DEP51 - Artículos de revistaDpto. Matemática Aplicada - Artículos de revistahttp://uvadoc.uva.es/handle/10324/13592019-07-19T20:53:07Z2019-07-19T20:53:07ZPrincipal Floquet subspaces and exponential separations of type II with applications to random delay differential equationsMierczynski, JanuszNovo, SylviaObaya, Rafaelhttp://uvadoc.uva.es/handle/10324/334852019-06-14T12:27:21Z2018-01-01T00:00:00ZThis paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type II, important for its application to random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type II in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.
2018-01-01T00:00:00ZNumerical analysis of a cell dwarfism modelAbia Llera, Luis MaríaAngulo Torga, ÓscarLópez Marcos, Juan CarlosLópez Marcos, Miguel Ángelhttp://uvadoc.uva.es/handle/10324/323412019-06-14T12:33:53Z2019-01-01T00:00:00ZIn this work, we study numerically a model which describes cell dwarfism. It consists in a pure initial value problem for a first order partial differential equation, that can be applied to the description of the evolution of diseases as thalassemia. We design two numerical methods that prevent the use of the characteristic curve x = 0, and derive their optimal rates of convergence. Numerical experiments are also reported in order to demonstrate the predicted accuracy of the schemes. Finally, a comparison study on their efficiency is presented.
2019-01-01T00:00:00ZA numerical study on the estimation of the stable size distribution for a cell population balance modelAbia Llera, Luis MaríaAngulo Torga, ÓscarLópez Marcos, Juan CarlosLópez Marcos, Miguel Ángelhttp://uvadoc.uva.es/handle/10324/323402019-06-14T12:33:54Z2018-01-01T00:00:00ZThe presence of a steady-state distribution is an important issue in the modelization of cell populations. In this paper,we analyse, froma numerical point of view, the approach to the stable size distribution for a size-structured balance model with an asymmetric division rate. To this end, we introduce a second-order numerical method on the basis of the integration along the characteristic curves over the natural grid. We validate the interest of the scheme by means of a
detailed analysis of convergence.
2018-01-01T00:00:00ZPalindromic 3-stage splitting integrators, a roadmapMartínez Campos, CédricSanz Serna, Jesús Maríahttp://uvadoc.uva.es/handle/10324/289292019-06-19T13:11:19Z2017-01-01T00:00:00ZThe implementation of multi-stage splitting integrators is essentially the same as the implementation of the familiar Strang/Verlet method. Therefore multi-stage formulas may be easily incorporated into software that now uses the Strang/Verlet integrator. We study in detail the two-parameter family of palindromic, three-stage splitting formulas and identify choices of parameters that may outperform the Strang/Verlet method. One of these choices leads to a method of effective order four suitable to integrate in time some partial differential equations. Other choices may be seen as perturbations of the Strang method that increase efficiency in molecular dynamics simulations and in Hybrid Monte Carlo sampling.
2017-01-01T00:00:00ZHigher-order exponential integrators for quasi-linear parabolic problems. Part II: ConvergenceGonzález Fernández, CesáreoThalhammer, Mechthildhttp://uvadoc.uva.es/handle/10324/289122019-06-14T12:33:53Z2016-01-01T00:00:00ZIn this work, the convergence analysis of explicit exponential time integrators based on general linear methods for quasi-linear parabolic initial boundary value problems is pursued. Compared to other types of exponential integrators encountering rather severe order reductions, in general, the considered class of exponential general linear methods provides the possibility of constructing schemes that retain higher-order accuracy in time when applied to quasi-linear parabolic problems. In view of practical applications, the case of variable time step sizes is incorporated. The convergence analysis is based upon two fundamental ingredients. The needed stability bounds, obtained under mild restrictions on the ratios of subsequent time step sizes, have been deduced in the recent work [C. González and M. Thalhammer, SIAM J. Numer. Anal., 53 (2015), pp. 701--719]. The core of the present work is devoted to the derivation of suitable local and global error representations. In conjunction with the stability bounds, a convergence result is established.
2016-01-01T00:00:00ZBogdanov–Takens resonance in time-delayed systemsCoccolo, MattiaZhu, BeibeiSanjuán, Miguel A. F.Sanz Serna, Jesús Maríahttp://uvadoc.uva.es/handle/10324/289112019-06-14T12:33:53Z2018-01-01T00:00:00ZWe analyze the oscillatory dynamics of a time-delayed dynamical system subjected to a periodic external forcing. We show that, for certain values of the delay, the response can be greatly enhanced by a very small forcing amplitude. This phenomenon is related to the presence of a Bogdanov–Takens bifurcation and displays some analogies to other resonance phenomena, but also substantial differences.
2018-01-01T00:00:00ZComputing normal forms and formal invariants of dynamical systems by means of word seriesMurua, AnderSanz Serna, Jesús Maríahttp://uvadoc.uva.es/handle/10324/289102019-06-14T12:33:53Z2016-01-01T00:00:00ZWe show how to use extended word series in the reduction of continuous and discrete dynamical systems to normal form and in the computation of formal invariants of motion in Hamiltonian systems. The manipulations required involve complex numbers rather than vector fields or diffeomorphisms. More precisely we construct a group G¯ and a Lie algebra g¯ in such a way that the elements of G¯ and g¯ are families of complex numbers; the operations to be performed involve the multiplication ★ in G¯ and the bracket of g¯ and result in universal coefficients that are then applied to write the normal form or the invariants of motion of the specific problem under consideration.
2016-01-01T00:00:00ZWord Series for Dynamical Systems and Their Numerical IntegratorsMurua, AnderSanz Serna, Jesús Maríahttp://uvadoc.uva.es/handle/10324/289042019-06-14T12:33:53Z2017-01-01T00:00:00ZWe study word series and extended word series, classes of formal series for the analysis of some dynamical systems and their discretizations. These series are similar to but more compact than B-series. They may be composed among themselves by means of a simple rule. While word series have appeared before in the literature, extended word series are introduced in this paper. We exemplify the use of extended word series by studying the reduction to normal form and averaging of some perturbed integrable problems. We also provide a detailed analysis of the behavior of splitting numerical methods for those problems.
2017-01-01T00:00:00ZVibrational resonance: a study with high-order word-series averagingMurua, AnderSanz Serna, Jesús Maríahttp://uvadoc.uva.es/handle/10324/289032019-06-14T12:33:53Z2016-01-01T00:00:00ZWe study a model problem describing vibrational resonance by means of a high-order averaging technique based on so-called word series. With the technique applied here, the tasks of constructing the averaged system and the associated change of variables are divided into two parts. It is first necessary to build recursively a set of so-called word basis functions and, after that, all the required manipulations involve only scalar coefficients that are computed by means of simple recursions. As distinct from the situation with other approaches, with word-series, high-order averaged systems may be derived without having to compute the associated change of variables. In the system considered here, the construction of high-order averaged systems makes it possible to obtain very precise approximations to the true dynamics.
2016-01-01T00:00:00ZSymplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control and moreSanz Serna, Jesús Maríahttp://uvadoc.uva.es/handle/10324/289022019-06-14T12:33:52Z2016-01-01T00:00:00ZThe study of the sensitivity of the solution of a system of differential equations with respect to changes in the initial conditions leads to the introduction of an adjoint system, whose discretization is related to reverse accumulation in automatic differentiation. Similar adjoint systems arise in optimal control and other areas, including classical mechanics. Adjoint systems are introduced in such a way that they exactly preserve a relevant quadratic invariant (more precisely, an inner product). Symplectic Runge--Kutta and partitioned Runge--Kutta methods are defined through the exact conservation of a differential geometric structure, but may be characterized by the fact that they preserve exactly quadratic invariants of the system being integrated. Therefore, the symplecticness (or lack of symplecticness) of a Runge--Kutta or partitioned Runge--Kutta integrator should be relevant to understanding its performance when applied to the computation of sensitivities, to optimal control problems, and in other applications requiring the use of adjoint systems. This paper examines the links between symplectic integration and those applications and presents in a new, unified way a number of results currently scattered among or implicit in the literature. In particular, we show how some common procedures, such as the direct method in optimal control theory and the computation of sensitivities via reverse accumulation, imply, probably unbeknownst to the user, “hidden” integrations with symplectic partitioned Runge--Kutta schemes.
2016-01-01T00:00:00ZA technique for studying strong and weak local errors of splitting stochastic integratorsÁlamo Zapatero, AlfonsoSanz Serna, Jesús Maríahttp://uvadoc.uva.es/handle/10324/288592019-06-14T12:33:52Z2016-01-01T00:00:00ZWe present a technique, based on so-called word series, to write down in a systematic way expansions of the strong and weak local errors of splitting algorithms for the integration of Stratonovich stochastic differential equations. Those expansions immediately lead to the corresponding order conditions. Word series are similar to, but simpler than, the B-series used to analyze Runge--Kutta and other one-step integrators. The suggested approach makes it unnecessary to use the Baker--Campbell--Hausdorff formula. As an application, we compare two splitting algorithms recently considered by Leimkuhler and Matthews to integrate the Langevin equations. The word series method clearly bears out reasons for the advantages of one algorithm over the other.
2016-01-01T00:00:00ZRandomized Hamiltonian Monte CarloBou-Rabee, NawafSanz Serna, Jesús Maríahttp://uvadoc.uva.es/handle/10324/288502019-06-14T12:33:52Z2017-01-01T00:00:00ZTuning the durations of the Hamiltonian flow in Hamiltonian Monte Carlo (also called Hybrid Monte Carlo) (HMC) involves a tradeoff between computational cost and sampling quality, which is typically challenging to resolve in a satisfactory way. In this article, we present and analyze a randomized HMC method (RHMC), in which these durations are i.i.d. exponential random variables whose mean is a free parameter. We focus on the small time step size limit, where the algorithm is rejection-free and the computational cost is proportional to the mean duration. In this limit, we prove that RHMC is geometrically ergodic under the same conditions that imply geometric ergodicity of the solution to underdamped Langevin equations. Moreover, in the context of a multidimensional Gaussian distribution, we prove that the sampling efficiency of RHMC, unlike that of constant duration HMC, behaves in a regular way. This regularity is also verified numerically in non-Gaussian target distributions. Finally, we suggest variants of RHMC for which the time step size is not required to be small.
2017-01-01T00:00:00ZNumerical generation of periodic traveling wave solutions of some nonlinear dispersive wave systems.Álvarez López, JorgeDurán Martín, Ángelhttp://uvadoc.uva.es/handle/10324/258282019-06-14T12:33:52Z2017-01-01T00:00:00ZIn this paper a numerical procedure to generate approximations to periodic traveling wave profiles of some nonlinear dispersive wave systems is introduced. The method is based on a suitable modification of a fixed point algorithm of Petviashvili type and solves several drawbacks of some previous algorithms presented in the literature. By way of illustration, the method is applied to generate numerically periodic traveling waves of two problems of interest: the fractional KdV type equations and the extended Boussinesq system.
2017-01-01T00:00:00ZPetviashvili type methods for traveling wave computations: II. Acceleration with vector extrapolation methods.Álvarez López, JorgeDurán Martín, Ángelhttp://uvadoc.uva.es/handle/10324/258262019-06-14T12:33:52Z2016-01-01T00:00:00ZIn this paper a family of fixed point algorithms, generalizing the Petviashvili method, is considered. A previous work studied the convergence of the methods. Presented here is a second part of the analysis, concerning the introduction of acceleration techniques, based on vector extrapolation, into the iterative procedures. The purpose of the research is two-fold: one is improving the performance of the methods in case of convergence and the second one is widening their application when generating traveling waves in nonlinear dispersive wave equations, transforming some divergent into convergent cases. A comparative study through
several numerical experiments is carried out.
2016-01-01T00:00:00ZCharacterization of cocycles attractors for nonautonomous reaction-diffusion equationsCardoso, CarlosLanga, JuanObaya, Rafaelhttp://uvadoc.uva.es/handle/10324/258012019-06-14T12:27:12Z2016-01-01T00:00:00ZIn this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutio
2016-01-01T00:00:00ZExponential stability for nonautonomous functional differential equations with state-dependent delayMaroto, IsmaelNúñez, CarmenObaya, Rafaelhttp://uvadoc.uva.es/handle/10324/257592019-06-14T12:27:14Z2017-01-01T00:00:00ZThe properties of stability of a compact set $K$ which is positively invariant for a semiflow $(\W\times W^{1,\infty}([-r,0],\mathbb{R}^n),\Pi,\mathbb{R}^+)$ determined by a family of nonautonomous FDEs with state-dependent delay taking values in $[0,r]$ are analyzed. The solutions of the variational equation through the orbits of $K$ induce linear skew-product semiflows on the bundles $K\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and $\mK\times C([-r,0],\mathbb{R}^n)$. The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fundamental tool to prove that the strictly negative character of this upper-Lyapunov exponent is equivalent to the exponential stability of $K$ in
$\W\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and also to the exponential stability of this compact set when the supremum norm is taken in $W^{1,\infty}([-r,0],\mathbb{R}^n)$. In particular, the existence of a uniformly exponentially stable solution of a uniformly almost periodic FDE ensures the existence of exponentially stable almost periodic solutions.
2017-01-01T00:00:00Z