Fluid approximation for generalized Jackson network with vacations

Using a bounding technique, we prove that the fluid model of generalized Jackson network (GJN) with vacations is the same as a GJN without vacations, which means that vacation mechanism does not affect the dynamic performance of GJN under fluid approximation. Furthermore, in order to present the impact of vacation on the performance of GJN, we show that exponential rate of convergence for fluid approximation only holds for large N, which is different from a GJN without vacations. The results on fluid approximation and convergence rate are embodied by the queue length, workload, and busy time processes.     

Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data

The aim of this paper is to answer an important issue in quantum mechanics, namely to estimate the purity of a quantum state of a light beam. Estimation of the purity is based on the results of quantum homodyne measurements performed on independent identically prepared quantum systems. The quantum state of the light is entirely characterized by the Wigner function, which can take negative values and must satisfy certain constraints of positivity imposed by quantum physics. We estimate the integrated squared Wigner function by a kernel-based second order U — statistic. This quadratic functional is a physical measure of the purity of the state. We also give an adaptive estimator, which does not depend on the smoothness parameters. We establish upper bounds of the minimax risk over a class of infinitely differentiable functions.      

Stabilization method for continuous approximation of transient convection problem

The aim of this work is to study a new finite element (FE) formulation for the approximation of nonsteady convection equation. Our approximation scheme is based on the Streamline Upwind Petrov Galerkin (SUPG) method for space variable, x, and a modified of the Euler implicit method for time variable, t. The most interest for this scheme lies in its application to resolve by continuous (FE) method the complex of viscoelastic fluid flow obeying an Oldroyd‐B differential model; this constituted our aim motivation and allows us to treat the constitutive law equation, which expresses the relation between the stress tensor and the velocity gradient and includes tensorial transport term. To make the analysis of the method more clear, we first study, in this article this modified method for the advection equation. We point out the stability of this new method and the error estimate of the approximation solution is discussed. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Dynamic permeability of an assemblage of soft spherical particles

Under oscillatory Stokes flow, dynamic permeability of assemblage of soft spherical particles is derived. For the bed of soft particles, the fluid‐particle system is represented as an assemblage of uniform permeable spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Oscillatory Stokes equations are employed inside the fluid envelope, and oscillatory Brinkman equations are used inside the porous region. Four known boundary conditions namely: Happel's, Kuwabara's, Kvashnin's, and Cunningham's are considered on the outer boundary and results are compared. The behavior of dynamic permeability is analyzed with various parameters such as Darcy number (Da), frequency parameter (?), porosity (φ), and viscosity ratio (δ). Copyright © 2013 John Wiley & Sons, Ltd.     

A graphon approach to limiting spectral distributions of Wigner‐type matrices

We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner‐type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes transforms explicitly with weaker assumptions on the convergence of variance profiles than previous results. As applications, we give a new proof of the semicircle law for generalized Wigner matrices and determine the limiting spectral distributions for three sparse inhomogeneous random graph models with sparsity ω(1/n): inhomogeneous random graphs with roughly equal expected degrees, W‐random graphs and stochastic block models with a growing number of blocks. Furthermore, we show our theorems can be applied to random Gram matrices with a variance profile for which we can find the limiting spectral distributions under weaker assumptions than previous results.     

Kinetic Theory of Quantum Electrodynamic Plasma in a Strong Electromagnetic Field: II. The Covariant Mean-Field Approximation

A covariant kinetic equation for the matrix Wigner function is derived in the mean-field approximation from a general kinetic equation for the fermionic subsystem of a quantum electrodynamic plasma. We show that in the semiclassical limit, the equations for the components of the Wigner function can be transformed into closed kinetic equations for the Lorentz-invariant distribution functions of particles and antiparticles.     

Dynamic Routing and Admission Control in High-Volume Service Systems: Asymptotic Analysis via Multi-Scale Fluid Limits

Motivated by applications in telephone call centers, we consider a service system model with m customer classes and r server pools. The model is one with doubly stochastic arrivals, which means that the m-vector λ of instantaneous arrival rates is allowed to vary both temporally and stochastically. Two levels of dynamic control are considered: customers may be either blocked or accepted at the time of their arrival, and then accepted customers of each class must be routed, either immediately upon acceptance or after some period of waiting, to a server pool that is qualified to handle that class. Customers who are made to wait before commencement of their service are liable to defect. The objective is to minimize the expected sum of blocking costs, waiting costs and defection costs over a fixed and finite planning horizon. We consider an asymptotic parameter regime in which (i) the arrival rates, service rates and defection rates are uniformly accelerated by a large factor κ, then (ii) arrival rates are increased by an additional factor g(κ), and the number of servers in each pool is increased by g(κ) as well. This produces a separation of time scales, justifying a pointwise stationary stochastic fluid approximation for our original system model. In the stochastic fluid approximation, optimal admission control and routing decisions are determined by a simple linear program that uses the current arrival rate vector λ as data. We explain how to implement the fluid model's optimal control policy in our original service system context, and prove that the proposed implementation is asymptotically optimal in the first-order sense. AMS subject classification: 60K30, 90B15, 90B36     

Numerical analysis of a locking‐free mixed finite element method for a bending moment formulation of Reissner‐Mindlin plate model

This article deals with the approximation of the bending of a clamped plate, modeled by Reissner‐Mindlin equations. It is known that standard finite element methods applied to this model lead to wrong results when the thickness t is small. Here, we propose a mixed formulation based on the Hellinger‐Reissner principle which is written in terms of the bending moments, the shear stress, the rotations and the transverse displacement. To prove that the resulting variational formulation is well posed, we use the Babu?ka‐Brezzi theory with appropriate t ‐dependent norms. The problem is discretized by standard mixed finite elements without the need of any reduction operator. Error estimates are proved. These estimates have an optimal dependence on the mesh size h and a mild dependence on the plate thickness t. This allows us to conclude that the method is locking‐free. The proposed method yields direct approximation of the bending moments and the shear stress. A local postprocessing leading to H¹ ‐type approximations of transverse displacement and rotations is introduced. Moreover, we propose a hybridization procedure, which leads to solving a significantly smaller positive definite system. Finally, we report numerical experiments which allow us to assess the performance of the method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Accurate pressure post-process of a finite element method for elastoacoustics

Summary. This paper deals with a post-process to obtain a more accurate approximation of the fluid pressure from a finite element computation of the vibration modes of a fluid-structure coupled system. The underlying finite element method, based on a displacement formulation for both media, consists of using Raviart-Thomas elements for the fluid combined with standard continuous elements for the solid. An easy to compute post-process of the pressure is derived. The relation between this post-process and an alternative finite element approximation of the problem based on discretizing the fluid pressure by enriched Crouzeix-Raviart elements is studied. Higher order estimates for the L² norm of the post-processed pressure are proved by exploiting this relation. As a by-product, higher order L² estimates for the solid displacements obtained with the original method are also proved.Member of CIC, Provincia de Buenos Aires, ArgentinaMember of CONICET, Argentina. Partially supported by FONDECYT 7.990.075 and FONDAP in Applied Mathematics, ChilePartially supported by FONDECYT 1.990.346 and FONDAP in Applied Mathematics, Chile     

Cross approximation in tensor electron density computations

We propose new tensor approximation algorithms for certain discrete functions related with Hartree–Fock/Kohn–Sham equations. Given a canonical tensor representation for the electron density function (for example, produced by quantum chemistry packages such as MOLPRO), we obtain its Tucker approximation with much fewer parameters than the input data and the Tucker approximation for the cubic root of this function, which is part of the Kohn–Sham exchange operator. The key idea is in the fast and accurate prefiltering of possibly large‐scale factors of the canonical tensor input. The new algorithms are based on the incomplete cross approximation method applied to matrices and tensors of order 3 and outperform other tools for the same purpose. First, we show that the cross approximation method is robust and much faster than the singular value decomposition‐based approach. As a consequence, it becomes possible to increase the resolution of grid and the complexity of molecules that can be handled by the Hartree–Fock chemical models. Second, we propose a new fast approximation method for f^1/3(x, y, z), based on the factor prefiltering method for f(x, y, z) and certain mimic approximation hypothesis. Third, we conclude that the Tucker format has advantages in the storage and computation time compared with the ubiquitous canonical format. Copyright © 2009 John Wiley & Sons, Ltd.     

On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures

Two dimensional diffuse interface model for a chemically reacting incompressible binary fluid in a bounded domain is considered. The corresponding evolution system consists of the Navier–Stokes equations for the (averaged) fluid velocity that are nonlinearly coupled with a convective Cahn–Hilliard–Oono type equation for the difference ψ of two fluid concentrations. The effects of a (reversible) chemical reaction is represented in the latter equation by an additional term of the form ε(ψ ? c₀), ε > 0. Here, c₀ is the stationary spatial average of ψ, provided that, for example, no‐slip and no‐flux boundary conditions are considered. The mass is not necessarily conserved unless the spatial average of the initial datum for ψ coincides with c₀. When ε = 0 (i.e., no chemical reaction), the model reduces to the well‐known Cahn–Hilliard–Navier–Stokes system, which has been investigated by several authors. Here, we want to show that the global dynamic behavior of the system is robust with respect to ε. More precisely, we construct a family of exponential attractors, which is continuous with respect to ε. Copyright © 2013 John Wiley & Sons, Ltd.     

On the advantage over a random assignment

We initiate the study of a new measure of approximation. This measure compares the performance of an approximation algorithm to the random assignment algorithm. This is a useful measure for optimization problems where the random assignment algorithm is known to give essentially the best possible polynomial time approximation. In this paper, we focus on this measure for the optimization problems Max‐Lin‐2 in which we need to maximize the number of satisfied linear equations in a system of linear equations modulo 2, and Max‐k‐Lin‐2, a special case of the above problem in which each equation has at most k variables. The main techniques we use, in our approximation algorithms and inapproximability results for this measure, are from Fourier analysis and derandomization. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Approximation of the incompressible non‐Newtonian fluid equations by the artificial compressibility method

This paper studies the approximation of the non‐Newtonian fluid equations by the artificial compressibility method. We first introduce a family of perturbed compressible non‐Newtonian fluid equations (depending on a positive parameter ε) that approximates the incompressible equations as ε → 0⁺. Then, we prove the unique existence and convergence of solutions for the compressible equations to the solutions of the incompressible equations. Copyright © 2012 John Wiley & Sons, Ltd.     

Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors

We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared pulses. The state is represented through the Wigner function, a “quasi-probability density” on ℝ² which may take negative values and must satisfy intrinsic positivity constraints imposed by quantum physics. The data consists of n i.i.d. observations from a probability density equal to the Radon transform of the Wigner function. We construct an estimator for the Wigner function and prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions. A similar result was previously derived by Cavalier in the context of positron emission tomography. Our work extends this result to the space of smooth Wigner functions, which is the relevant parameter space for quantum homodyne tomography.      

Weyl quantization of Lebesgue spaces

We study boundedness and compactness properties for the Weyl quantization with symbols in L^q (?^2d ) acting on L^p (?^d ). This is shown to be equivalent, in suitable Banach space setting, to that of the Wigner transform. We give a short proof by interpolation of Lieb's sufficient conditions for the boundedness of the Wigner transform, proving furthermore that these conditions are also necessary. This yields a complete characterization of boundedness for Weyl operators in L^p setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L_*^q (?^2d ) and L_?^q (R^2d ), respectively smaller and larger than the L^q (?^2d ),and showing that the Weyl correspondence is bounded on L_*^q (R^2d ) (and yields compact operators), whereas it is not on L_?^q (R^2d ). We conclude with a remark on weak‐type L^p boundedness (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sampling and multilevel coarsening algorithms for fast matrix approximations

This paper addresses matrix approximation problems for matrices that are large, sparse, and/or representations of large graphs. To tackle these problems, we consider algorithms that are based primarily on coarsening techniques, possibly combined with random sampling. A multilevel coarsening technique is proposed, which utilizes a hypergraph associated with the data matrix and a graph coarsening strategy based on column matching. We consider a number of standard applications of this technique as well as a few new ones. Among standard applications, we first consider the problem of computing partial singular value decomposition, for which a combination of sampling and coarsening yields significantly improved singular value decomposition results relative to sampling alone. We also consider the column subset selection problem, a popular low‐rank approximation method used in data‐related applications, and show how multilevel coarsening can be adapted for this problem. Similarly, we consider the problem of graph sparsification and show how coarsening techniques can be employed to solve it. We also establish theoretical results that characterize the approximation error obtained and the quality of the dimension reduction achieved by a coarsening step, when a proper column matching strategy is employed. Numerical experiments illustrate the performances of the methods in a few applications.     

Mathematical and numerical modellization of large‐scale oceanic waves

This paper is devoted to the theoretical and numerical study of a method which computes the variability of current and density in an oceanic domain. The equations are of Navier–Stokes type for the velocity and of transport‐diffusion type for the density. They are linearized around a given mean circulation and modified by physical assumptions including hydrostatic approximation. The existence and uniqueness of a solution are proved for two sets of equations: first the three‐dimensional problem and then the two‐dimensional cyclic problem derived by assuming a sinusoïdal x‐dependence for the perturbation of the mean flow. The latter corresponds to a modellization of tropical instability waves which are illustrated by the ‘El Nino’ phenomenon. These two problems differ from classical ones because of hydrostatic approximation, boundary conditions imposed by the oceanic domain and complex‐valued functions for the cyclic case. A numerical model is developed for the two‐dimensional cyclic equations. Time discretization is performed by the characteristics method; space discretization uses Q₁ finite elements. Numerical results are presented in a realistic case corresponding to the tropical Pacific Ocean. Copyright © 1999 John Wiley & Sons. Ltd.     

On the Spectral Distribution of Gaussian Random Matrices

We consider the empirical spectral distribution （ESD） of a random matrix from the Gaussian Unitary Ensemble. Based on the Plancherel-Rotaeh approximation formula for Hermite polynomials, we prove that the expected empirical spectral distribution converges at the rate of O（n^-1） to the Wigner distribution function uniformly on every compact intervals [u,v] within the limiting support （-1, 1）. Furthermore, the variance of the ESD for such an interval is proved to be （πn）^-2 logn asymptotically which surprisingly enough, does not depend on the details （e.g. length or location） of the interval, This property allows us to determine completely the covariance function between the values of the ESD on two intervals.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

Efficient approximation of random fields for numerical applications

This article is dedicated to the rapid computation of separable expansions for the approximation of random fields. We consider approaches based on techniques from the approximation of non‐local operators on the one hand and based on the pivoted Cholesky decomposition on the other hand. Especially, we provide an a posteriori error estimate for the pivoted Cholesky decomposition in terms of the trace. Numerical examples are provided to validate and quantify the presented methods. Copyright © 2015 John Wiley & Sons, Ltd.