126 research outputs found

### 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

### Analysis of the random heat equation via approximate density functions

[EN] In this paper we study the randomized heat equation with homogeneous boundary conditions. The diffusion coefficient is assumed to be a random variable and the initial condition is treated as a stochastic process. The solution of this randomized partial differential equation problem is a stochastic process, which is given by a random series obtained via the classical method of separation of variables. Any stochastic process is determined by its finite-dimensional joint distributions. In this paper, the goal is to obtain approximations to the probability density function of the solution (the first finite-dimensional distributions) under mild conditions. Since the solution is expressed as a random series, we perform approximations to its probability density function. Several illustrative examples are shown.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.Calatayud, J.; Cortés, J. (2021). Analysis of the random heat equation via approximate density functions. Romanian Reports in Physics. 73(2):1-10. http://hdl.handle.net/10251/181144S11073

### 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

### 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

### Probabilistic analysis of a class of 2D-random heat equations via densities

We give new probabilistic results for a class of random two-dimensional homogeneous heat equations with mixed homogeneous Dirichlet and Neumann boundary conditions and an arbitrary initial condition on a rectangular domain. The diffusion coefficient is assumed to be an arbitrary second-order random variable, while the initial condition is a stochastic process admitting a Karhunen-Loève expansion. We then construct pointwise convergent approximations for the main moments and the density of the solution. The theoretical results are numerically illustrated.J.-C. Cortés has been supported by the Spanish Agencia Estatal de Investigación, Spain grant PID2020-115270GB-I00. V. Bevia has been supported by the Programa de Ayudas de Investigación y Desarrollo (PAID), Spain

### 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

### On the random wave equation within the mean square context

This paper deals with the random wave equation on a bounded domain with Dirichlet boundary conditions. Randomness arises from the velocity wave, which is a positive random variable, and the two initial conditions, which are regular stochastic processes. The aleatory nature of the inputs is mainly justified from data errors when modeling the motion of a vibrating string. Uncertainty is propagated from these inputs to the output, so that the solution becomes a smooth random field. We focus on the mean square contextualization of the problem. Existence and uniqueness of the exact series solution, based upon the classical method of separation of variables, are rigorously established. Exact series for the mean and the variance of the solution process are obtained, which converge at polynomial rate. Some numerical examples illustrate these facts

### A theoretical study of a short rate model

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2016, Director: José Manuel Corcuera ValverdeThe goal of this project is to do a theoretical study of a short interest rate model under the risk neutral probability, which is able to represent long range dependence. In order to do this, it will be explained the necessary literature to understand the model. Furthermore, we will expose the consequences of adapting this model for evaluating bonds and derivatives. In order to do this, we will use ambit processes which in general are not semimartingales. Our purpose is to see if these new models can capture the features of the bond market by extending popular models like the Vasicek model