene (Spiro-DPSP^2). It was found that the field-effect mobilities of charge carriers in thin films of symmetrically spiro-linked compounds and their corresponding parent compounds are in the same order of magnitude (~10^-5 cm^2/Vs). However, the thin films of the parent compounds were easily crystallized after the samples have been exposed in ambient atmosphere and at room temperature for three days. In contrast, the thin films and the transistor characteristics of symmetrically spiro-linked compound did not change significantly after the samples have been stored in ambient atmosphere and at room temperature for several months. Furthermore, temperature dependence of the mobility was analyzed in two models, namely the Arrhenius model and the Gaussian Disorder model. The Arrhenius model tends to give a high value of the prefactor mobility. However, it is difficult to distinguish whether the temperature behaviors of the material under consideration follows the Arrhenius model or the Gaussian Disorder model due to the narrow accessible range of the temperatures. For the first time, phototransistors have been fabricated and demonstrated by using organic materials. In this case, asymmetrically spiro-linked compounds are used as active materials. Intramolecular charge transfer between a bis(diphenylamino)biphenyl unit and a sexiphenyl unit leads to an increase in charge carrier density, providing the amplification effect. The operational responsivity of better than 1 A/W can be obtained for ultraviolet light at 370 nm, making the device interesting for sensor applications. This result offers a new potential application of organic thin film phototransistors as low-light level and low-cost visible blind ultraviolet photodetectors.

A Filter for Visual Tracking Based on a Stochastic Model for Driver Behaviour

A driver controls a car by turning the steering wheel or by pressing on the accelerator or the brake. These actions are modelled by Gaussian processes, leading to a stochastic model for the motion of the car. The stochastic model is the basis of a new filter for tracking and predicting the motion of the car, using measurements obtained by fitting a rigid 3D model to a monocular sequence of video images. Experiments show that the filter easily outperforms traditional filters.

The representation of conditional expectations of non-Gaussian noise

For Wiener spaces conditional expectations and $L^{2}$-martingales w.r.t. the natural filtration have a natural representation in terms of chaos expansion. In this note an extension to larger classes of processes is discussed. In particular, it is pointed out that orthogonality of the chaos expansion is not required.

Stress field control of eruption dynamics at a rift volcano: Nyamuragira, D.R.Congo

Using the record of 30 flank eruptions over the last 110 years at Nyamuragira, we have tested the relationship between the eruption dynamics and the local stress field. There are two groups of eruptions based on their duration (< 80days >) that are also clustered in space and time. We find that the eruptions fed by dykes parallel to the East African Rift Valley have longer durations (and larger volumes) than those eruptions fed by dykes with other orientations. This is compatible with a model for compressible magma transported through an elastic-walled dyke in a differential stress field from an over-pressured reservoir (Woods et al., 2006). The observed pattern of eruptive fissures is consistent with a local stress field modified by a northwest-trending, right lateral slip fault that is part of the northern transfer zone of the Kivu Basin rift segment. We have also re-tested with new data the stochastic eruption models for Nyamuragira of Burt et al. (1994). The time-predictable, pressure-threshold model remains the best fit and is consistent with the typically observed declining rate of sulphur dioxide emission during the first few days of eruption with lava emission from a depressurising, closed, crustal reservoir. The 2.4-fold increase in long-term eruption rate that occurred after 1977 is confirmed in the new analysis. Since that change, the record has been dominated by short-duration eruptions fed by dykes perpendicular to the Rift. We suggest that the intrusion of a major dyke during the 1977 volcano-tectonic event at neighbouring Nyiragongo volcano inhibited subsequent dyke formation on the southern flanks of Nyamuragira and this may also have resulted in more dykes reaching the surface elsewhere. Thus that sudden change in output was a result of a changed stress field that forced more of the deep magma supply to the surface. Another volcano-tectonic event in 2002 may also have changed the magma output rate at Nyamuragira.

Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.

Direct Computation of Stochastic Flow in Reservoirs with Uncertain Parameters

A direct method is presented for determining the uncertainty in reservoir pressure, flow, and net present value (NPV) using the time-dependent, one phase, two- or three-dimensional equations of flow through a porous medium. The uncertainty in the solution is modelled as a probability distribution function and is computed from given statistical data for input parameters such as permeability. The method generates an expansion for the mean of the pressure about a deterministic solution to the system equations using a perturbation to the mean of the input parameters. Hierarchical equations that define approximations to the mean solution at each point and to the field covariance of the pressure are developed and solved numerically. The procedure is then used to find the statistics of the flow and the risked value of the field, defined by the NPV, for a given development scenario. This method involves only one (albeit complicated) solution of the equations and contrasts with the more usual Monte-Carlo approach where many such solutions are required. The procedure is applied easily to other physical systems modelled by linear or nonlinear partial differential equations with uncertain data.

On the long time behavior of free stochastic Schrödinger evolutions

We discuss the time evolution of the wave function which is the solution of a stochastic Schrödinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and uniqueness of solutions. We observe that there exist three time regimes: the collapse regime, the classical regime and the diffusive regime. Concerning the latter, we assert that the general solution converges almost surely to a diffusing Gaussian wave function having a finite spread both in position as well as in momentum. This paper corrects and completes earlier works on this issue.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.