Asymptotic Convergence Behavior Research Articles

Spurious logarithms and the KPSS statistic

This paper analyzes the asymptotic behavior of two types of so-called KPSS tests when a logarithm transformation has been applied spuriously to data that are themselves an integrated time series. Although a different limit distribution is obtained, the asymptotic convergence behavior of the KPSS statistic is reminiscent of that of integrated time series, and it is shown that the KPSS test cannot distinguish consistently between an integrated time series and the logarithm of an integrated time series.

The Chebyshev iteration revisited

Compared to Krylov space methods based on orthogonal or oblique projection, the Chebyshev iteration does not require inner products and is therefore particularly suited for massively parallel computers with high communication cost. Here, six different algorithms that implement this method are presented and compared with respect to roundoff effects, in particular, the ultimately achievable accuracy. Two of these algorithms replace the three-term recurrences by more accurate coupled two-term recurrences and seem to be new. It is also shown that, for real data, the classical three-term Chebyshev iteration is never seriously affected by roundoff, in contrast to the corresponding version of the conjugate gradient method. Even for complex data, strong roundoff effects are seen to be limited to very special situations where convergence is anyway slow. The Chebyshev iteration is applicable to symmetric definite linear systems and to nonsymmetric matrices whose eigenvalues are known to be confined to an elliptic domain that does not include the origin. Also considered is a corresponding stationary 2-step method, which has the same asymptotic convergence behavior and is additionally suitable for mildly nonlinear problems.

Stiff Well-Posedness and Asymptotic Convergence for a Class of Linear Relaxation Systems in a Quarter Plane

In this paper we study the asymptotic equivalence of a general linear system of 1-dimensional conservation laws and the corresponding relaxation model proposed by S. Jin and Z. Xin (1995, Comm. Pure Appl. Math.48, 235–277) in the limit of small relaxation rate. The main interest is this asymptotic equivalence in the presence of physical boundaries. We identify and rigorously justify a necessary and sufficient condition (which we call the Stiff Kreiss Condition, or SKC in short) on the boundary condition to guarantee the uniform well-posedness of the initial boundary value problem for the relaxation system independent of the rate of relaxation. The SKC is derived and simplified by using a normal mode analysis and a conformal mapping theorem. The asymptotic convergence and boundary layer behavior are studied by the Laplace transform and a matched asymptotic analysis. An optimal rate of convergence is obtained.

Asymptotic convergence for iterative optimization in electronic structure

There have recently been a number of proposals for solving large electronic structure problems (local-density approximation, Hartree-Fock, and tight-binding methods) iteratively with a computational effort proportional to the size of the system. The effort needed to perform a single iteration in these schemes is well understood but the convergence rate has been an empirical matter. This paper will show that many of the proposed methods have a single underlying geometrical structure, which has a specific asymptotic convergence behavior, and that behavior can be understood in terms of some simple condition numbers based on the spectrum of the Hamiltonian.

Stokes flow in an elastic tube—a large-displacement fluid-structure interaction problem

SUMMARY Viscous flow in elastic (collapsible) tubes is a large-displacement fluid-structure interaction problem frequently encountered in biomechanics. This paper presents a robust and rapidly converging procedure for the solution of the steady three-dimensional Stokes equations, coupled to the geometrically non-linear shell equations which describe the large deformations of the tube wall. The fluid and solid equations are coupled in a segregated method whose slow convergence is accelerated by an extrapolation procedure based on the scheme’s asymptotic convergence behaviour. A displacement control technique is developed to handle the system’s snap-through behaviour. Finally, results for the tube’s post-buckling deformation and for the flow in the strongly collapsed tube are shown. © 1998 John Wiley & Sons, Ltd.

Asymptotically optimal row-action methods for generalized least squares problems

Recently row-action methods have been frequently used in actual applications, for instance, image reconstruction. The asymptotic convergence behavior of row-action methods has been studied by lusem and De Pierro (1990) and Mandel (1984). Here asymptotically optimal behavior of row-action methods are investigated. Some asymptotically optimal row-action methods, for example, asymptotically optimal Kaczmarz and Hildreth methods, are proposed. Several numerical algorithms of asymptotically optimal row-action methods for generalized least squares problems are given.

Exact Solution to the Moment Problem for the XY Chain

We present the exact solution to the moment problem for the spin-1/2 isotropic antiferromagnetic XY chain with explicit forms for the moments with respect to the Néel state, the cumulant generating function, and the Resolvent Operator. We verify the correctness of the Horn–Weinstein Theorems, but the analytic structure of the generating function <e-tH> in the complex t-plane is quite different from that assumed by the t-Expansion and the Connected Moments Expansion due to the vanishing gap. This function has a finite radius of convergence about t=0, and for large t has a leading descending algebraic series E(t)-E0~ At-2. The Resolvent has a branch cut and essential singularity near the ground state energy of the form G(s)/s~ B|s+1|-3/4 exp (C|s+1|1/2). Consequently extrapolation strategies based on these assumptions are flawed and in practise we find that the CMX methods are pathological, and cannot be applied, while numerical evidence for two of the t-expansion methods indicates a clear asymptotic convergence behaviour with truncation order.

A note on conjugate gradient convergence

The one-dimensional discrete Poisson equation on a uniform grid with \(n\) points produces a linear system of equations with a symmetric, positive-definite coefficient matrix. Hence, the conjugate gradient method can be used, and standard analysis gives an upper bound of \(O(n\)) on the number of iterations required for convergence. This paper introduces a systematically defined set of solutions dependent on a parameter \(\beta\), and for several values of \(\beta\), presents exact analytic expressions for the number of steps \(k(\beta,\tau,n\)) needed to achieve accuracy \(\tau\). The asymptotic behavior of these expressions has the form \(O(n^{\alpha\))} as \(n \rightarrow \infty\) and \(O(\tau^{\gamma\))} as \(\tau \rightarrow 0\). In particular, two choices of \(\beta\) corresponding to nonsmooth solutions give \(\alpha = 0\), i.e., iteration counts independent of \(n\); this is in contrast to the standard bounds. The standard asymptotic convergence behavior, \(\alpha = 1\), is seen for a relatively smooth solution. Numerical examples illustrate and supplement the analysis.

Smoothing of elliptic differential inclusions and its iterative treatment

In this paper some properties of the embedding of nonlinear elliptic boundary value problems with discontinuities into families of smooth problems are investigated. The regularizations which are studied with respect to the related rate of convergence of the solutions of the smooth auxiliary problems to the the solution of the original discontinuous elliptic boundary value problem. In contrast to embedding methods for regular problems, naturally here an ill-conditioning occurs due to the approximation of boundary value problems with discontinuities. This yields a specific asymptotic convergence behaviour of Newton's method when applied to the regularized problems. The region of contraction of Newton's method is investigated and parameter selection rules are proposed such that truncated Newton iterations on each parameter level will generate an overall convergence of the resulting path-following technique. In addition some principles for the construction of monotone embedding methods for discontinuous elliptic boundary value problems are considered.

Asymptotic convergence analysis of the projection approximation subspace tracking algorithms

Subspace tracking plays an important role in a variety of adaptive subspace methods. In this paper, we present a theoretical convergence analysis of two recently proposed projection approximation subspace tracking algorithms (PAST and PASTd). By invoking Ljung's ordinary differential equation approach, we derive a pair of coupled matrix differential equations, whose trajectories describe the asymptotic convergence behavior of the subspace tracking algorithms. We discuss properties of the matrix differential equations and determine their asymptotically stable equilibrium states and domain of attraction. It turns out that, under weak conditions, both PAST and PASTd globally converge to the desired signal subspace or signal eigenvectors and eigenvalues with probability one. Numerical examples are also included to illustrate the asymptotic convergence rate of the algorithms.

Density estimation with laguerre series and censored samples

In this paper we investigate the nonparametric density estimation by means of the orthonormal system of the Laguerre series on the positive half line. We work with stochastically right censored samples, discuss the asymptotic convergence behaviour in this case and give convergence rates either of the mean integrated square error and the mean square error. With an extra assumption on the distribution functions we are in position to obtain the same convergence order as in the case of uncensored samples.

ODE Method Versus Martingale Convergence Theory

During the last few years two methods, namely The Ordinary Differential Equation method, due to Ljung and Martingale convergence Theory which was used by several authors have been proposed for the convergence analysis of the processes governed by stochastic difference equations. The idea of the ODE method is to associate the set of stochastic difference equations governing the process dynamics with a set of ordinary differential equations whose stability properties contain the necessary information about the asymptotic convergence behaviour of the algorithm. An alternative analysis can be done by means of the Martingale convergence theory, what is a so called “stochastic version of the Ljapunov stability theorem”. Both methods are applicable to the stochastic convergence analysis of recursive parameter estimation methods and adaptive systems. The purpose of this work is to compare both methods with respect to the assumptions needed and to the conclusions obtained.

An Analysis of Cut-Off Rules for Optimization Algorithms

The problem of optimal stopping has long bee of intrest to various scientific disciplines. Most notably, the approacbes taken have been based on sequential decision theory and modeling techniqes. Aiming at optimization algorithms, we propose in this paper to combine the present approaches and study the cut-off rule problem, using asymptotic convergence behavior of sequences. A unifying framework for the design and analysis of stopping rules for algorithms generating a monotonically convergent sequence is presented while methodology for optimal design is discussed. The concavity structure should be observed to play an important role in our analysis.