Uncertainty quantification for random Hamiltonian systems by using polynomial expansions and geometric integrators

Recent advances in the field of uncertainty quantification are based on achieving suitable functional representations of the solutions to random systems. This aims at improving the performance of Monte Carlo simulation, at least for low-dimensional problems and moderately large independent variable. One of these functional representations are the so-called generalized polynomial chaos (gPC) expansions, based upon orthogonal polynomial decompositions. When the input random parameters are independent (a germ), a Galerkin projection technique applied to the truncated gPC expansion is usually employed. This approach exhibits fast mean-square convergence for smooth dynamics, whenever applicable. However, the main difficulty arises when solving the Galerkin system for the gPC coefficients, which may rely on different solvers (algorithms and codes) than those for the original system. A recent contribution noticed that, for random Hamiltonian systems, the Galerkin system is Hamiltonian too. Thus, the well-known symplectic integrators can be applied. The present paper investigates random Hamiltonian systems in general, when the input random parameters may be non-independent. In such a case, polynomial expansions based on the canonical basis and an imitation of the Galerkin projection technique are proposed. The Hamiltonian structure of the original system is unfortunately not conserved, but volume preservation is. Hence volume-preserving integrators are of use. Numerical experiments suggest that the proposed polynomial expansions may be useful for fast and accurate uncertainty quantification

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 the convergence of the mild random walk algorithm to generate random one-factorizations of complete graphs

The complete graph Kn, for n even, has a one-factorization (proper edge coloring) with n – 1 colors. In the recent contribution [Dotan M., Linial N. (2017). ArXiv:1707.00477v2], the authors raised a conjecture on the convergence of the mild random walk on the Markov chain whose nodes are the colorings of Kn. The mild random walk consists in moving from a coloring C to a recoloring C′ if and only if ϕ(C′) ≤ ϕ(C), where ϕ is the potential function that takes its minimum at one-factorizations. We show the validity of such algorithm with several numerical experiments that demonstrate convergence in all cases (not just asymptotically) with polynomial cost. We prove several results on the mild random walk, we study deeply the properties of local minimum colorings, we give a detailed proof of the convergence of the algorithm for K4 and K6, and we raise new conjectures. We also present an alternative to the potential measure ϕ by consider the Shannon entropy, which has a strong parallelism with ϕ from the numerical standpoint

Analytic solution to the generalized delay diffusion equation with uncertain inputs in the random Lebesgue sense

[EN] In this paper, we deal with the randomized generalized diffusion equation with delay:u(t)(t, x) = a(2)u(xx)(t, x) + b(2)u(xx)(t - tau, x),t > tau,0 = 0;u(t,x)=phi(t,x),0 0andl > 0are constant. The coefficientsa(2)andb(2)are nonnegative random variables, and the initial condition phi(t, x)and the solutionu(t, x)are random fields. The separation of variables method develops a formal series solution. We prove that the series satisfies the delay diffusion problem in the random Lebesgue sense rigorously. By truncating the series, the expectation and the variance of the random-field solution can be approximated.Secretaria de Estado de Investigacion, Desarrollo e Innovacion, Grant/Award Number: MTM2017-89664-P; Spanish Ministerio de Economia, Industria y Competitividad (MINECO); Agencia Estatal de Investigacion (AEI); Fondo Europeo de Desarrollo Regional (FEDER UE), Grant/Award Number: MTM2017-89664-PCortés, J.; Jornet, M. (2021). Analytic solution to the generalized delay diffusion equation with uncertain inputs in the random Lebesgue sense. Mathematical Methods in the Applied Sciences. 44(2):2265-2272. https://doi.org/10.1002/mma.6921S22652272442Smith, H. (2011). An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics. doi:10.1007/978-1-4419-7646-8Driver, R. D. (1977). Ordinary and Delay Differential Equations. Applied Mathematical Sciences. doi:10.1007/978-1-4684-9467-9Kolmanovskii, V., & Myshkis, A. (1999). Introduction to the Theory and Applications of Functional Differential Equations. doi:10.1007/978-94-017-1965-0Wu, J. (1996). Theory and Applications of Partial Functional Differential Equations. Applied Mathematical Sciences. doi:10.1007/978-1-4612-4050-1Diekmann, O., Verduyn Lunel, S. M., van Gils, S. A., & Walther, H.-O. (1995). Delay Equations. Applied Mathematical Sciences. doi:10.1007/978-1-4612-4206-2Hale, J. K. (1977). Theory of Functional Differential Equations. Applied Mathematical Sciences. doi:10.1007/978-1-4612-9892-2Travis, C. C., & Webb, G. F. (1974). Existence and stability for partial functional differential equations. Transactions of the American Mathematical Society, 200, 395-395. doi:10.1090/s0002-9947-1974-0382808-3Bocharov, G. A., & Rihan, F. A. (2000). Numerical modelling in biosciences using delay differential equations. Journal of Computational and Applied Mathematics, 125(1-2), 183-199. doi:10.1016/s0377-0427(00)00468-4Jackson, M., & Chen-Charpentier, B. M. (2017). Modeling plant virus propagation with delays. Journal of Computational and Applied Mathematics, 309, 611-621. doi:10.1016/j.cam.2016.04.024Chen-Charpentier, B. M., & Diakite, I. (2016). A mathematical model of bone remodeling with delays. Journal of Computational and Applied Mathematics, 291, 76-84. doi:10.1016/j.cam.2014.11.025ErneuxT.Applied Delay Differential Equations Surveys and Tutorials in the Applied Mathematical Sciences Series:Springer New York;2009.Kyrychko, Y. N., & Hogan, S. J. (2010). On the Use of Delay Equations in Engineering Applications. Journal of Vibration and Control, 16(7-8), 943-960. doi:10.1177/1077546309341100Matsumoto, A., & Szidarovszky, F. (2009). Delay Differential Nonlinear Economic Models. Nonlinear Dynamics in Economics, Finance and Social Sciences, 195-214. doi:10.1007/978-3-642-04023-8_11Harding, L., & Neamţu, M. (2016). A Dynamic Model of Unemployment with Migration and Delayed Policy Intervention. Computational Economics, 51(3), 427-462. doi:10.1007/s10614-016-9610-3Xiu, D. (2010). Numerical Methods for Stochastic Computations. doi:10.2307/j.ctv7h0skvLe Maître, O. P., & Knio, O. M. (2010). Spectral Methods for Uncertainty Quantification. Scientific Computation. doi:10.1007/978-90-481-3520-2NeckelT RuppF.Random Differential Equations in Scientific Computing. Walter de Gruyter;2013.Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Calatayud, J., Cortés, J.-C., & Jornet, M. (2019). Improving the Approximation of the First- and Second-Order Statistics of the Response Stochastic Process to the Random Legendre Differential Equation. Mediterranean Journal of Mathematics, 16(3). doi:10.1007/s00009-019-1338-6Licea, J. A., Villafuerte, L., & Chen-Charpentier, B. M. (2013). Analytic and numerical solutions of a Riccati differential equation with random coefficients. Journal of Computational and Applied Mathematics, 239, 208-219. doi:10.1016/j.cam.2012.09.040Burgos, C., Calatayud, J., Cortés, J.-C., & Villafuerte, L. (2018). Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series. Applied Mathematics Letters, 78, 95-104. doi:10.1016/j.aml.2017.11.009Calatayud, J., Cortés, J.-C., & Jornet, M. (2019). Random differential equations with discrete delay. Stochastic Analysis and Applications, 37(5), 699-707. doi:10.1080/07362994.2019.1608833Calatayud, J., Cortés, J.-C., & Jornet, M. (2019). $$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay. Mediterranean Journal of Mathematics, 16(4). doi:10.1007/s00009-019-1370-6Martín, J. A., Rodríguez, F., & Company, R. (2004). Analytic solution of mixed problems for thegeneralized diffusion equation with delay. Mathematical and Computer Modelling, 40(3-4), 361-369. doi:10.1016/j.mcm.2003.10.046Martínez-Cervantes, G. (2016). Riemann integrability versus weak continuity. Journal of Mathematical Analysis and Applications, 438(2), 840-855. doi:10.1016/j.jmaa.2016.01.054Cortés, J. C., Sevilla-Peris, P., & Jódar, L. (2005). Analytic-numerical approximating processes of diffusion equation with data uncertainty. Computers & Mathematics with Applications, 49(7-8), 1255-1266. doi:10.1016/j.camwa.2004.05.015Khusainov, D. Y., Ivanov, A. F., & Kovarzh, I. V. (2009). Solution of one heat equation with delay. Nonlinear Oscillations, 12(2), 260-282. doi:10.1007/s11072-009-0075-3Calatayud, J., Cortés, J. C., & Jornet, M. (2018). 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.5333BotevZ RidderA.Variance reductionWiley StatsRef: Statistics Reference Online;2017:1–6.https://doi.org/10.1002/9781118445112.stat0797

Lp-solution to the random linear delay differential equation with stochastic forcing term

[EN] This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay tau > 0, by adding a random forcing term f(t) that varies with time: x'(t) = ax(t) + bx(t-tau) + f(t), t >= 0, with initial condition x(t) = g(t), -tau <= t <= 0. The coefficients a and b are assumed to be random variables, while the forcing term f(t) and the initial condition g(t) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space L-p of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an L-p-solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz's integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay tau tends to 0, the random delay equation tends in L-p to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Cortés, J.; Jornet, M. (2020). Lp-solution to the random linear delay differential equation with stochastic forcing term. Mathematics. 8(6):1-16. https://doi.org/10.3390/math8061013S11686Xiu, D., & Karniadakis, G. E. (2004). Supersensitivity due to uncertain boundary conditions. International Journal for Numerical Methods in Engineering, 61(12), 2114-2138. doi:10.1002/nme.1152Casabán, 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.2015.08.009Strand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Cortés, J.-C., Jódar, L., Roselló, M.-D., & Villafuerte, L. (2012). Solving initial and two-point boundary value linear random differential equations: A mean square approach. Applied Mathematics and Computation, 219(4), 2204-2211. doi:10.1016/j.amc.2012.08.066Calatayud, J., Cortés, J.-C., Jornet, M., & Villafuerte, L. (2018). Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties. Advances in Difference Equations, 2018(1). doi:10.1186/s13662-018-1848-8Calatayud, J., Cortés, J.-C., & Jornet, M. (2019). Improving the Approximation of the First- and Second-Order Statistics of the Response Stochastic Process to the Random Legendre Differential Equation. Mediterranean Journal of Mathematics, 16(3). doi:10.1007/s00009-019-1338-6Licea, J. A., Villafuerte, L., & Chen-Charpentier, B. M. (2013). Analytic and numerical solutions of a Riccati differential equation with random coefficients. Journal of Computational and Applied Mathematics, 239, 208-219. doi:10.1016/j.cam.2012.09.040Burgos, C., Calatayud, J., Cortés, J.-C., & Villafuerte, L. (2018). Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series. Applied Mathematics Letters, 78, 95-104. doi:10.1016/j.aml.2017.11.009Nouri, K., & Ranjbar, H. (2014). Mean Square Convergence of the Numerical Solution of Random Differential Equations. Mediterranean Journal of Mathematics, 12(3), 1123-1140. doi:10.1007/s00009-014-0452-8Calatayud, J., Cortés, J.-C., & Jornet, M. (2019). Random differential equations with discrete delay. Stochastic Analysis and Applications, 37(5), 699-707. doi:10.1080/07362994.2019.1608833Calatayud, J., Cortés, J.-C., & Jornet, M. (2019). $$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay. Mediterranean Journal of Mathematics, 16(4). doi:10.1007/s00009-019-1370-6Caraballo, T., Cortés, J.-C., & Navarro-Quiles, A. (2019). Applying the random variable transformation method to solve a class of random linear differential equation with discrete delay. Applied Mathematics and Computation, 356, 198-218. doi:10.1016/j.amc.2019.03.048Zhou, T. (2014). A Stochastic Collocation Method for Delay Differential Equations with Random Input. Advances in Applied Mathematics and Mechanics, 6(4), 403-418. doi:10.4208/aamm.2012.m38Shi, W., & Zhang, C. (2017). Generalized polynomial chaos for nonlinear random delay differential equations. Applied Numerical Mathematics, 115, 16-31. doi:10.1016/j.apnum.2016.12.004Khusainov, D. Y., Ivanov, A. F., & Kovarzh, I. V. (2009). Solution of one heat equation with delay. Nonlinear Oscillations, 12(2), 260-282. doi:10.1007/s11072-009-0075-3Shaikhet, L. (2016). Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations. International Journal of Robust and Nonlinear Control, 27(6), 915-924. doi:10.1002/rnc.3605Benhadri, M., & Zeghdoudi, H. (2018). Mean square asymptotic stability in nonlinear stochastic neutral Volterra-Levin equations with Poisson jumps and variable delays. Functiones et Approximatio Commentarii Mathematici, 58(2). doi:10.7169/facm/1657Santonja, F.-J., & Shaikhet, L. (2012). Analysing Social Epidemics by Delayed Stochastic Models. Discrete Dynamics in Nature and Society, 2012, 1-13. doi:10.1155/2012/530472Liu, L., & Caraballo, T. (2018). Analysis of a Stochastic 2D-Navier–Stokes Model with Infinite Delay. Journal of Dynamics and Differential Equations, 31(4), 2249-2274. doi:10.1007/s10884-018-9703-xLupulescu, V., & Abbas, U. (2011). Fuzzy delay differential equations. Fuzzy Optimization and Decision Making, 11(1), 99-111. doi:10.1007/s10700-011-9112-7Krapivsky, P. L., Luck, J. M., & Mallick, K. (2011). On stochastic differential equations with random delay. Journal of Statistical Mechanics: Theory and Experiment, 2011(10), P10008. doi:10.1088/1742-5468/2011/10/p10008GARRIDO-ATIENZA, M. J., OGROWSKY, A., & SCHMALFUSS, B. (2011). RANDOM DIFFERENTIAL EQUATIONS WITH RANDOM DELAYS. Stochastics and Dynamics, 11(02n03), 369-388. doi:10.1142/s0219493711003358Cortés, J.-C., Villafuerte, L., & Burgos, C. (2017). A Mean Square Chain Rule and its Application in Solving the Random Chebyshev Differential Equation. Mediterranean Journal of Mathematics, 14(1). doi:10.1007/s00009-017-0853-6Cortés, J. C., Jódar, L., & Villafuerte, L. (2007). Numerical solution of random differential equations: A mean square approach. Mathematical and Computer Modelling, 45(7-8), 757-765. doi:10.1016/j.mcm.2006.07.017Braumann, C. A., Cortés, J.-C., Jódar, L., & Villafuerte, L. (2018). On the random gamma function: Theory and computing. Journal of Computational and Applied Mathematics, 335, 142-155. doi:10.1016/j.cam.2017.11.045Khusainov, D. Y., & Pokojovy, M. (2015). Solving the Linear 1D Thermoelasticity Equations with Pure Delay. International Journal of Mathematics and Mathematical Sciences, 2015, 1-11. doi:10.1155/2015/47926

Spatio-temporal stochastic differential equations for crime incidence modeling

We propose a methodology for the quantitative fitting and forecasting of real spatio-temporal crime data, based on stochastic differential equations. The analysis is focused on the city of Valencia, Spain, for which 90247 robberies and thefts with their latitude-longitude positions are available for a span of eleven years (2010–2020) from records of the 112-emergency phone. The incidents are placed in the 26 zip codes of the city (46001–46026), and monthly time series of crime are built for each of the zip codes. Their annual-trend components are modeled by Itoˆ diffusion, with jointly correlated noises to account for district-level relations. In practice, this study may help simulate spatio-temporal situations and identify risky areas and periods from present and past data.Funding for open access charge: CRUE-Universitat Jaume

A phenomenological model for COVID-19 data taking into account neighboring-provinces effect and random noise

We model the incidence of the COVID-19 disease during the first wave of the epidemic in Castilla-Leon (Spain). Within-province dynamics may be governed by a generalized logistic map, but this lacks of spatial structure. To couple the provinces, we relate the daily new infec- tions through a density-independent parameter that entails positive spatial correlation. Pointwise values of the input parameters are fitted by an optimization procedure. To accommodate the significant variability in the daily data, with abruptly increasing and decreasing magnitudes, a random noise is incorporated into the model, whose parameters are calibrated by maximum like- lihood estimation. The calculated paths of the stochastic response and the probabilistic regions are in good agreement with the data

Improving Kernel Methods for Density Estimation in Random Differential Equations Problems

[EN] Kernel density estimation is a non-parametric method to estimate the probability density function of a random quantity from a finite data sample. The estimator consists of a kernel function and a smoothing parameter called the bandwidth. Despite its undeniable usefulness, the convergence rate may be slow with the number of realizations and the discontinuity and peaked points of the target density may not be correctly captured. In this work, we analyze the applicability of a parametric method based on Monte Carlo simulation for the density estimation of certain random variable transformations. This approach has important applications in the setting of differential equations with input random parameters.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Cortés, J.; Jornet Sanz, M. (2020). Improving Kernel Methods for Density Estimation in Random Differential Equations Problems. Mathematical and Computational Applications (Online). 25(2):1-9. https://doi.org/10.3390/mca25020033S19252Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2020). Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. Applied Numerical Mathematics, 151, 413-424. doi:10.1016/j.apnum.2020.01.012Calatayud, J., Cortés, J.-C., & Jornet, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A: Statistical Mechanics and its Applications, 512, 261-279. doi:10.1016/j.physa.2018.08.024Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2019). Density function of random differential equations via finite difference schemes: a theoretical analysis of a random diffusion-reaction Poisson-type problem. Stochastics, 92(4), 627-641. doi:10.1080/17442508.2019.1645849Calatayud, J., Cortés, J.-C., Dorini, F. A., & Jornet, M. (2020). Extending the study on the linear advection equation subject to stochastic velocity field and initial condition. Mathematics and Computers in Simulation, 172, 159-174. doi:10.1016/j.matcom.2019.12.014Jornet, M., Calatayud, J., Le Maître, O. P., & Cortés, J.-C. (2020). Second order linear differential equations with analytic uncertainties: Stochastic analysis via the computation of the probability density function. Journal of Computational and Applied Mathematics, 374, 112770. doi:10.1016/j.cam.2020.112770Tang, K., Wan, X., & Liao, Q. (2020). Deep density estimation via invertible block-triangular mapping. Theoretical and Applied Mechanics Letters, 10(3), 143-148. doi:10.1016/j.taml.2020.01.023Botev, Z., & Ridder, A. (2017). Variance Reduction. Wiley StatsRef: Statistics Reference Online, 1-6. doi:10.1002/9781118445112.stat0797

A modified perturbation method for mathematical models with randomness: An analysis through the steady-state solution to Burgers’ PDE

The variability of the data and the incomplete knowledge of the true physics require the incorporation of randomness into the formulation of mathematical models. In this setting, the deterministic numerical methods cannot capture the propagation of the uncertainty from the inputs to the model output. For some problems, such as the Burgers' equation (simplification to understand properties of the Navier–Stokes equations), a small variation in the parameters causes nonnegligible changes in the output. Thus, suitable techniques for uncertainty quantification must be used. The generalized polynomial chaos (gPC) method has been successfully applied to compute the location of the transition layer of the steady-state solution, when a small uncertainty is incorporated into the boundary. On the contrary, the classical perturbation method does not give reliable results, due to the uncertainty magnitude of the output. We propose a modification of the perturbation method that converges and is comparable with the gPC approach in terms of efficiency and rate of convergence. The method is even applicable when the input random parameters are dependent random variables

Approximate solutions of randomized non-autonomous complete linear differential equations via probability density functions

Solving a random differential equation means to obtain an exact or approximate expression for the solution stochastic process, and to compute its statistical properties, mainly the mean and the variance functions. However, a major challenge is the computation of the probability density function of the solution. In this article we construct reliable approximations of the probability density function to the randomized non-autonomous complete linear differential equation by assuming that the diffusion coefficient and the source term are stochastic processes and the initial condition is a random variable. The key tools to construct these approximations are the random variable transformation technique and Karhunen-Lo`eve expansions. The study is divided into a large number of cases with a double aim: firstly, to extend the available results in the extant literature and, secondly, to embrace as many practical situations as possible. Finally, a wide variety of numerical experiments illustrate the potentiality of our findings