Combining Polynomial Chaos Expansions and the Random Variable Transformation Technique to Approximate the Density Function of Stochastic Problems, Including Some Epidemiological Models

[EN] In this paper, we deal with computational uncertainty quantification for stochastic models with one random input parameter. The goal of the paper is twofold: First, to approximate the set of probability density functions of the solution stochastic process, and second, to show the capability of our theoretical findings to deal with some important epidemiological models. The approximations are constructed in terms of a polynomial evaluated at the random input parameter, by means of generalized polynomial chaos expansions and the stochastic Galerkin projection technique. The probability density function of the aforementioned univariate polynomial is computed via the random variable transformation method, by taking into account the domains where the polynomial is strictly monotone. The algebraic/exponential convergence of the Galerkin projections gives rapid convergence of these density functions. The examples are based on fundamental epidemiological models formulated via linear and nonlinear differential and difference equations, where one of the input parameters is assumed to be a random variable.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Chen-Charpentier, BM.; Cortés, J.; Jornet-Sanz, M. (2019). Combining Polynomial Chaos Expansions and the Random Variable Transformation Technique to Approximate the Density Function of Stochastic Problems, Including Some Epidemiological Models. Symmetry (Basel). 11(1):1-28. https://doi.org/10.3390/sym11010043S128111Strand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Bharucha-Reid, A. T. (1964). 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Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function. Mathematical Methods in the Applied Sciences, 42(17), 5649-5667. doi:10.1002/mma.5333Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function. Applied Mathematics and Computation, 331, 33-45. doi:10.1016/j.amc.2018.02.051Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation, 32, 199-210. doi:10.1016/j.cnsns.2015.08.009Kegan, B., & West, R. W. (2005). Modeling the simple epidemic with deterministic differential equations and random initial conditions. Mathematical Biosciences, 194(2), 217-231. doi:10.1016/j.mbs.2005.02.002Crestaux, T., Le Maıˆtre, O., & Martinez, J.-M. (2009). Polynomial chaos expansion for sensitivity analysis. Reliability Engineering & System Safety, 94(7), 1161-1172. doi:10.1016/j.ress.2008.10.008Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety, 93(7), 964-979. doi:10.1016/j.ress.2007.04.002Chen-Charpentier, B. M., Cortés, J.-C., Romero, J.-V., & Roselló, M.-D. (2013). Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation. Applied Mathematics and Computation, 219(9), 4208-4218. doi:10.1016/j.amc.2012.11.007Ernst, O. G., Mugler, A., Starkloff, H.-J., & Ullmann, E. (2011). On the convergence of generalized polynomial chaos expansions. 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Computational methods for random differential equations: probability density function and estimation of the parameters

[EN] Mathematical models based on deterministic differential equations do not take into account the inherent uncertainty of the physical phenomenon (in a wide sense) under study. In addition, inaccuracies in the collected data often arise due to errors in the measurements. It thus becomes necessary to treat the input parameters of the model as random quantities, in the form of random variables or stochastic processes. This gives rise to the study of random ordinary and partial differential equations. The computation of the probability density function of the stochastic solution is important for uncertainty quantification of the model output. Although such computation is a difficult objective in general, certain stochastic expansions for the model coefficients allow faithful representations for the stochastic solution, which permits approximating its density function. In this regard, Karhunen-Loève and generalized polynomial chaos expansions become powerful tools for the density approximation. Also, methods based on discretizations from finite difference numerical schemes permit approximating the stochastic solution, therefore its probability density function. The main part of this dissertation aims at approximating the probability density function of important mathematical models with uncertainties in their formulation. Specifically, in this thesis we study, in the stochastic sense, the following models that arise in different scientific areas: in Physics, the model for the damped pendulum; in Biology and Epidemiology, the models for logistic growth and Bertalanffy, as well as epidemiological models; and in Thermodynamics, the heat partial differential equation. We rely on Karhunen-Loève and generalized polynomial chaos expansions and on finite difference schemes for the density approximation of the solution. These techniques are only applicable when we have a forward model in which the input parameters have certain probability distributions already set. When the model coefficients are estimated from collected data, we have an inverse problem. The Bayesian inference approach allows estimating the probability distribution of the model parameters from their prior probability distribution and the likelihood of the data. Uncertainty quantification for the model output is then carried out using the posterior predictive distribution. In this regard, the last part of the thesis shows the estimation of the distributions of the model parameters from experimental data on bacteria growth. To do so, a hybrid method that combines Bayesian parameter estimation and generalized polynomial chaos expansions is used.[ES] Los modelos matemáticos basados en ecuaciones diferenciales deterministas no tienen en cuenta la incertidumbre inherente del fenómeno físico (en un sentido amplio) bajo estudio. Además, a menudo se producen inexactitudes en los datos recopilados debido a errores en las mediciones. Por lo tanto, es necesario tratar los parámetros de entrada del modelo como cantidades aleatorias, en forma de variables aleatorias o procesos estocásticos. Esto da lugar al estudio de las ecuaciones diferenciales aleatorias. El cálculo de la función de densidad de probabilidad de la solución estocástica es importante en la cuantificación de la incertidumbre de la respuesta del modelo. Aunque dicho cálculo es un objetivo difícil en general, ciertas expansiones estocásticas para los coeficientes del modelo dan lugar a representaciones fieles de la solución estocástica, lo que permite aproximar su función de densidad. En este sentido, las expansiones de Karhunen-Loève y de caos polinomial generalizado constituyen herramientas para dicha aproximación de la densidad. Además, los métodos basados en discretizaciones de esquemas numéricos de diferencias finitas permiten aproximar la solución estocástica, por lo tanto, su función de densidad de probabilidad. La parte principal de esta disertación tiene como objetivo aproximar la función de densidad de probabilidad de modelos matemáticos importantes con incertidumbre en su formulación. Concretamente, en esta memoria se estudian, en un sentido estocástico, los siguientes modelos que aparecen en diferentes áreas científicas: en Física, el modelo del péndulo amortiguado; en Biología y Epidemiología, los modelos de crecimiento logístico y de Bertalanffy, así como modelos de tipo epidemiológico; y en Termodinámica, la ecuación en derivadas parciales del calor. Utilizamos expansiones de Karhunen-Loève y de caos polinomial generalizado y esquemas de diferencias finitas para la aproximación de la densidad de la solución. Estas técnicas solo son aplicables cuando tenemos un modelo directo en el que los parámetros de entrada ya tienen determinadas distribuciones de probabilidad establecidas. Cuando los coeficientes del modelo se estiman a partir de los datos recopilados, tenemos un problema inverso. El enfoque de inferencia Bayesiana permite estimar la distribución de probabilidad de los parámetros del modelo a partir de su distribución de probabilidad previa y la verosimilitud de los datos. La cuantificación de la incertidumbre para la respuesta del modelo se lleva a cabo utilizando la distribución predictiva a posteriori. En este sentido, la última parte de la tesis muestra la estimación de las distribuciones de los parámetros del modelo a partir de datos experimentales sobre el crecimiento de bacterias. Para hacerlo, se utiliza un método híbrido que combina la estimación de parámetros Bayesianos y los desarrollos de caos polinomial generalizado.[CA] Els models matemàtics basats en equacions diferencials deterministes no tenen en compte la incertesa inherent al fenomen físic (en un sentit ampli) sota estudi. A més a més, sovint es produeixen inexactituds en les dades recollides a causa d'errors de mesurament. Es fa així necessari tractar els paràmetres d'entrada del model com a quantitats aleatòries, en forma de variables aleatòries o processos estocàstics. Açò dóna lloc a l'estudi de les equacions diferencials aleatòries. El càlcul de la funció de densitat de probabilitat de la solució estocàstica és important per a quantificar la incertesa de la sortida del model. Tot i que, en general, aquest càlcul és un objectiu difícil d'assolir, certes expansions estocàstiques dels coeficients del model donen lloc a representacions fidels de la solució estocàstica, el que permet aproximar la seua funció de densitat. En aquest sentit, les expansions de Karhunen-Loève i de caos polinomial generalitzat esdevenen eines per a l'esmentada aproximació de la densitat. A més a més, els mètodes basats en discretitzacions mitjançant esquemes numèrics de diferències finites permeten aproximar la solució estocàstica, per tant la seua funció de densitat de probabilitat. La part principal d'aquesta dissertació té com a objectiu aproximar la funció de densitat de probabilitat d'importants models matemàtics amb incerteses en la seua formulació. Concretament, en aquesta memòria s'estudien, en un sentit estocàstic, els següents models que apareixen en diferents àrees científiques: en Física, el model del pèndol amortit; en Biologia i Epidemiologia, els models de creixement logístic i de Bertalanffy, així com models de tipus epidemiològic; i en Termodinàmica, l'equació en derivades parcials de la calor. Per a l'aproximació de la densitat de la solució, ens basem en expansions de Karhunen-Loève i de caos polinomial generalitzat i en esquemes de diferències finites. Aquestes tècniques només són aplicables quan tenim un model cap avant en què els paràmetres d'entrada tenen ja determinades distribucions de probabilitat. Quan els coeficients del model s'estimen a partir de les dades recollides, tenim un problema invers. L'enfocament de la inferència Bayesiana permet estimar la distribució de probabilitat dels paràmetres del model a partir de la seua distribució de probabilitat prèvia i la versemblança de les dades. La quantificació de la incertesa per a la resposta del model es fa mitjançant la distribució predictiva a posteriori. En aquest sentit, l'última part de la tesi mostra l'estimació de les distribucions dels paràmetres del model a partir de dades experimentals sobre el creixement de bacteris. Per a fer-ho, s'utilitza un mètode híbrid que combina l'estimació de paràmetres Bayesiana i els desenvolupaments de caos polinomial generalitzat.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017–89664–P.Calatayud Gregori, J. (2020). Computational methods for random differential equations: probability density function and estimation of the parameters [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/138396TESISPremios Extraordinarios de tesis doctorale

Mean square solutions of random linear models and computation of their probability density function

[EN] This thesis concerns the analysis of differential equations with uncertain input parameters, in the form of random variables or stochastic processes with any type of probability distributions. In modeling, the input coefficients are set from experimental data, which often involve uncertainties from measurement errors. Moreover, the behavior of the physical phenomenon under study does not follow strict deterministic laws. It is thus more realistic to consider mathematical models with randomness in their formulation. The solution, considered in the sample-path or the mean square sense, is a smooth stochastic process, whose uncertainty has to be quantified. Uncertainty quantification is usually performed by computing the main statistics (expectation and variance) and, if possible, the probability density function. In this dissertation, we study random linear models, based on ordinary differential equations with and without delay and on partial differential equations. The linear structure of the models makes it possible to seek for certain probabilistic solutions and even approximate their probability density functions, which is a difficult goal in general. A very important part of the dissertation is devoted to random second-order linear differential equations, where the coefficients of the equation are stochastic processes and the initial conditions are random variables. The study of this class of differential equations in the random setting is mainly motivated because of their important role in Mathematical Physics. We start by solving the randomized Legendre differential equation in the mean square sense, which allows the approximation of the expectation and the variance of the stochastic solution. The methodology is extended to general random second-order linear differential equations with analytic (expressible as random power series) coefficients, by means of the so-called Fröbenius method. A comparative case study is performed with spectral methods based on polynomial chaos expansions. On the other hand, the Fröbenius method together with Monte Carlo simulation are used to approximate the probability density function of the solution. Several variance reduction methods based on quadrature rules and multilevel strategies are proposed to speed up the Monte Carlo procedure. The last part on random second-order linear differential equations is devoted to a random diffusion-reaction Poisson-type problem, where the probability density function is approximated using a finite difference numerical scheme. The thesis also studies random ordinary differential equations with discrete constant delay. We study the linear autonomous case, when the coefficient of the non-delay component and the parameter of the delay term are both random variables while the initial condition is a stochastic process. It is proved that the deterministic solution constructed with the method of steps that involves the delayed exponential function is a probabilistic solution in the Lebesgue sense. Finally, the last chapter is devoted to the linear advection partial differential equation, subject to stochastic velocity field and initial condition. We solve the equation in the mean square sense and provide new expressions for the probability density function of the solution, even in the non-Gaussian velocity case.[ES] Esta tesis trata el análisis de ecuaciones diferenciales con parámetros de entrada aleatorios, en la forma de variables aleatorias o procesos estocásticos con cualquier tipo de distribución de probabilidad. En modelización, los coeficientes de entrada se fijan a partir de datos experimentales, los cuales suelen acarrear incertidumbre por los errores de medición. Además, el comportamiento del fenómeno físico bajo estudio no sigue patrones estrictamente deterministas. Es por tanto más realista trabajar con modelos matemáticos con aleatoriedad en su formulación. La solución, considerada en el sentido de caminos aleatorios o en el sentido de media cuadrática, es un proceso estocástico suave, cuya incertidumbre se tiene que cuantificar. La cuantificación de la incertidumbre es a menudo llevada a cabo calculando los principales estadísticos (esperanza y varianza) y, si es posible, la función de densidad de probabilidad. En este trabajo, estudiamos modelos aleatorios lineales, basados en ecuaciones diferenciales ordinarias con y sin retardo, y en ecuaciones en derivadas parciales. La estructura lineal de los modelos nos permite buscar ciertas soluciones probabilísticas e incluso aproximar su función de densidad de probabilidad, lo cual es un objetivo complicado en general. Una parte muy importante de la disertación se dedica a las ecuaciones diferenciales lineales de segundo orden aleatorias, donde los coeficientes de la ecuación son procesos estocásticos y las condiciones iniciales son variables aleatorias. El estudio de esta clase de ecuaciones diferenciales en el contexto aleatorio está motivado principalmente por su importante papel en la Física Matemática. Empezamos resolviendo la ecuación diferencial de Legendre aleatorizada en el sentido de media cuadrática, lo que permite la aproximación de la esperanza y la varianza de la solución estocástica. La metodología se extiende al caso general de ecuaciones diferenciales lineales de segundo orden aleatorias con coeficientes analíticos (expresables como series de potencias), mediante el conocido método de Fröbenius. Se lleva a cabo un estudio comparativo con métodos espectrales basados en expansiones de caos polinomial. Por otro lado, el método de Fröbenius junto con la simulación de Monte Carlo se utilizan para aproximar la función de densidad de probabilidad de la solución. Para acelerar el procedimiento de Monte Carlo, se proponen varios métodos de reducción de la varianza basados en reglas de cuadratura y estrategias multinivel. La última parte sobre ecuaciones diferenciales lineales de segundo orden aleatorias estudia un problema aleatorio de tipo Poisson de difusión-reacción, en el que la función de densidad de probabilidad es aproximada mediante un esquema numérico de diferencias finitas. En la tesis también se tratan ecuaciones diferenciales ordinarias aleatorias con retardo discreto y constante. Estudiamos el caso lineal y autónomo, cuando el coeficiente de la componente no retardada i el parámetro del término retardado son ambos variables aleatorias mientras que la condición inicial es un proceso estocástico. Se demuestra que la solución determinista construida con el método de los pasos y que involucra la función exponencial retardada es una solución probabilística en el sentido de Lebesgue. Finalmente, el último capítulo lo dedicamos a la ecuación en derivadas parciales lineal de advección, sujeta a velocidad y condición inicial estocásticas. Resolvemos la ecuación en el sentido de media cuadrática y damos nuevas expresiones para la función de densidad de probabilidad de la solución, incluso en el caso de velocidad no Gaussiana.[CA] Aquesta tesi tracta l'anàlisi d'equacions diferencials amb paràmetres d'entrada aleatoris, en la forma de variables aleatòries o processos estocàstics amb qualsevol mena de distribució de probabilitat. En modelització, els coeficients d'entrada són fixats a partir de dades experimentals, les quals solen comportar incertesa pels errors de mesurament. A més a més, el comportament del fenomen físic sota estudi no segueix patrons estrictament deterministes. És per tant més realista treballar amb models matemàtics amb aleatorietat en la seua formulació. La solució, considerada en el sentit de camins aleatoris o en el sentit de mitjana quadràtica, és un procés estocàstic suau, la incertesa del qual s'ha de quantificar. La quantificació de la incertesa és sovint duta a terme calculant els principals estadístics (esperança i variància) i, si es pot, la funció de densitat de probabilitat. En aquest treball, estudiem models aleatoris lineals, basats en equacions diferencials ordinàries amb retard i sense, i en equacions en derivades parcials. L'estructura lineal dels models ens fa possible cercar certes solucions probabilístiques i inclús aproximar la seua funció de densitat de probabilitat, el qual és un objectiu complicat en general. Una part molt important de la dissertació es dedica a les equacions diferencials lineals de segon ordre aleatòries, on els coeficients de l'equació són processos estocàstics i les condicions inicials són variables aleatòries. L'estudi d'aquesta classe d'equacions diferencials en el context aleatori està motivat principalment pel seu important paper en Física Matemàtica. Comencem resolent l'equació diferencial de Legendre aleatoritzada en el sentit de mitjana quadràtica, el que permet l'aproximació de l'esperança i la variància de la solució estocàstica. La metodologia s'estén al cas general d'equacions diferencials lineals de segon ordre aleatòries amb coeficients analítics (expressables com a sèries de potències), per mitjà del conegut mètode de Fröbenius. Es duu a terme un estudi comparatiu amb mètodes espectrals basats en expansions de caos polinomial. Per altra banda, el mètode de Fröbenius juntament amb la simulació de Monte Carlo són emprats per a aproximar la funció de densitat de probabilitat de la solució. Per a accelerar el procediment de Monte Carlo, es proposen diversos mètodes de reducció de la variància basats en regles de quadratura i estratègies multinivell. L'última part sobre equacions diferencials lineals de segon ordre aleatòries estudia un problema aleatori de tipus Poisson de difusió-reacció, en què la funció de densitat de probabilitat és aproximada mitjançant un esquema numèric de diferències finites. En la tesi també es tracten equacions diferencials ordinàries aleatòries amb retard discret i constant. Estudiem el cas lineal i autònom, quan el coeficient del component no retardat i el paràmetre del terme retardat són ambdós variables aleatòries mentre que la condició inicial és un procés estocàstic. Es prova que la solució determinista construïda amb el mètode dels passos i que involucra la funció exponencial retardada és una solució probabilística en el sentit de Lebesgue. Finalment, el darrer capítol el dediquem a l'equació en derivades parcials lineal d'advecció, subjecta a velocitat i condició inicial estocàstiques. Resolem l'equació en el sentit de mitjana quadràtica i donem noves expressions per a la funció de densitat de probabilitat de la solució, inclús en el cas de velocitat no Gaussiana.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017–89664–P. I acknowledge the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.Jornet Sanz, M. (2020). Mean square solutions of random linear models and computation of their probability density function [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/138394TESI

On particle Gibbs Markov chain Monte Carlo models

This article analyses a new class of advanced particle Markov chain Monte Carlo algorithms recently introduced by Andrieu, Doucet, and Holenstein (2010). We present a natural interpretation of these methods in terms of well known unbiasedness properties of Feynman-Kac particle measures, and a new duality with many-body Feynman-Kac models. This perspective sheds a new light on the foundations and the mathematical analysis of this class of methods. A key consequence is the equivalence between the backward and ancestral particle Markov chain Monte Carlo methods, and Gibbs sampling of a many-body Feynman-Kac target distribution. Our approach also presents a new stochastic differential calculus based on geometric combinatorial techniques to derive explicit non-asymptotic Taylor type series of the semigroup of a class of particle Markov chain Monte Carlo models around their invariant measures with respect to the population size of the auxiliary particle sampler. These results provide sharp quan- titative estimates of the convergence properties of conditional particle Markov chain models with respect to the time horizon and the size of the systems. We illustrate the implication of these results with sharp estimates of the contraction coefficient and the Lyapunov exponent of conditional particle samplers, and explicit and non-asymptotic Lp-mean error decompositions of the law of the random states around the limiting invariant measure. The abstract framework developed in the article also allows the design of natural extensions to island (also called SMC2) type particle methodologies

Mathematical Modelling in Engineering & Human Behaviour 2018

This book includes papers in cross-disciplinary applications of mathematical modelling: from medicine to linguistics, social problems, and more. Based on cutting-edge research, each chapter is focused on a different problem of modelling human behaviour or engineering problems at different levels. The reader would find this book to be a useful reference in identifying problems of interest in social, medicine and engineering sciences, and in developing mathematical models that could be used to successfully predict behaviours and obtain practical information for specialised practitioners. This book is a must-read for anyone interested in the new developments of applied mathematics in connection with epidemics, medical modelling, social issues, random differential equations and numerical methods

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

VI Workshop on Computational Data Analysis and Numerical Methods: Book of Abstracts

The VI Workshop on Computational Data Analysis and Numerical Methods (WCDANM) is going to be held on June 27-29, 2019, in the Department of Mathematics of the University of Beira Interior (UBI), Covilhã, Portugal and it is a unique opportunity to disseminate scientific research related to the areas of Mathematics in general, with particular relevance to the areas of Computational Data Analysis and Numerical Methods in theoretical and/or practical field, using new techniques, giving especial emphasis to applications in Medicine, Biology, Biotechnology, Engineering, Industry, Environmental Sciences, Finance, Insurance, Management and Administration. The meeting will provide a forum for discussion and debate of ideas with interest to the scientific community in general. With this meeting new scientific collaborations among colleagues, namely new collaborations in Masters and PhD projects are expected. The event is open to the entire scientific community (with or without communication/poster)

Updating the Lambda modes of a nuclear power reactor

[EN] Starting from a steady state configuration of a nuclear power reactor some situations arise in which the reactor configuration is perturbed. The Lambda modes are eigenfunctions associated with a given configuration of the reactor, which have successfully been used to describe unstable events in BWRs. To compute several eigenvalues and its corresponding eigenfunctions for a nuclear reactor is quite expensive from the computational point of view. Krylov subspace methods are efficient methods to compute the dominant Lambda modes associated with a given configuration of the reactor, but if the Lambda modes have to be computed for different perturbed configurations of the reactor more efficient methods can be used. In this paper, different methods for the updating Lambda modes problem will be proposed and compared by computing the dominant Lambda modes of different configurations associated with a Boron injection transient in a typical BWR reactor. (C) 2010 Elsevier Ltd. All rights reserved.This work has been partially supported by the Spanish Ministerio de Educacion y Ciencia under projects ENE2008-02669 and MTM2007-64477-AR07, the Generalitat Valenciana under project ACOMP/2009/058, and the Universidad Politecnica de Valencia under project PAID-05-09-4285.González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ. (2011). Updating the Lambda modes of a nuclear power reactor. Mathematical and Computer Modelling. 54(7):1796-1801. https://doi.org/10.1016/j.mcm.2010.12.013S1796180154

Data based identification and prediction of nonlinear and complex dynamical systems

We thank Dr. R. Yang (formerly at ASU), Dr. R.-Q. Su (formerly at ASU), and Mr. Zhesi Shen for their contributions to a number of original papers on which this Review is partly based. This work was supported by ARO under Grant No. W911NF-14-1-0504. W.-X. Wang was also supported by NSFC under Grants No. 61573064 and No. 61074116, as well as by the Fundamental Research Funds for the Central Universities, Beijing Nova Programme.Peer reviewedPostprin

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding