The Experts below are selected from a list of 207 Experts worldwide ranked by ideXlab platform

Zhan Huashui - One of the best experts on this subject based on the ideXlab platform.

Juan Pablo Agnelli - One of the best experts on this subject based on the ideXlab platform.

  • a posteriori error estimates for elliptic problems with Dirac Measure terms in weighted spaces
    Mathematical Modelling and Numerical Analysis, 2014
    Co-Authors: Juan Pablo Agnelli, Eduardo M Garau, Pedro Morin
    Abstract:

    In this article we develop a posteriori error estimates for general second order elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error Measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clement or Scott-Zhang interpolation operators, without need of suitable modications, and makes use of weighted estimates for fractional integrals and maximal functions. Numerical experiments with an adaptive algorithm yield optimal meshes and very good eectivity indices.

José M. F. Moura - One of the best experts on this subject based on the ideXlab platform.

  • Distributed Kalman Filtering Over Massive Data Sets: Analysis Through Large Deviations of Random Riccati Equations
    IEEE Transactions on Information Theory, 2015
    Co-Authors: Di Li, José M. F. Moura, Vincent H. Poor
    Abstract:

    This paper studies the convergence of the estimation error process and the characterization of the corresponding invariant Measure in distributed Kalman filtering for potentially unstable and large linear dynamic systems. A gossip network protocol termed modified gossip interactive Kalman filtering (M-GIKF) is proposed, where sensors exchange their filtered states (estimates and error covariances) and propagate their observations via intersensor communications of rate γ̅; γ̅ is defined as the averaged number of intersensor message passages per signal evolution epoch. The filtered states are interpreted as stochastic particles swapped through local interaction. This paper shows that the conditional estimation error covariance sequence at each sensor under M-GIKF evolves as a random Riccati equation (RRE) with Markov modulated switching. By formulating the RRE as a random dynamical system, it is shown that the network achieves weak consensus, i.e., the conditional estimation error covariance at a randomly selected sensor converges weakly (in distribution) to a unique invariant Measure. Further, it is proved that as γ̅ → ∞ this invariant Measure satisfies the large deviation (LD) upper and lower bounds, implying that this Measure converges exponentially fast (in probability) to the Dirac Measure δP*, where P* is the stable error covariance of the centralized (Kalman) filtering setup. The LD results answer a fundamental question on how to quantify the rate at which the distributed scheme approaches the centralized performance as the intersensor communication rate increases.

  • Moderate Deviations of a Random Riccati Equation
    IEEE Transactions on Automatic Control, 2012
    Co-Authors: José M. F. Moura
    Abstract:

    The paper characterizes the invariant filtering Measures resulting from Kalman filtering with intermittent observations in which the observation arrival is modeled as a Bernoulli process with packet arrival probability γ̅. Our prior work showed that, for γ̅ >; 0 , the sequence of random conditional error covariance matrices converges weakly to a unique invariant distribution μγ̅. This paper shows that, as γ̅ approaches one, the family {μγ̅}γ̅ >; 0 satisfies a moderate deviations principle with good rate function I (·): (1) as γ̅ ↑ 1 , the family {μγ̅} converges weakly to the Dirac Measure δP* concentrated on the fixed point of the associated discrete time Riccati operator; (2) the probability of a rare event (an event bounded away from P*) under μγ̅ decays to zero as a power law of (1-γ̅) as γ̅↑ 1; and, (3) the best power law decay exponent is obtained by solving a deterministic variational problem involving the rate function I (·). For specific scenarios, the paper develops computationally tractable methods that lead to efficient estimates of rare event probabilities under μγ̅.

Wei Gong - One of the best experts on this subject based on the ideXlab platform.

  • adaptive finite element method for parabolic equations with Dirac Measure
    Computer Methods in Applied Mechanics and Engineering, 2018
    Co-Authors: Wei Gong
    Abstract:

    Abstract In this paper we study the adaptive finite element method for parabolic equations with Dirac Measure. Two kinds of problems with separate Measure data in time and Measure data in space are considered. It is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the Dirac Measure, and thus fit to adaptive FEM for space discretization and variable time steps for time discretization. For both cases we use piecewise linear and continuous finite elements for the space discretization and backward Euler scheme, or equivalently piecewise constant discontinuous Galerkin method, for the time discretization, the a posteriori error estimates based on energy and L 2 norms for the fully discrete problems are then derived to guide the adaptive procedure. Numerical results are provided at the end of the paper to support our theoretical findings.

Catalina Pesce - One of the best experts on this subject based on the ideXlab platform.

  • blow up for the 3 dimensional axially symmetric harmonic map flow into begin document s 2 end document
    Discrete and Continuous Dynamical Systems- Series A, 2019
    Co-Authors: Juan Davila, Manuel Del Pino, Catalina Pesce
    Abstract:

    We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere \begin{document}$ S^2 $\end{document} , \begin{document}$ \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u & = u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) & = u_0 \quad \text{in } \Omega , \end{align*} $\end{document} with \begin{document}$ u(x,t): \bar \Omega\times [0,T) \to S^2 $\end{document} . Here \begin{document}$ \Omega $\end{document} is a bounded, smooth axially symmetric domain in \begin{document}$ \mathbb{R}^3 $\end{document} . We prove that for any circle \begin{document}$ \Gamma \subset \Omega $\end{document} with the same axial symmetry, and any sufficiently small \begin{document}$ T>0 $\end{document} there exist initial and boundary conditions such that \begin{document}$ u(x,t) $\end{document} blows-up exactly at time \begin{document}$ T $\end{document} and precisely on the curve \begin{document}$ \Gamma $\end{document} , in fact \begin{document}$ | {\nabla} u(\cdot ,t)|^2 \rightharpoonup | {\nabla} u_*|^2 + 8\pi \delta_\Gamma \quad\mbox{as}\quad t\to T . $\end{document} for a regular function \begin{document}$ u_*(x) $\end{document} , where \begin{document}$ \delta_\Gamma $\end{document} denotes the Dirac Measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [ 5 , 6 ].

  • blow up for the 3 dimensional axially symmetric harmonic map flow into s2
    arXiv: Analysis of PDEs, 2019
    Co-Authors: Juan Davila, Manuel Del Pino, Catalina Pesce
    Abstract:

    We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= u_b \quad \text{on } \partial \Omega\times(0,T) \\ u(\cdot,0) &= u_0 \quad \text{in } \Omega , \end{align*} with $u(x,t): \bar \Omega\times [0,T) \to S^2$. Here $\Omega$ is a bounded, smooth axially symmetric domain in $\mathbb{R}^3$. We prove that for any circle $\Gamma \subset \Omega$ with the same axial symmetry, and any sufficiently small $T>0$ there exist initial and boundary conditions such that $u(x,t)$ blows-up exactly at time $T$ and precisely on the curve $\Gamma$, in fact $$ |\nabla u(\cdot ,t)|^2 \rightharpoonup |\nabla u_*|^2 + 8\pi \delta_\Gamma \text{ as } t\to T . $$ for a regular function $u_*(x)$, where $\delta_\Gamma$ denotes the Dirac Measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng.