Basque Center for Applied Mathematics

Derio, Spain

Basque Center for Applied Mathematics

Derio, Spain

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Henao D.,University Pierre and Marie Curie | Mora-Corral C.,Basque Center for Applied Mathematics
Archive for Rational Mechanics and Analysis | Year: 2010

In this paper, we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture. The main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface. Accordingly, we define a functional that measures the area of the created surface. This functional has relationships with the theory of Cartesian currents. We show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient, and that the weak limit of one-to-one almost everywhere deformations is also one-to-one almost everywhere. We then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy, taking into account orientation-preserving and non-interpenetration conditions. © 2009 Springer-Verlag.

Munch A.,University Blaise Pascal | Zuazua E.,Basque Center for Applied Mathematics
Inverse Problems | Year: 2010

The numerical approximation of exact or trajectory controls for the wave equation is known to be a delicate issue, since the pioneering work of Glowinski-Lions in the nineties, because of the anomalous behavior of the high-frequency spurious numerical waves. Various efficient remedies have been developed and analyzed in the last decade to filter out these highfrequency components: Fourier filtering, Tychonoff's regularization, mixed finite-element methods, multi-grid strategies, etc. Recently convergence rate results have also been obtained. This work is devoted to analyzing this issue for the heat equation, which is the opposite paradigm because of its strong dissipativity and smoothing properties. The existing analytical results guarantee that, at least in some simple situations, as in the finite-difference scheme in 1 - d, the null or trajectory controls for numerical approximation schemes converge. This is due to the intrinsic high-frequency damping of the heat equation that is inherited by its numerical approximation schemes. But when developing numerical simulations the topic appears to be much more subtle and difficult. In fact, efficiently computing the null control for a numerical approximation scheme of the heat equation is a difficult problem in itself. The difficulty is strongly related to the regularizing effect of the heat kernel. The controls of minimal L2-norm are characterized as minima of quadratic functionals on the solutions of the adjoint heat equation, or its numerical versions. These functionals are shown to be coercive in very large spaces of solutions, sufficient to guarantee the L2 character of controls, but very far from being identifiable as energy spaces for the adjoint system. The very weak coercivity of the functionals under considerationmakes the approximation problem exponentially ill-posed and the functional framework far from being well adapted to standard techniques in numerical analysis. In practice, the controls of the minimal L2-norm exhibit a singular highly oscillatory behavior near the final controllability time, which cannot be captured numerically. Standard techniques, such as Tychonoff's regularization or quasi-reversibility methods, allow a slight smoothing of the singularities but significantly reduce the quality of the approximation. In this paper we develop some more involved and less-standard approaches which turn out to be more efficient. We first discuss the advantages of using controls with compact support with respect to the time variable or the effect of adding numerical dissipative singular terms. We also develop the numerical version of the so-called transmutation method that allows writing the control of a heat process in terms of the corresponding control of the associated wave process, by means of a 'time convolution' with a one-dimensional controlled fundamental heat solution. This method, although it can be proved to converge, is also subtle in its computational implementation. Indeed, it requires using convergent numerical schemes for the control of the wave equation, a problem that, as mentioned above, is delicate in itself. But we also need to compute an accurate approximation of a controlled fundamental heat solution, an issue that requires its own analysis and significant numerical and computational new developments. These methods are thoroughly illustrated and discussed in the paper, accompanied by some numerical experiments in one space dimension that show the subtlety of the issue. These experiments allow one to compare the efficiency of the various methods. This is done in the case where the control is distributed in some subdomain of the domain where the heat process evolves but similar results and numerical experiments could be derived for other cases, such as the one in which the control acts on the boundary. The techniques we employ here can also be adapted to the multi-dimensional case. © 2010 IOP Publishing Ltd.

Henao D.,University Pierre and Marie Curie | Mora-Corral C.,Basque Center for Applied Mathematics
Archive for Rational Mechanics and Analysis | Year: 2011

Motivated by nonlinear elasticity theory, we study deformations that are approximately differentiable, orientation-preserving and one-to-one almost everywhere, and in addition have finite surface energy. This surface energy e{open} was used by the authors in a previous paper, and has connections with the theory of currents. In the present paper we prove that e{open} measures exactly the area of the surface created by the deformation. This is done through a proper definition of created surface, which is related to the set of discontinuity points of the inverse of the deformation. In doing so, we also obtain an SBV regularity result for the inverse. © 2011 Springer-Verlag.

Beauchard K.,Ecole Normale Superieure de Cachan | Zuazua E.,Basque Center for Applied Mathematics
Archive for Rational Mechanics and Analysis | Year: 2011

This work is concerned with (n-component) hyperbolic systems of balance laws in m space dimensions. First, we consider linear systems with constant coefficients and analyze the possible behavior of solutions as t → ∞. Using the Fourier transform, we examine the role that control theoretical tools, such as the classical Kalman rank condition, play. We build Lyapunov functionals allowing us to establish explicit decay rates depending on the frequency variable. In this way we extend the previous analysis by Shizuta and Kawashima under the so-called algebraic condition (SK). In particular, we show the existence of systems exhibiting more complex behavior than the one that the (SK) condition allows. We also discuss links between this analysis and previous literature in the context of damped wave equations, hypoellipticity and hypocoercivity. To conclude, we analyze the existence of global solutions around constant equilibria for nonlinear systems of balance laws. Our analysis of the linear case allows proving existence results in situations that the previously existing theory does not cover. © 2010 Springer-Verlag.

Pagnini G.,Basque Center for Applied Mathematics | Pagnini G.,Ikerbasque
Physica A: Statistical Mechanics and its Applications | Year: 2014

In the present Short Note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non-Markovian master equations. Particle trajectories are assumed to be solely Markovian and described by the Continuous Time Random Walk model. But, as a consequence of the complexity of the medium, each trajectory is supposed to scale in time according to a particular random timescale. The link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed. In particular, when a stationary distribution is considered, the timescale distribution is uniquely determined as a function related to the fundamental solution of the space-time fractional diffusion equation. In contrast, when the non-stationary case is considered, the timescale distribution is no longer unique. Two distributions are here computed: one related to the M-Wright/Mainardi function, which is Green's function of the time-fractional diffusion equation, and another related to the Mittag-Leffler function, which is the solution of the fractional-relaxation equation. © 2014 Elsevier B.V. All rights reserved.

Lu Q.,Basque Center for Applied Mathematics | Lu Q.,University of Electronic Science and Technology of China
Inverse Problems | Year: 2012

In this paper, we establish a global Carleman estimate for stochastic parabolic equations. Based on this estimate, we study two inverse problems for stochastic parabolic equations. One is concerned with a determination problem of the history of a stochastic heat process through the observation at the final time T for which we obtain a conditional stability estimate. The other is an inverse source problem with observation on the lateral boundary. We derive the uniqueness of the source. © 2012 IOP Publishing Ltd.

Basque Center For Applied Mathematics | Date: 2012-07-18

Scheduling method, so that several jobs (n) can share the resources of at least one processor (1) in a computer system or data-processing system, wherein jobs (n) are grouped in queues or classes (i) known to share the same service requirement distribution function F_(i)(x), the method being characterized in that it comprises the steps of estimating the expected remaining service time of each job (n) in each class (i), selecting at least one job (n) where highest priority is given to the job (n) with the minimum quotient of the estimated expected remaining service time and a preference value assigned to the job (n), removing the selected job (n) from the class (i), executing the selected job (n) for a predetermined time quantum, and, once the time quantum is finished, if the job (n) is not completed, putting the job (n) back in the queue or class (i).

Nakata Y.,Basque Center for Applied Mathematics
Nonlinear Analysis: Real World Applications | Year: 2011

We study the global dynamics of a time delayed epidemic model proposed by Liu et al. (2008) [J. Liu, J. Wu, Y. Zhou, Modeling disease spread via transport-related infection by a delay differential equation, Rocky Mountain J. Math. 38 (5) (2008) 15251540] describing disease transmission dynamics among two regions due to transport-related infection. We prove that if an endemic equilibrium exists then it is globally asymptotically stable for any length of time delay by constructing a Lyapunov functional. This suggests that the endemic steady state for both regions is globally asymptotically stable regardless of the length of the travel time when the disease is transferred between two regions by human transport. © 2011 Elsevier Ltd. All rights reserved.

Method for selecting a transmission channel, within a series of transmission channels in a time division multiple access (TDMA) communications system, to which assign a specific timeslot for communication. As conventional methods, the method comprises the steps of receiving the quality of transmission of each channel C in each time slot, storing C_(i) during a predefined time window T comprising a specific number of timeslots. Contrary to conventional methods, the method of the invention further comprises the step of selecting at least one channel that has the best current quality of transmission C with respect to a function of the total decrease of quality of transmission between each timeslot of the time window T and the current timeslot. Therefore, for a channel to be chosen, the channel must have an acceptable current quality of transmission and must not have had its quality decreased significantly during the time window T.

The invention refers to a method for scaling the speed of operation of at least one processor unit in a virtualized resource-sharing system in order to reduce power consumption, wherein the virtualized system comprises at least one server and a total number of N virtual machines (VMs) running on Na switched-on processor units, by adjusting the speed of at least one processor unit (and preferably of all processor units) to its maximum speed multiplied by a factor that depends on the number of active VMs. Estimations of performance and power consumption are also part of the invention.

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