Estimation of second-order properties from jittered time series

This paper considers spectral and autocovariance estimation for a zero-mean, band-limited, stationary process that has been sampled at time points jittered from a regular, equi-interval, sampling scheme. The case of interest is where the sampling scheme is near regular so that the jitter standard deviation is small compared to the sampling interval. Such situations occur with many time series collected in the physical sciences including, in particular, oceanographic profiles.Spectral estimation procedures are developed for the case of independent jitter and autocovariance estimation procedures for both independent and dependent jitter. These are typically modifications of general estimation procedures proposed elsewhere, but tailored to the particular jittered sampling scheme considered. The theoretical properties of these estimators are developed and their relative efficiencies compared.The properties of the jittered sampling point process are also developed. These lead to a better understanding, in this situation, of more general techniques available for processes sampled by stationary point processes.     

Nonequilibrium stationary states of noninteracting electrons in a one-dimensional lattice

Nonequilibrium stationary states of a one-dimensional quantum conductor placed between two reservoirs are investigated. Applying the theory of C^*-algebra, as t→+∞, any state including the degrees of freedom of reservoirs is shown to weakly evolve to a quasifree stationary state with nonvanishing currents. The stationary state exhibits transports which are consistent with nonequilibrium thermodynamics and, in this sense, it has broken time symmetry. Particularly, the electric and energy currents are shown to be expressed by two-probe Landauer-type formulas and they reduce to the results by Sivan–Imry and Bagwell–Orlando in appropriate regimes. As a consequence of the time reversal symmetry, there exists another stationary state with anti-thermodynamical transports, which is the t→−∞ limit of the initial state. The consistency between the dynamical reversibility and the irreversibility of the evolution of states is discussed as well.     

The choice of spectral element basis functions in domains with an axis of symmetry

New spectral element basis functions are constructed for problems possessing an axis of symmetry. In problems defined in domains with an axis of symmetry there is a potential problem of degeneracy of the system of discrete equations corresponding to nodes located on the axis of symmetry. The standard spectral element basis functions are modified so that the axial conditions are satisfied identically. The modified basis is employed only in spectral elements that are adjacent to the axis of symmetry. This modification of the spectral element method ensures that the nodes are the same in each element, which is not the case in other methods that have been proposed to tackle the problem along the axis of symmetry, and that there are no nodes along the axis of symmetry. The problems of Stokes flow past a confined cylinder and sphere are considered and the performance of the original and modified basis functions are compared.     

On stochastic processes in a multilevel control bulk queueing system

This article analyzes some stochastic processes that arise in a bulk single server queue with continuously operating server, state dependent compound Poisson input flow and general state dependent service process. The authors treat the queueing process as a semi–regenerative process and obtain the invariant probability measure and the transient distribution for the embedded Markov chain. The stationary probability measure for the queueing process with continuous time parameter is found by using semi-regenerative techniques. The authors also study the input and output processes and establish ergodic theorems for some functionals of these processes. The results are obtained in terms of the invariant probability measure for the embedded process and the stationary measure for the queueing process with continuous time parameter     

Parallel Symmetric Attack on NTRU using Non-Deterministic Lattice Reduction

Currently, the most efficient passive attack on the NTRU public-key cryptosystem, proposed by Coppersmith and Shamir [1], is based on finding a short enough vector in an integral lattice. An NTRU lattice possesses a cyclic automorphism group whose symmetry may be exploited. We have designed methods for reducing bases of NTRU integral lattices based on this symmetry. In addition to these methods, we use hill-descending techniques to combine new and proposed lattice-reduction algorithms. This approach includes deterministic and non-deterministic components which may be efficiently parallelized.     

Spectral tail processes and max-stable approximations of multivariate regularly varying time series

A regularly varying time series as introduced in Basrak and Segers (2009) is a (multivariate) time series such that all finite dimensional distributions are multivariate regularly varying. The extremal behavior of such a process can then be described by the index of regular variation and the so-called spectral tail process, which is the limiting distribution of the rescaled process, given an extreme event at time 0. As shown in Basrak and Segers (2009), the stationarity of the underlying time series implies a certain structure of the spectral tail process, informally known as the “time change formula”. In this article, we show that on the other hand, every process which satisfies this property is in fact the spectral tail process of an underlying stationary max-stable process. The spectral tail process and the corresponding max-stable process then provide two complementary views on the extremal behavior of a multivariate regularly varying stationary time series.     

Parameterized families of finite difference schemes for the wave equation

This article is devoted to an analysis of simple families of finite difference schemes for the wave equation. These families are dependent on several free parameters, and methods for obtaining stability bounds as a function of these parameters are discussed in detail. Access to explicit stability bounds such as those derived here may, it is hoped, lead to optimization techniques for so‐called spectral‐like methods, which are difference schemes dependent on many free parameters (and for which maximizing the order of accuracy may not be the defining criterion). Though the focus is on schemes for the wave equation in one dimension, the analysis techniques are extended to two dimensions; implicit schemes such as ADI methods are examined in detail. Numerical results are presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 463–480, 2004.     

Hypothesis Tests of Convergence in Markov Chain Monte Carlo

Abstract

Deciding when a Markov chain has reached its stationary distribution is a major problem in applications of Markov Chain Monte Carlo methods. Many methods have been proposed ranging from simple graphical methods to complicated numerical methods. Most such methods require a lot of user interaction with the chain which can be very tedious and time-consuming for a slowly mixing chain. This article describes a system to reduce the burden on the user in assessing convergence. The method uses simple nonparametric hypothesis testing techniques to examine the output of several independent chains and so determines whether there is any evidence against the hypothesis of convergence. We illustrate the proposed method on some examples from the literature.     

A combined BDF‐semismooth Newton approach for time‐dependent Bingham flow

This article is devoted to the numerical simulation of time‐dependent convective Bingham flow in cavities. Motivated by a primal‐dual regularization of the stationary model, a family of regularized time‐dependent problems is introduced. Well posedness of the regularized problems is proved, and convergence of the regularized solutions to a solution of the original multiplier system is verified. For the numerical solution of each regularized multiplier system, a fully discrete approach is studied. A stable finite element approximation in space together with a second‐order backward differentiation formula for the time discretization are proposed. The discretization scheme yields a system of Newton differentiable nonlinear equations in each time step, for which a semismooth Newton algorithm is used. We present two numerical experiments to verify the main properties of the proposed approach. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Sequential Estimation for a Functional of the Spectral Density of a Gaussian Stationary Process

Integral functional of the spectral density of stationary process is an important index in time series analysis. In this paper we consider the problem of sequential point and fixed-width confidence interval estimation of an integral functional of the spectral density for Gaussian stationary process. The proposed sequential point estimator is based on the integral functional replaced by the periodogram in place of the spectral density. Then it is shown to be asymptotically risk efficient as the cost per observation tends to zero. Next we provide a sequential interval estimator, which is asymptotically efficient as the width of the interval tends to zero. Finally some numerical studies will be given.     

The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case

In this article we consider the 3D Primitive Equations (PEs) of the ocean, without viscosity and linearized around a stratified flow. As recalled in the Introduction, the PEs without viscosity ought to be supplemented with boundary conditions of a totally new type which must be nonlocal. In this article a set of boundary conditions is proposed for which we show that the linearized PEs are well-posed. The proposed boundary conditions are based on a suitable spectral decomposition of the unknown functions. Noteworthy is the rich structure of the Primitive Equations without viscosity. Our study is based on a modal decomposition in the vertical direction; in this decomposition, the first mode is essentially a (linearized) Euler flow, then a few modes correspond to a stationary problem partly elliptic and partly hyperbolic; finally all the other modes correspond to a stationary problem fully hyperbolic.     

Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近[英文]

本文研究了Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近问题，构造了定常Navier-Stokes方程对称破坏分歧点扩充系统及其谱Galerkin逼近扩充系统，证明了谱Galerkin逼扩充系统解的存在性和收敛性，从而给出了Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近，并给出了逼近的误差估计。     

On the spectral radius of the Jacobi iteration matrix for a rectangular region with two different media

The spectral radius of the Jacobi iteration matrix plays an important role to estimate the optimum relaxation factor, when the successive overrelaxation (SOR) method is used for solving a linear system. The specific systems are finite difference forms of the Laplace equation satisfied on a rectanglar region with two different media. Though the potential function for the inhomogeneous closed region is continuous, the first order derivative is not continuous. So this requires internal boundary conditions or interface conditions. In this paper, the spectral radius of the Jacobi iteration matrix for the inhomogeneous rectangular region is formulated and the approximation for the explicit formula, suitable for the computation of the spectral radius, is deduced. It is also found by the proposed formula that the spectral radius and the optimum relaxation factor rigorously depend on the inhomogeneity or the internal boundary conditions in the closed region, and especially vary with the position of the internal boundary. These findings are also confirmed by the numerical results of the power method.The stationary iterative method using the proposed formula for calculating estimates of the spectral radius of the Jacobi iteration matrix is compared with Carré's method, Kulstrud's method and the stationary iterative method using Frankel's theoretical formula, all for the case of some numerical models with two different media. According to the results our stationary iterative method gives the best results ffor the estimate of the spectral radius of the Jacobi iteration matrix, for the required number of iterations to calculate solutions, and for the accuracy of the solutions.As a numerical example the microstrip transmission line is taken, the propating mode of which can be approximated by a TEM mode. The cross section includes inhomogeneous media and a strip conductor. Upper and lower bounds of the spectral radius of the Jacobi iteration matrix are estimated. Our method using these estimates is also compared with the other methods. The upper bound of the spectral radius of the Jacobi iteration matrix for more general closed regions with two different media might be given by the proposed formula.     

The classification of Hamiltonian stationary Lagrangian tori in {{\mathbb {CP}}^2} by their spectral data

It is known that all weakly conformal Hamiltonian stationary Lagrangian immersions of tori in ${{\mathbb {CP}}^2}$ may be constructed by methods from integrable systems theory. This article describes the precise details of a construction which leads to a form of classification. The immersion is encoded as spectral data in a similar manner to the case of minimal Lagrangian tori in ${{\mathbb {CP}}^2}$ , but the details require a careful treatment of both the ??dressing construction?? and the spectral data to deal with a loop of flat connexions which is quadratic in the loop parameter.     

A posteriori error estimates for discontinuous Galerkin method to the elasticity problem

This work concerns with the discontinuous Galerkin (DG) method for the time‐dependent linear elasticity problem. We derive the a posteriori error bounds for semidiscrete and fully discrete problems, by making use of the stationary elasticity reconstruction technique which allows to estimate the error for time‐dependent problem through the error estimation of the associated stationary elasticity problem. For fully discrete scheme, we make use of the backward‐Euler scheme and an appropriate space‐time reconstruction. The technique here can be applicable for a variety of DG methods as well.     

Generation of strongly non-Gaussian stochastic processes by iterative scheme upgrading phase and amplitude contents

Random excitations, such as wind velocity, always exhibit non-Gaussian features. Sample realisations of stochastic processes satisfying given features should be generated, in order to perform the dynamical analysis of structures under stochastic loads based on the Monte Carlo simulation. In this paper, an efficient method is proposed to generate stationary non-Gaussian stochastic processes. It involves an iterative scheme that produces a class of sample processes satisfying the following conditions. (1) The marginal cumulative distribution function of each sample process is perfectly identical to the prescribed one. (2) The ensemble-averaged power spectral density function of these non-Gaussian sample processes is as close to the prescribed target as possible. In this iterative scheme, the underlying processes are generated by means of the spectral representation method that recombines the upgraded power spectral density function with the phase contents of the new non-Gaussian processes in the latest iteration. Numerical examples are provided to demonstrate the capabilities of the proposed approach for four typical non-Gaussian distributions, some of which deviate significantly from the Gaussian distribution. It is found that the estimated power spectral density functions of non-Gaussian processes are close to the target ones, even for the extremely non-Gaussian case. Furthermore, the capability of the proposed method is compared to two other methods. The results show that the proposed method performs well with convergence speed, accuracy, and random errors of power spectral density functions.     

Rotation Techniques in Asymmetric Multidimensional Scaling

Multidimensional scaling (MDS) is a set of techniques, used especially in behavioral and social sciences, that enable a researcher to visualize proximity data in a multidimensional space. This article focuses on a particular class of MDS models proposed to deal with proximities which describe asymmetric relationships (i.e., trade indices for a set of countries, brand switching data, occupational mobility tables, and so on). They are based on the decomposition of the relationships into a symmetric and a skew-symmetric part. In this way the objects are represented as points in a multidimensional space and the intensity of their relationships as scalar products (symmetry) or triangle areas (skew-symmetry). These models are seen as special cases of a general model and their rotational indeterminacy is investigated. The aim is to propose a rotation method that makes easier the visual inspection of the graphical representation, highlighting the simple structure of the data. In particular an orthomax-like family of rotation methods and a general algorithm are proposed. Advantages of the proposal are illustrated by analysis of import-export data.     

Numerical methods for a fixed domain formulation of the glacier profile problem with alternative boundary conditions

In this paper we develop a set of numerical techniques for the simulation of the profile evolution of a valley glacier in the framework of isothermal shallow ice approximation models. The different mathematical formulations are given in terms of a highly nonlinear parabolic equation. A first nonlinearity comes from the free boundary problem associated with the unknown basal extension of the glacier region. This feature is treated using a fixed domain complementarity formulation which is solved numerically by a duality method. The nonlinear diffusive term is explicitly treated in the time marching scheme. A convection dominated problem arises, so a characteristic scheme is proposed for the time discretization, while piecewise linear finite elements are used for the spatial discretization. The presence of infinite slopes in polar regimes motivates an alternative formulation based on a prescribed flux boundary condition at the head of the glacier instead a homogeneous Dirichlet one. Finally, several numerical examples illustrate the performance of the proposed methods.     

WEB‐spline based mesh‐free finite element analysis for the heat equation and the time‐dependent Navier–Stokes equation: A survey

In this article, we give some numerical techniques and error estimates using web‐spline based mesh‐free finite element method for the heat equation and the time‐dependent Navier–Stokes equations on bounded domains. The web‐spline method uses weighted extended B‐splines on a regular grid as basis functions and does not require any grid generation. We demonstrate the method by providing numerical results for the Poisson's and stationary Stokes equation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On bounded solutions of transport equation solved with a spectral method

In this article, a spectral method accompanied by finite difference method has been proposed for solving a boundary value problem that accompanies a stationary transport equation. We also prove that the solution is bounded by a value that depends of the source function. The accuracy and computational efficiency of the proposed method are verified with the help of a numerical example. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012