Hybrid PDE solver for data-driven problems and modern branching

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European Journal of Applied Mathematics (EJAM

Parallelization of multidimensional hyperbolic partial differential equation on détente instantanée contrôlée dehydration process

The purpose of this research is to propose some new modified mathematical models to enhance the previous model in simulating, visualizing and predicting the heat and mass transfer in dehydration process using instant controlled pressure drop (DIC) technique. The main contribution of this research is the mathematical models which are formulated from the regression model (Haddad et al., 2007) to multidimensional hyperbolic partial differential equation (HPDE) involving dependent parameters; moisture content, temperature, and pressure, and independent parameters; time and dimension of region. The HPDE model is performed in multidimensional; one, two and three dimensions using finite difference method with central difference formula is used to discretize the mathematical models. The implementation of numerical methods such as Alternating Group Explicit with Brian (AGEB) and Douglas-Rachford (AGED) variances, Red Black Gauss Seidel (RBGS) and Jacobi (JB) method to solve the system of linear equation is another contribution of this research. The sequential algorithm is developed by using Matlab R2011a software. The numerical results are analyzed based on execution time, number of iterations, maximum error, root mean square error, and computational complexity. The grid generation process involved a fine grained large sparse data by minimizing the size of interval, increasing the dimension of the model and level of time steps. Another contribution is the implementation of the parallel algorithm to increase the speedup of computation and to reduce computational complexity problem. The parallelization of the mathematical model is run on Matlab Distributed Computing Server with Linux operating system. The parallel performance evaluation of multidimensional simulation in terms of execution time, speedup, efficiency, effectiveness, temporal performance, granularity, computational complexity and communication cost are analyzed for the performance of parallel algorithm. As a conclusion, the thesis proved that the multidimensional HPDE is able to be parallelized and PAGEB method is the alternative solution for the large sparse simulation. Based on the numerical results and parallel performance evaluations, the parallel algorithm is able to reduce the execution time and computational complexity compared to the sequential algorithm

On fractional probabilistic mean value theorems, fractional counting processes and related results

2015 - 2016The thesis collects the outcomes of the author’s research carried out in the research group Probability Theory and Mathematical Statistics at the Department of Mathematics, University of Salerno, during the doctoral programme “Mathematics, Physics and Applications”. The results are at the interface between Fractional Calculus and Probability Theory. While research in probability and applied fields is now well established and enthusiastically supported, the subject of fractional calculus, i.e. the study of an extension of derivatives and integrals to any arbitrary real or complex order, has achieved widespread popularity only during the past four decades or so, because of its applications in several fields of science, engineering and finance. Moreover, the application of the fractional paradigm to probability theory has been carefully but partially explored over the years, especially from the point of view of stochastic processes. The aim of the thesis is to prove some new theorems at the interface between Mathematical Analysis and Probability Theory, and to study rigorously certain new stochastic processes and statistical models constructed on top of some well-known classical results and then generalized by means of fractional calculus. The dissertation is organized as follows. In Chapter 1 we give an overview about the main ideas that inspire fractional calculus and about the mathematical techniques for dealing with fractional operators and the related special functions and probability distributions. In order to develop certain fractional probabilistic analogues of Taylor’s theorem 1 and mean value theorem, in Chapter 2 we introduce the nth-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the nth-order fractional equilibrium distribution is identical to the starting distribution if and only if it is exponential. The nth-order fractional equilibrium density is then used to prove a fractional probabilistic Taylor’s theorem based on derivatives of Riemann-Liouville type. A fractional analogue of the probabilistic mean value theorem is thus developed for pairs of nonnegative random variables ordered according to the survival bounded stochastic order. We also provide some related results, both involving the normalized moments and a fractional extension of the variance, and a formula of interest to actuarial science. In conclusion, we discuss the probabilistic Taylor’s theorem based on fractional Caputo derivatives. In Chapter 3 we consider a fractional counting process with jumps of integer amplitude 1,2,...,k, whose probabilities satisfy a suitable system of fractional differencedifferential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first-passage time of the fractional counting process in an integral form. We then show that the ratios given by the powers of the fractional Poisson process and of the counting process over their means tend to 1 in probability. In Chapter 4 we propose a generalization of the alternating Poisson process from the point of view of fractional calculus. We consider the system of differential equations governing the state probabilities of the alternating Poisson process and replace the ordinary derivative with a fractional one (in the Caputo sense). This produces a fractional 2-state point process, whose probability mass is expressed in terms of the (two-parameter) Mittag-Leffler function. We then show that it can be recovered also by means of renewal theory arguments. We study the limit state probability, and certain proportions involving the fractional moments of the sub-renewal periods of the process. In order to derive new Mittag-Leffler-like distributions related to the considered process, we then exploit a transformation acting on pairs of stochastically ordered random variables, which is an extension of the equilibrium operator and deserves interest in the analysis of alternating stochastic processes. In Chapter 5 we analyse a jump-telegraph process by replacing the classical exponential distribution of the interarrival times which separate consecutive velocity changes (and jumps) with a generalized Mittag-Leffler distribution. Such interarrival times constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based arguments, we obtain the forward and backward transition 2 densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. We conclude the chapter by investigating the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds. Chapter 6 is dedicated to a stochastic model for competing risks involving the MittagLeffler distribution, inspired by fractional random growth phenomena. We prove the independence between the time to failure and the cause of failure, and investigate some properties of the related hazard rates and ageing notions. We also face the general problem of identifying the underlying distribution of latent failure times when their joint distribution is expressed in terms of copulas and the time transformed exponential model. The special case concerning the Mittag-Leffler distribution is approached by means of numerical treatment. We finally adapt the proposed model to the case of a random number of independent competing risks. This leads to certain mixtures of Mittag-Leffler distributions, whose parameters are estimated through the method of moments for fractional moments. [edited by author]La tesi raccoglie i risultati dell’attivit`a di ricerca condotta dall’autore nel gruppo di ricerca Calcolo delle Probabilit`a e Statistica Matematica, presso il Dipartimento di Matematica dell’Universita` di Salerno, nell’ambito del Corso di Dottorato in “Matematica, Fisica e Applicazioni”, XXIX ciclo. I risultati si collocano all’interfaccia tra Calcolo delle Probabilit`a e Calcolo Frazionario. Mentre la ricerca in probabilit`a `e oggi ben consolidata e supportata, il calcolo frazionario, cio`e lo studio della possibilita` di generalizzare il calcolo integrale e il calcolo differenziale classici ad un ordine arbitrario, reale o complesso, ha acquisito notevole popolarita` e importanza nel corso degli ultimi quattro decenni, soprattutto in virtu` delle sue applicazioni in numerosi campi delle scienze e dell’ingegneria. Inoltre, le intersezioni tra calcolo delle probabilita` e calcolo frazionario sono state esplorate con attenzione, ma parzialmente, nel corso degli anni, soprattutto dal punto di vista dei processi stocastici. Lo scopo della tesi `e quello di dimostrare alcuni nuovi teoremi che si collocano all’interfaccia tra l’Analisi Matematica e il Calcolo delle Probabilita`, e di studiare con rigore certi nuovi processi stocastici e modelli statistici costruiti a partire da risultati classici ben noti e poi modificati mediante le tecniche del calcolo frazionario. La tesi `e strutturata come segue. Nel primo capitolo si richiamano alcune nozioni di base e le proprieta` dei principali operatori e delle funzioni del calcolo frazionario, l’integrale di Riemann-Liouville, le derivate di Riemann-Liouville e di Caputo, la funzione di Mittag-Leffler. 1 Nel capitolo 2, al fine di ricavare alcuni analoghi probabilistici di tipo frazionario dei teoremi di Taylor e di Lagrange, `e stata introdotta la distribuzione di equilibrio frazionaria di ordine n definita in termini dell’integrale di Weyl e ne sono state indagate le propriet`a principali. In particolare, si dimostra che la distribuzione di equilibrio frazionaria di ordine n costruita a partire da un’assegnata distribuzione di probabilita`, coincide con questa se e solo se essa `e esponenziale. La distribuzione introdotta viene utilizzata per dimostrare una versione frazionaria dei teoremi di Taylor e del valore medio probabilistici, quest’ultimo applicabile a coppie di variabili aleatorie opportunamente ordinate. Inoltre, si forniscono sia risultati che coinvolgono i momenti normalizzati e un’estensione frazionaria della varianza, sia una formula di interesse nelle scienze attuariali. In conclusione, si discute il teorema di Taylor probabilistico basato sulla derivata frazionaria nel senso di Caputo. Nel terzo capitolo `e stata considerata una generalizzazione frazionaria del processo di Poisson con salti di ampiezza arbitraria, esprimendo la legge di probabilita` mediante funzioni di tipo Mittag- Leffler. L’evoluzione del processo `e guidata da equazioni differenziali e alle differenze finite frazionarie. Dopo aver studiato due rappresentazioni equivalenti del processo considerato, particolare attenzione `e stata posta al problema del tempo di primo passaggio, alla determinazione dei tempi di attesa ed a problemi di tipo asintotico. Tra le altre cose, si `e mostrato che il tempo di prima occorrenza di un salto di ampiezza i, i ∈{1,2,...,k}, k ∈N, `e distribuito come il tempo di prima occorrenza di un evento di un processo di Poisson frazionario di parametro λi > 0, generalizzando, quindi, una importante proprieta` valida nel caso classico. Nel quarto capitolo si propone una generalizzazione del processo di Poisson alternante dal punto di vista del calcolo frazionario, ottenuta sostituendo nel sistema di equazioni differenziali che governa la funzione di probabilita` del processo di Poisson alternante la derivata ordinaria con la derivata frazionaria (nel senso di Caputo) o, equivalentemente, mediante argomenti di teoria del rinnovo. La massa di probabilit`a del nuovo processo `e espressa in termini della funzione di Mittag-Leffler con due parametri. Abbiamo studiato il comportamento asintotico delle probabilit`a di stato e alcune proporzioni che coinvolgono i momenti frazionari dei periodi di rinnovo del processo. Infine, sono state ricavate nuove distribuzioni di tipo Mittag-Leffler relative al processo considerato sfruttando una trasformazione agente su coppie di variabili casuali ordinate stocasticamente, che estende l’operatore equilibrio, di interesse per l’analisi di processi stocastici alternanti. Nel Capitolo 5 si studia un processo stocastico unidimensionale che descrive un moto aleatorio caratterizzato dall’alternarsi di due diverse velocit`a in direzioni opposte. Il processo che regola i cambi di velocit`a (e di direzione) `e il processo di Poisson alternante di tipo frazionario studiato nel capitolo 4. In particolare, nell’istante in cui si verifica un evento di tale processo si compie un salto di ampiezza non aleatoria e quindi il cambiamento di direzione. Pertanto, il processo in esame `e una generalizzazione del processo del telegrafo integrato con salti. Le densit`a di transizione in avanti e all’indietro del moto sono espresse come serie uniformemente convergenti 2 di funzioni di Mittag-Leffler. Particolare attenzione `e stata dedicata al caso di salti di ampiezza costante e uguale distribuzione dei tempi di rinnovo. La distribuzione del tempo di primo passaggio attraverso una barriera costante `e espressa in modo implicito. Tuttavia, in alcuni casi `e data la forma esplicita. L’analisi viene eseguita anche mediante un approccio computazionale. Partendo da fenomeni di crescita di tipo frazionario, nel capitolo 6 abbiamo costruito un modello statistico a rischi competitivi che coinvolge la distribuzione di MittagLeffler. Abbiamo dimostrato l’indipendenza tra il tempo e la causa del fallimento, ed abbiamo indagato alcune propriet`a dei tassi di rischio e delle nozioni di invecchiamento relativi. Abbiamo trattato il problema dell’individuazione della distribuzione sottostante dei tempi di guasto latenti quando la loro distribuzione congiunta `e espressa in termini di copule e mediante il modello TTE (Time Transformed Exponential). Il caso particolare riguardante la distribuzione Mittag-Leffler `e stato trattato numericamente. Il modello proposto `e stato adattato al caso di un numero casuale di rischi in competizione indipendenti. Questo porta ad alcune misture di distribuzioni di tipo Mittag-Leffler, i cui parametri sono stati stimati mediante il metodo dei momenti per momenti frazionari. [a cura dell'autore]XV n.s

Stirring and mixing : 1999 Program of Summer Study in Geophysical Fluid Dynamics

The central theme of the 1999 GFD Program was the stirring, transport, reaction and mixing of passive and active tracers in turbulent, stratified, rotating fluids. The problem of mixing in fluids has applications in areas ranging from oceanography to engineering and astrophysics. In geophysical settings, mixing spans and unites a broad range of scales -- from micrometers to megameters. The mixing of passive tracers is of fundamental importance in environmental and industrial problems, such as pollution, and in determining the large-scale heat and salt balance of the worlds oceans. The transport of active tracers, on the other hand, such as vorticity, plays a key role in the turbulence that occurs in most geophysical and astrophysical fluids. William R. Young (Scripps Institution of Oceanography) gave a series of principal lectures, the notes of which as taken by the fellows, appear in this volume. Report of the projects of the student fellows makes up the second half of this volume.Funding was provided by the National Science Foundation under Grant No. OCE-9810647 and the Office of Naval Research under Grant No. NOO0l4-97-1-0934

Modeling and analysis of single-molecule experiments

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemistry, 2005.Vita.Includes bibliographical references (p. 281-311).Single molecule experiments offer a unique window into the molecular world. This window allows us to distinguish the behaviors of individual molecules from the behavior of bulk by observing rare events and heterogeneity in the dynamics. This thesis discusses both models for single molecule experiments, including the stretching of DNA in hydrodynamic flows and the diffusion of tracer particles in heterogeneous environments, and methods to analyze single molecule data to allow determination of properties and models for single molecule experiments. These methods of analysis are based on combining information theory and Bayesian methods with physical insight and are applied to several experimental situations.by James B. Witkoskie.Ph.D

Monetary Misconceptions

The paper identifies a number of misconceptions about the monetary policy process and the monetary transmission mechanism in the UK. Among the misconceptions about the process are the alleged lack of regional and sectoral representativeness of the Monetary Policy Committee and the view that operational central bank independence means that monetary and fiscal policy are not properly coordinated. Among the transmission mechanism misconceptions, the "New Paradigm" figures prominently. Among the New Paradigm changes in the British economy that have been given prominence are the following: increasing openness; lower global inflation; lower profit margins, reflecting stronger competitive pressures; buoyant stock markets; a lower natural rate of unemployment; and a higher trend rate of growth of productivity. I argue that the New Paradigm has been over-hyped and misunderstood as regards its implications for monetary policy. Other misconceptions include the 'death of inflation', the 'end of boom and bust', a couple of Neanderthal Keynesian fallacies and the monetary fine tuning fallacy.Monetary policy, inflation targeting, New Paradigm, stabilisation policy

Theory of optical rectification in a travelling wave structure

This thesis is concerned with the interaction of an optical wave with a microwave in a waveguiding structure coupled by a second order nonlinearity. Emphasis is laid upon the generation of ultrashort electrical transients via optical rectification (OR) as well as cascading effects due to the interplay of OR and the linear electro-optic effect. A simple transmission line model is introduced to explain qualitatively the basic physical mechanisms of an externally induced polarisation in a travelling wave structure. For a quantitative description, evolution equations for the overall interaction between the microwave and the optical wave based on a coupled mode formalism are developed. The basic properties of the structure under consideration are discussion and techniques for their evaluation are introduced. A set of corresponding parameters for typical structures is estimated and used for calculations throughout the thesis. The generation of electrical signals from optical waves via OR is discussed in detail for the cases of single and mixed polarization optical modes in the structure. The self phase modulation due to cascading of OR and the electro-optic effect is elucidated. It is shown that continuous wave solutions of the conservative system are modulationally unstable in a large range of relevant system parameters. The possibility of formation of solitary waves due to the mutual interaction of optical wave and microwave is considered in the context of long wave short wave interaction. Basic properties of bright stationary solutions and their excitation are discussed. The possibility of formation of solitons due to microwave self-interaction is illuminated. The linear stability of bright solitary waves is investigated. The observed oscillations and radiation of perturbed propagated bound states are explained by the existence of discrete, quasi-bond internal modes of the stationary solutions. Collision scenarios are addressed

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