A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.     

Nonlinear time-series modeling and prediction using correlation analysis

We present a novel way for time series prediction. The method is based on the correlation analysis and allows for handling nonlinearities of different type and character. The presented approach results in an approximation model that combines nonlinear units taken from radial basis functions (RBF) and from multilayer perceptrons (MLP). The approach leads to a low mean error of the approximation with a number of parameters significantly smaller when compared to RBF and MLP. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.     

Step size control in the numerical solution of stochastic differential equations

We introduce a variable step size algorithm for the pathwise numerical approximation of solutions to stochastic ordinary differential equations. The algorithm is based on a new pair of embedded explicit Runge-Kutta methods of strong order 1.5(1.0), where the method of strong order 1.5 advances the numerical computation and the difference between approximations defined by the two methods is used for control of the local error. We show that convergence of our method is preserved though the discretization times are not stopping times any more, and further, we present numerical results which demonstrate the effectiveness of the variable step size implementation compared to a fixed step size implementation.     

Adaptive Mixture Modeling Metropolis Methods for Bayesian Analysis of Nonlinear State-Space Models

We describe a strategy for Markov chain Monte Carlo analysis of nonlinear, non-Gaussian state-space models involving batch analysis for inference on dynamic, latent state variables and fixed model parameters. The key innovation is a Metropolis–Hastings method for the time series of state variables based on sequential approximation of filtering and smoothing densities using normal mixtures. These mixtures are propagated through the nonlinearities using an accurate, local mixture approximation method, and we use a regenerating procedure to deal with potential degeneracy of mixture components. This provides accurate, direct approximations to sequential filtering and retrospective smoothing distributions, and hence a useful construction of global Metropolis proposal distributions for simulation of posteriors for the set of states. This analysis is embedded within a Gibbs sampler to include uncertain fixed parameters. We give an example motivated by an application in systems biology. Supplemental materials provide an example based on a stochastic volatility model as well as MATLAB code.     

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments.     

Weighted norms in subspace‐based methods for time series analysis

Many modern approaches of time series analysis belong to the class of methods based on approximating high‐dimensional spaces by low‐dimensional subspaces. A typical method would embed a given time series into a structured matrix and find a low‐dimensional approximation to this structured matrix. The purpose of this paper is twofold: (i) to establish a correspondence between a class of SVD‐compatible matrix norms on the space of Hankel matrices and weighted vector norms (and provide methods to construct this correspondence) and (ii) to motivate the importance of this for problems in time series analysis. Examples are provided to demonstrate the merits of judiciously selecting weights on imputing missing data and forecasting in time series. Copyright © 2016 John Wiley & Sons, Ltd.     

A local least-squares method for solving nonlinear partial differential equations of second order

In this paper a mesh-free method for the treatment of time-independent and time-dependent nonlinear PDEs of second order is presented. The basic idea of the discretization is a local least-squares approximation, similar to the moving least-squares approach in data approximation. However, in our approach the PDE is incorporated as an additional minimization constraint. The discretization leads to a fixed-point problem, which is solved by iteration. Because of the local nature of the method only small dimensional matrix inversions have to be done. The approximation error of the discretization—even on unstructured meshes—is comparable to respective versions of finite elements. As a by-product the method provides an a posteriori measure for the local approximation error. We discuss implementational aspects and present numerical simulations.     

On orthogonal polynomial approximation with the dimensional expanding technique for precise time integration in transient analysis

We use four orthogonal polynomial series, Legendre, Chebyshev, Hermite and Laguerre series, to approximate the non-homogeneous term for the precise time integration and incorporate them with the dimensional expanding technique. They are applied to various structures subjected to transient dynamic loading together with Fourier and Taylor approximation proposed in previous works. Numerical examples show that all six methods are efficient and have reasonable precision. In particular, Legendre approximation has much higher precision and better convergence; Chebyshev approximation is also good, but only slightly inferior to Legendre approximation. The other four approximation methods usually produce results with errors hundreds of thousands of times larger. Hermite and Laguerre approximation may be useful for some special non-homogeneous terms, but do not work sufficiently well in our numerical examples. Other contributions of this paper include, a Dynamic Programming scheme for computing series coefficients, a general formula to find the assistant matrix for any polynomial series.     

Local Lyapunov exponents computed from observed data

Summary We develop methods for determining local Lyapunov exponents from observations of a scalar data set. Using average mutual information and the method of false neighbors, we reconstruct a multivariate time series, and then use local polynomial neighborhood-to-neighborhood maps to determine the phase space partial derivatives required to compute Lyapunov exponents. In several examples we demonstrate that the methods allow one to accurately reproduce results determined when the dynamics is known beforehand. We present a new recursive QR decomposition method for finding the eigenvalues of products of matrices when that product is severely ill conditioned, and we give an argument to show that local Lyapunov exponents are ambiguous up to order 1/L in the number of steps due to the choice of coordinate system. Local Lyapunov exponents are the critical element in determining the practical predictability of a chaotic system, so the results here will be of some general use.     

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure

In this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration. Compared with other combined methods, this method can improve the efficiency of computation, deal with the discontinuous problem well and keep local mass balance. We study the convergence of this method and give the corresponding optimal error estimates in L^∞(L²) for velocity and concentration and the super convergence in L^∞(H¹) for pressure. Finally, we also present some numerical examples to confirm our theoretical analysis.     

Variational Bayes Estimation of Discrete-Margined Copula Models With Application to Time Series

We propose a new variational Bayes (VB) estimator for high-dimensional copulas with discrete, or a combination of discrete and continuous, margins. The method is based on a variational approximation to a tractable augmented posterior and is faster than previous likelihood-based approaches. We use it to estimate drawable vine copulas for univariate and multivariate Markov ordinal and mixed time series. These have dimension rT, where T is the number of observations and r is the number of series, and are difficult to estimate using previous methods. The vine pair-copulas are carefully selected to allow for heteroscedasticity, which is a feature of most ordinal time series data. When combined with flexible margins, the resulting time series models also allow for other common features of ordinal data, such as zero inflation, multiple modes, and under or overdispersion. Using six example series, we illustrate both the flexibility of the time series copula models and the efficacy of the VB estimator for copulas of up to 792 dimensions and 60 parameters. This far exceeds the size and complexity of copula models for discrete data that can be estimated using previous methods. An online appendix and MATLAB code implementing the method are available as supplementary materials.     

Un exemple d'application des grandes déviations à l'approximation stochastique

We present an application of the large deviations theory to stochastic approximation, for the one-stage Robbins-Monro algorithm, as well as for the averaging method (Polyak-Ruppert). The study deals with a continuous time linear regression model where the noise is a continuous local martingale.     

Numerical solutions of the Laplace’s equation

This paper uses the sinc methods to construct a solution of the Laplace’s equation using two solutions of the heat equation. A numerical approximation is obtained with an exponential accuracy. We also present a reliable algorithm of Adomian decomposition method to construct a numerical solution of the Laplace’s equation in the form a rapidly convergence series and not at grid points. Numerical examples are given and comparisons are made to the sinc solution with the Adomian decomposition method. The comparison shows that the Adomian decomposition method is efficient and easy to use.     

A UNIFIED FRAMEWORK FOR SOME INEXACT PROXIMAL POINT ALGORITHMS*

We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximal point subproblems. Our development further enhances the constructive approximation approach of the recently proposed hybrid projection–proximal and extragradient–proximal methods. Specifically, we introduce an even more flexible error tolerance criterion, as well as provide a unified view of these two algorithms. Our general method possesses global convergence and local (super)linear rate of convergence under standard assumptions, while using a constructive approximation criterion suitable for a number of specific implementations. For example, we show that close to a regular solution of a monotone system of semismooth equations, two Newton iterations are sufficient to solve the proximal subproblem within the required error tolerance. Such systems of equations arise naturally when reformulating the nonlinear complementarity problem.

     

An improved local error estimator for symmetric time-stepping schemes

We propose a symmetrized version of the defect to be used in the estimation of the local time-stepping error of symmetric one-step methods for the time propagation of linear autonomous evolution equations. Using the anti-commutator of the numerical flow and the right-hand side operator in the definition of the defect of the numerical approximation, a local error estimator is obtained which has higher accuracy asymptotically than an established version using the common defect. This theoretical result is illustrated for a splitting method applied to a linear Schrödinger equation.     

The Optimal Discretization of Stochastic Differential Equations

We study pathwise approximation of scalar stochastic differential equations. The mean squared L₂-error and the expected number n of evaluations of the driving Brownian motion are used for the comparison of arbitrary methods. We introduce an adaptive discretization that reflects the local properties of every single trajectory. The corresponding error tends to zero like c·n^−1/2, where c is the average of the diffusion coefficient in space and time. Our method is justified by the matching lower bound for arbitrary methods that are based on n evaluations on the average. Hence the adaptive discretization is asymptotically optimal. The new method is very easy to implement, and about 7 additional arithmetical operations are needed per evaluation of the Brownian motion. Hereby we can determine the complexity of pathwise approximation of stochastic differential equations. We illustrate the power of our method already for moderate accuracies by means of a simulation experiment.     

On zero locations of predictor polynomials

Predictor polynomials are often used in linear prediction methods mainly for extracting properties of physical systems which are described by time series. The aforementioned properties are associated with a few zeros of large polynomials and for this reason the zero locations of those polynomials must be analyzed. We present a linear algebra approach for determining the zero locations of predictor polynomials, which enables us to generalize some early results obtained by Kumaresan in the signal analysis field. We also present an analysis of zero locations for time series having multiple zeros. © 1997 by John Wiley & Sons, Ltd.     

Approximations of non-smooth integral type functionals of one dimensional diffusion processes

In this article, we obtain the weak and strong rates of convergence of time integrals of non-smooth functions of a one dimensional diffusion process. We propose the use of the exact simulation scheme to simulate the process at discretization points. In particular, we also present the rates of convergence for the weak and strong errors of approximation for the local time of a one dimensional diffusion process as an application of our method.     

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.