Asymptotic Convergence Rate Research Articles

Globalization of convergence of the constrained piecewise Levenberg–Marquardt method

We develop linesearch algorithms intended for globalization of convergence of the piecewise Levenberg–Marquardt method for constrained piecewise smooth equation. Conditions ensuring global convergence properties and asymptotic superlinear convergence rate are proposed. The peculiarities of the global convergence results in the piecewise smooth case are discussed and illustrated by examples. We also provide numerical results for unconstrained and constrained reformulations of nonlinear complementarity problems, comparing the performance of the globalized piecewise Levenberg–Marquardt algorithm with some relevant alternatives.

Variable Selection in Semi-Functional Partially Linear Regression Models with Time Series Data

This article investigates a variable selection method in semi-functional partially linear regression (SFPLR) models for strong α-mixing functional time series data. We construct penalized least squares estimators for unknown parameters and unknown link functions in our models. Under some regularity assumptions, we establish the asymptotic convergence rate and asymptotic distribution for the proposed estimators. Furthermore, we make a comparison of our variable selection method with the oracle method without variable selection in simulation studies and an electricity consumption data analysis. Simulation experiments and real data analysis results indicate that the variable selection method performs well at extracting the primary information and reducing dimensionality.

Estimation in functional partially linear spatial autoregressive model

Functional regression has been a hot topic in statistical research. However, not much work has been done when response variables are cross-sectionally dependent variables and explanatory variables contain a real-valued scalar variable and a functional-valued random variable. In this paper, we consider a new functional partially linear spatial autoregressive model. Based on the functional principal components analysis and basis function approximation, we obtain the estimators of the unknown parameter and functions through the instrumental variables estimation method. The asymptotic normality and convergence rates of estimators are proved under some mild conditions. In addition, we illustrate the finite sample performance of the proposed estimation method through simulation study and a real data analysis.

A robust cut‐cell finite element method for Poisson's equation in three dimensions

SummaryThis article documents a cut‐cell finite element method for solving Poisson's equation in smooth three‐dimensional domains using a uniform, Cartesian axis‐aligned grid. Neumann boundary conditions are imposed weakly by way of a Delaunay triangulation, while Dirichlet boundary conditions are imposed strongly using a projection method. A set of numerical simulations demonstrates the proposed method is robust and preserves the asymptotic rate of convergence expected of corresponding body‐fitted methods.

Asymptotic stability of rarefaction wave and boundary layer for outflow problem on the viscous vasculogenesis model *

In this paper, we are concerned with the outflow problem on a simplified viscous vasculogenesis model in the half-line R+ . Firstly, we establish the global-in-time asymptotic stability of the rarefaction wave. Secondly, we obtain the unique existence and decay property of the boundary layer by using stable manifold theorem. Moreover, the asymptotic stability and convergence rates of solution towards boundary layer are obtained. The appearance of concentration makes the stationary problem more difficult than Navier–Stokes equations or Navier–Stokes–Poisson equations.

Application of efficient algorithm based on block Newton method to elastoplastic problems with nonlinear kinematic hardening

PurposeThe purpose of this study is to present the effectiveness and robustness of a numerical algorithm based on the block Newton method for the nonlinear kinematic hardening rules adopted in modeling ductile materials.Design/methodology/approachElastoplastic problems can be defined as a coupled problem of the equilibrium equation for the overall structure and the yield equations for the stress state at every material point. When applying the Newton method to the coupled residual equations, the displacement field and the internal variables, which represent the plastic deformation, are updated simultaneously.FindingsThe presented numerical scheme leads to an explicit form of the hardening behavior, which includes the evolution of the equivalent plastic strain and the back stress, with the internal variables. The features of the present approach allow the displacement field and the hardening behavior to be updated straightforwardly. Thus, the scheme does not have any local iterative calculations and enables us to simultaneously decrease the residuals in the coupled boundary value problems.Originality/valueA pseudo-stress for the local residual and an algebraically derived consistent tangent are applied to elastic-plastic boundary value problems with nonlinear kinematic hardening. The numerical procedure incorporating the block Newton method ensures a quadratic rate of asymptotic convergence of a computationally efficient solution scheme. The proposed algorithm provides an efficient and robust computation in the elastoplastic analysis of ductile materials. Numerical examples under elaborate loading conditions demonstrate the effectiveness and robustness of the numerical scheme implemented in the finite element analysis.

Comparative study on excess distribution estimation in iid settings

. This study considers excess distribution estimation in iid settings. There are two ways for the estimation; the fitting to the generalized Pareto distribution and the fully non parametric estimation. The fitting estimator is justified by the approximation proven in the extreme value theory; however, the accuracy depends on how extremely large the target is. The non parametric estimator does not need an approximation and has the advantage of wide applicability. This study conducts both theoretical and numerical comparative study on excess distribution estimation. Asymptotic convergence rates of two estimators are obtained, and the mean integrated squared errors are numerically surveyed by simulation study. An illustrative example of Abisko rainfall amount is presented.

Computational Requirements for Modeling Thermal Conduction in Polymeric Phase-Change Materials: Periodic Hard Spheres Case.

This research focuses on modeling heat transfer in heterogeneous media composed of stacked spheres of paraffin as a perspective polymeric phase-change material. The main goal is to study the requirements of the numerical scheme to correctly predict the thermal conductivity in a periodic system composed of an indefinitely repeated configuration of spherical particles subjected to a temperature gradient. Based on OpenFOAM, a simulation platform is created with which the resolution requirements for accurate heat transfer predictions were inferred systematically. The approach is illustrated for unit cells containing either a single sphere or a configuration of two spheres. Asymptotic convergence rates confirming the second-order accuracy of the method are established in case the grid is fine enough to have eight or more grid cells covering the distance of the diameter of a sphere. Configurations with two spheres can be created in which small gaps remain between these spheres. It was found that even the under-resolution of these small gaps does not yield inaccurate numerical solutions for the temperature field in the domain, as long as one adheres to using eight or more grid cells per sphere diameter. Overlapping and (barely) touching spheres in a configuration can be simulated with high fidelity and realistic computing costs. This study further extends to examine the effective thermal conductivity of the unit cell, particularly focusing on the volume fraction of paraffin in cases with unit cells containing a single sphere. Finally, we explore the dependence of the effective thermal conductivity for unit cells containing two spheres at different distances between them.

On the discrepancies between POD and Fourier modes on aperiodic domains

The application of Fourier analysis in combination with the Proper Orthogonal Decomposition (POD) is investigated. In this approach to turbulence decomposition, which has recently been termed Spectral POD (SPOD), Fourier modes are considered as solutions to the corresponding Fredholm integral equation of the second kind along homogeneous-periodic or homogeneous coordinates. In the present work, the notion that the POD modes formally converge to Fourier modes for increasing domain length is challenged. A family of analytical correlation functions parametrized by the Taylor macro/micro scale ratio (MMSR) is investigated numerically. The results of the analysis indicate that the discrepancy between POD and Fourier modes along locally translationally invariant coordinates is coupled to the MMSR of the flow. Increasing discrepancies are observed for smaller MMSRs, which are characteristic of low Reynolds number flows. The asymptotic convergence rate of the eigenspectrum matches the corresponding convergence rate of the exact analytical Fourier spectrum of the kernel in question, even for extremely small domains and small MMSRs where the corresponding DFT spectra suffer heavily from windowing effects. The Taylor micro scales are consistently underestimated when reconstructed using Fourier modes—failing to converge to the correct value even if all Fourier modes are used—while these are accurately reconstructed using POD modes.

Proof of Reliability Convergence to 1 at Rate of Spearman–Brown Formula for Random Test Forms and Irrespective of Item Pool Dimensionality

It is shown that the psychometric test reliability, based on any true-score model with randomly sampled items and uncorrelated errors, converges to 1 as the test length goes to infinity, with probability 1, assuming some general regularity conditions. The asymptotic rate of convergence is given by the Spearman–Brown formula, and for this it is not needed that the items are parallel, or latent unidimensional, or even finite dimensional. Simulations with the 2-parameter logistic item response theory model reveal that the reliability of short multidimensional tests can be positively biased, meaning that applying the Spearman–Brown formula in these cases would lead to overprediction of the reliability that results from lengthening a test. However, test constructors of short tests generally aim for short tests that measure just one attribute, so that the bias problem may have little practical relevance. For short unidimensional tests under the 2-parameter logistic model reliability is almost unbiased, meaning that application of the Spearman–Brown formula in these cases of greater practical utility leads to predictions that are approximately unbiased.

Dissipation-based proper orthogonal decomposition of turbulent Rayleigh–Bénard convection flow

We present a formulation of proper orthogonal decomposition (POD) producing a velocity–temperature basis optimized with respect to an H1 dissipation norm. This decomposition is applied, along with a conventional POD optimized with respect to an L2 energy norm, to a dataset generated from a direct numerical simulation of Rayleigh–Bénard convection in a cubic cell (Ra=107, Pr=0.707). The dataset is enriched using symmetries of the cell, and we formally link symmetrization to degeneracies and to the separation of the POD bases into subspaces with distinct symmetries. We compare the two decompositions, demonstrating that each of the 20 lowest dissipation modes is analogous to one of the 20 lowest energy modes. Reordering of modes between the decompositions is limited, although a corner mode known to be crucial for reorientations of the large-scale circulation is promoted in the dissipation decomposition, indicating suitability of the dissipation decomposition for capturing dynamically important structures. Dissipation modes are shown to exhibit enhanced activity in boundary layers. Reconstructing kinetic and thermal energy, viscous and thermal dissipation, and convective heat flux, we show that the dissipation decomposition improves overall convergence of each quantity in the boundary layer. Asymptotic convergence rates are nearly constant among the quantities reconstructed globally using the dissipation decomposition, indicating that a range of dynamically relevant scales is efficiently captured. We discuss the implications of the findings for using the dissipation decomposition in modeling and argue that the H1 norm allows for a better modal representation of the flow dynamics.

Estimates of constants in the limit theorems for chaotic dynamical systems

In a vast area of probabilistic limit theorems for dynamical systems with chaotic behaviors always only functional form (exponential, power, etc.) of the asymptotic laws and of convergence rates were studied. However, for basically all applications, e.g., for computer simulations, development of algorithms to study chaotic dynamical systems numerically, as well as for design and analysis of real (e.g., in physics) experiments, the exact values (or at least estimates) of constants (parameters) of the functions, which appear in the asymptotic laws and rates of convergence, are of primary interest. In this paper, we provide such estimates of constants (parameters) in the central limit theorem, large deviations principle, law of large numbers and the rate of correlations decay for strongly chaotic dynamical systems.

Performance estimation of switched linear systems via n×(n+2) generalized coordinate transformations

AbstractFor a continuous‐time switched linear system, the spectral abscissa, that is, the maximum Lyapunov exponent, essentially characterizes the worst‐case asymptotic divergence or convergence rate under arbitrary switching. In this study, based on the generalized coordinate transformation method, a computational algorithm is proposed to get the performance estimation of the switched linear system under arbitrary switching by estimating the spectral abscissa of the system. Simulation examples are presented to illustrate the effectiveness of the proposed scheme, which gives a better estimation of the spectral abscissa than the existing results.

Improving Cointegration-Based Pairs Trading Strategy with Asymptotic Analyses and Convergence Rate Filters

A pairs trading strategy (PTS) constructs a mean-reverting portfolio whose logarithmic value moves back and forth around a mean price level. It makes profits by longing (or shorting) the portfolio when it is underpriced (overpriced) and closing the portfolio when its value converges to the mean price level. The cointegration-based PTS literature uses the historical sample mean and variance to establish their open/close thresholds, which results in bias thresholds and less converged trades. We derive the asymptotic mean around which the portfolio value oscillates. Revised open/close thresholds determined by our asymptotic mean and standard derivations significantly improve PTS performance. The derivations of asymptotic means can be extended to construct a convergence rate filter mechanism to remove stock pairs that are unlikely to be profitable from trading to further reduce trading risks. Moreover, the PTS literature oversimplifies the joint problem of examining a stock pair’s cointegration property and selecting the fittest vector error correction model (VECM). We propose a two-step model selection procedure to determine the cointegration rank and the fittest VECM via the trace and likelihood ratio tests. We also derive an approximate simple integral trading volume ratio to meet no-odd-lot trading constraints. Experiments from Yuanta/P-shares Taiwan Top 50 Exchange Traded Fund and Yuanta/P-shares Taiwan Mid-Cap 100 Exchange Traded Fund constituent stock tick-by-tick backtesting during 2015–2018 show remarkable improvements by adopting our approaches.

A stabilized finite element method for modeling dispersed multiphase flows using orthogonal subgrid scales

We propose a finite-element formulation for simulating multi-component flows occupying the same domain with spatially varying concentrations. Each constituent is assumed to behave as an incompressible Newtonian fluid, and solutions are sought for the velocities and volume fractions of each phase, as well as the common pressure. Stabilization terms are derived within the framework of the variational multiscale method based on an approximation of the finite-element residual to achieve control of the pressure and volume fractions. We utilize the concept of term-by-term stabilization in conjunction with orthogonal subgrid scales, thus incorporating only those terms of the residual essential to obtain stability and projecting them on a space orthogonal to the finite element space. The resulting system of equations is solved in a monolithic manner, requiring a small number of nonlinear iterations. Several benchmark tests have been performed to confirm the stability and optimal asymptotic convergence rates for linear and higher-order elements using the proposed formulation.

How do kernel-based sensor fusion algorithms behave under high-dimensional noise?

Abstract We study the behavior of two kernel based sensor fusion algorithms, nonparametric canonical correlation analysis (NCCA) and alternating diffusion (AD), under the nonnull setting that the clean datasets collected from two sensors are modeled by a common low-dimensional manifold embedded in a high-dimensional Euclidean space and the datasets are corrupted by high-dimensional noise. We establish the asymptotic limits and convergence rates for the eigenvalues of the associated kernel matrices assuming that the sample dimension and sample size are comparably large, where NCCA and AD are conducted using the Gaussian kernel. It turns out that both the asymptotic limits and convergence rates depend on the signal-to-noise ratio (SNR) of each sensor and selected bandwidths. On one hand, we show that if NCCA and AD are directly applied to the noisy point clouds without any sanity check, it may generate artificial information that misleads scientists’ interpretation. On the other hand, we prove that if the bandwidths are selected adequately, both NCCA and AD can be made robust to high-dimensional noise when the SNRs are relatively large.

On asymptotic convergence rate of random search

This paper presents general theoretical studies on asymptotic convergence rate (ACR) for finite dimensional optimization. Given the continuous problem function and discrete time stochastic optimization process, the ACR is the optimal constant for control of the asymptotic behaviour of the expected approximation errors. Under general assumptions, condition ACR<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$<1$$\\end{document} implies the linear behaviour of the expected time of hitting the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}- optimal sublevel set with ε→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0^+ $$\\end{document} and determines the upper bound for the convergence rate of the trajectories of the process. This paper provides general characterization of ACR and, in particular, shows that some algorithms cannot converge linearly fast for any nontrivial continuous optimization problem. The relation between asymptotic convergence rate in the objective space and asymptotic convergence rate in the search space is provided. Examples and numerical simulations with use of a (1+1) self-adaptive evolution strategy and other algorithms are presented.

Nonparametric estimation for SDE with sparsely sampled paths: An FDA perspective

We consider the problem of nonparametric estimation of the drift and diffusion coefficients of a Stochastic Differential Equation (SDE), based on n independent replicates Xi(t):t∈[0,1]1≤i≤n, observed sparsely and irregularly on the unit interval, and subject to additive noise corruption. By sparse we intend to mean that the number of measurements per path can be arbitrary (as small as two), and can remain constant with respect to n. We focus on time-inhomogeneous SDE of the form dX(t)=μ(t)X(t)αdt+σ(t)X(t)βdB(t), where α∈{0,1} and β∈{0,1/2,1}, which includes prominent examples such as Brownian motion, Ornstein–Uhlenbeck process, geometric Brownian motion and Brownian bridge. Our estimators are constructed by relating the local (drift/diffusion) parameters of the SDE to its global parameters (mean/covariance, and their derivatives) by means of an apparently novel Partial Differential Equation (PDE). This allows us to use methods inspired by functional data analysis, and pool information across the sparsely measured paths. The methodology we develop is fully non-parametric and avoids any functional form specification on the time-dependency of either the drift function or the diffusion function. We establish almost sure uniform asymptotic convergence rates of the proposed estimators as the number of observed paths n grows to infinity. Our rates are non-asymptotic in the number of measurements per path, explicitly reflecting how different sampling frequency might affect the speed of convergence. Our framework suggests possible further fruitful interactions between FDA and SDE methods in problems with replication.

Correction: On the asymptotic rate of convergence of Stochastic Newton algorithms and their Weighted Averaged versions

Correction: On the asymptotic rate of convergence of Stochastic Newton algorithms and their Weighted Averaged versions

A von Kármán-type model for two-layer laminated glass plates, with applications to buckling and free vibration under in-plane edge loads

This article presents a layerwise geometrically nonlinear two-dimensional model for the undamped dynamical response of thin two-layer plates with partial shear interaction. The model is tailored for the analysis of laminated glass plates of the type most commonly used in building structures (two glass layers bonded by a polymeric interlayer). The geometrical nonlinearities are introduced via the so-called von Kármán strains. The process of dimensional reduction is based on Podio-Guidugli’s method of internal constraints. An unconventional choice of generalised displacements highlights several direct similarities with the single-layer plate theories of von Kármán and Mindlin.The model is used to study the flexural buckling and free vibration of two-layer plates under in-plane edge loads. Analytical solutions are obtained for rectangular plates with (i) all edges simply supported without shear restraint and (ii) a two-parameter system of uniform in-plane normal edge loads. These solutions degenerate into those of single-layer von Kármán plates in the two limiting cases of zero and full shear interaction. A conforming rectangular finite element is formulated and shown to perform effectively. When using uniform meshes, the observed asymptotic rate of convergence for the finite element approximation of the lowest buckling load and vibration frequency is quadratic.