Showing papers on "Rate of convergence published in 1988"

Nonlocal Continuum Damage, Localization Instability and Convergence

Abstract: A recent nonlocal damage formulation, in which the spatially averaged quantity was the energy dissipated due to strain-softening, is extended to a more general form in which the strain remains local while any variable that controls strain-softening is nonlocal. In contrast to the original imbricate nonlocal model for strain-softening, the stresses which figure in the constitutive relation satisfy the differential equations of equilibrium and boundary conditions of the usual classical form, and no zero-energy spurious modes of instability are encountered. However, the field operator for the present formulation is in general nonsymmetric, although not for the elastic part of response. It is shown that the energy dissipation and damage cannot localize into regions of vanishing volume. The static strain-localization instability, whose solution is reduced to an integral equation, is found to be controlled by the characteristic length of the material introduced in the averaging rule. The calculated static stability limits are close to those obtained in the previous nonlocal studies, as well as to those obtained by the crack band model in which the continuum is treated as local but the minimum size of the strain-softening region (localization region) is prescribed as a localization limiter. Furthermore, the rate of convergence of static finite-element solutions with nonlocal damage is studied and is found to be of a power type, almost quadratric. A smooth weighting function in the averaging operator is found to lead to a much better convergence than unsmooth functions.

On the dynamics of finite-strain rods undergoing large motions a geometrically exact approach

Abstract: The dynamics of a fully nonlinear rod model, capable of undergoing finite bending, shearing, and extension, is considered in detail. Unlike traditional nonlinear structural dynamics formulations, due to the effect of finite rotations the deformation map takes values in r 3 × SO(3), which is a differentiable manifold and not a linear space. An implicit time stepping algorithm that furnishes a canonical extension of the classical Newmark algorithm to the rotation group (SO(3)) is developed. In addition to second-order accuracy, the proposed algorithm reduces exactly to the plane formulation. Moreover, the exact linearization of the algorithm and associated configuration update is obtained in closed form, leading to a configuration-dependent nonsymmetric tangent inertia matrix. As a result, quadratic rate of convergence is attained in a Newton-Raphson iterative solution strategy. The generality of the proposed formulation is demonstrated through several numerical examples that include finite vibration, centrifugal stiffening of a fast rotating beam, dynamic instability and snap-through, and large overall motions of a free-free flexible beam. Complete details on implementation are given in three appendices.

Optimal Rates of Convergence for Deconvolving a Density

Abstract: Suppose that the sum of two independent random variables X and Z is observed, where Z denotes measurement error and has a known distribution, and where the unknown density f of X is to be estimated. One application is the estimation of a prior density for a sequence of location parameters. A second application arises in the errors-in-variables problem for nonlinear and generalized linear models, when one attempts to model the distribution of the true but unobservable covariates. This article shows that if Z is normally distributed and f has k bounded derivatives, then the fastest attainable convergence rate of any nonparametric estimator of f is only (log n)–k/2. Therefore, deconvolution with normal errors may not be a practical proposition. Other error distributions are also treated. Stefanski—Carroll (1987a) estimators achieve the optimal rates. The results given have versions for multiplicative errors, where they imply that even optimal rates are exceptionally slow.

Modified Broyden’s method for accelerating convergence in self-consistent calculations

Abstract: A modification to Broyden's method for obtaining stable and computationally efficient convergence in self-consistent calculations is developed and discussed. The method incorporates the advantages of two schemes proposed by Srivastava and by Vanderbilt and Louie without any increase in complexity. Its improvement over their methods is discussed. The present method is compared with two other widely used convergence methods, simple mixing and Anderson's method, for the case of the disordered binary alloy ${\mathrm{Ni}}_{0.35}$${\mathrm{Fe}}_{0.65}$ on the verge of a magnetic instability and is shown to be much improved in stability and rate of convergence.

Observers for linear systems with unknown inputs

Abstract: A direct design procedure of a full-order observer for a linear system with unknown inputs is presented, using straightforward matrix calculations. Some examples are given; in these examples a reduced-order observer is also derived. It is shown that this may restrict the rate of convergence of some state estimates. >

How Far are Automatically Chosen Regression Smoothing Parameters from their Optimum

Abstract: We address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation. Call the “optimal bandwidth” the minimizer of the average squared error. We consider several automatically selected bandwidths that approximate the optimum. How far are the automatically selected bandwidths from the optimum? The answer is studied theoretically and through simulations. The theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution. The convergence rate turns out to be excruciatingly slow. This is not too disappointing, because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error. In some simulations by John Rice, the selectors considered here performed quite differently from each other. We anticipated that these differences would be reflected in differ...

The hierarchical basis multigrid method

Abstract: We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauβ-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

Global convergence of a class of trust region algorithms for optimization with simple bounds

Abstract: This paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present. Weak conditions on the accuracy of the Hessian approximations are considered. It is also shown that, when the strict complementarily condition holds, the proposed algorithms reduce to an unconstrained calculation after finitely many iterations, allowing a fast asymptotic rate of convergence.

Adaptive techniques in the finite element method

Abstract: An effective h-version finite element adaptive strategy combined with mesh regeneration is presented. This is based on the error estimator developed in Reference 1. The rate of convergence of the adaptive procedure has been tested for some examples and very strong convergence observed. Unlike some existing h-version adaptive procedures, a nearly optimal mesh of predicted accuracy can be obtained in one or two adaptive process steps.

Variance Function Estimation in Regression: the Effect of Estimating the Mean

Abstract: : The authors consider estimation of a variance function g in regression problems. Such estimation requires simultaneous estimation of the mean function f. We obtain sharp results on the extent to which the smoothness of f influences best rates of convergence for estimating g. For example, in nonparametric regression with two derivatives on g, classical rates of convergence are possible if and only if the unknown f satisfies a Lipschitz condition of order 1/3 or more. If a parametric model is known for g, then g may be estimated n 1/2 - consistently if and only if f is Lipschitz of order 1/2 or more. Optimal rates of convergence are attained by kernel estimators.

Monotonicity of quadratic-approximation algorithms

Abstract: It is desirable that a numerical maximization algorithm monotonically increase its objective function for the sake of its stability of convergence. It is here shown how one can adjust the Newton-Raphson procedure to attain monotonicity by the use of simple bounds on the curvature of the objective function. The fundamental tool in the analysis is the geometric insight one gains by interpreting quadratic-approximation algorithms as a form of area approximation. The statistical examples discussed include maximum likelihood estimation in mixture models, logistic regression and Cox's proportional hazards regression.

Convergence and performance analysis of the normalized LMS algorithm with uncorrelated Gaussian data

Abstract: It is demonstrated that the normalized least mean square (NLMS) algorithm can be viewed as a modification of the widely used LMS algorithm. The NLMS is shown to have an important advantage over the LMS, which is that its convergence is independent of environmental changes. In addition, the authors present a comprehensive study of the first and second-order behavior in the NLMS algorithm. They show that the NLMS algorithm exhibits significant improvement over the LMS algorithm in convergence rate, while its steady-state performance is considerably worse. >

Numerical integration of stochastic differential equations

Abstract: Several numerical methods for treating stochastic differential equations are considered. Both the convergence in the mean square limit and the convergence of the moments is discussed and the generation of appropriate random numbers is treated. The necessity of simulations at various time steps with an extrapolation to time step zero is emphasized and demonstrated by a simple example.

An approximation method for tandem queues with blocking

Abstract: We propose an approximate analysis of open systems of tandem queues with blocking caused by finite buffers between servers. Our approach relies on the use of marginal probability distributions (“state equivalence”) coupled with an approximate evaluation of the conditional probabilities introduced through the equivalence. The method iterates over consecutive pairs of servers using the solution of a two-queue system as a building block. It produces performance measures for individual servers as well as an approximation to joint queue-length probability distributions for pairs of neighboring stations. Experience indicates that the number of iterations required for the method grows moderately with the number of nodes in the network. We give examples to demonstrate the accuracy and the convergence properties of the proposed approximation.

On the Rate of Convergence in the Central Limit Theorem for Martingales with Discrete and Continuous Time

Abstract: Heyde and Brown (1970) established a bound on the rate of convergence in the central limit theorem for discrete time martingales having finite moments of order $2 + 2\delta$ with $0 0$. Moreover, an example is constructed demonstrating that this bound is asymptotically exact for all $\delta > 0$. The result for discrete time martingales is then used to derive the corresponding bound on the rate of convergence in the central limit theorem for locally square integrable martingales with continuous time.

On the Convergence Rate in the Central Limit Theorem for Associated Processes

Abstract: We give uniform rates of convergence in the central limit theorem for associated processes with finite third moment. No stationarity is required. Using a coefficient $u(n)$ which describes the covariance structure of the process, we obtain a convergence rate $O(n^{-1/2}\log^2n)$ if $u(n)$ exponentially decreases to 0. An example shows that such a rate can no longer be obtained if $u(n)$ decreases only as a power.

On error-saturation nonlinearities in LMS adaptation

Abstract: The effect of a saturation-type error nonlinearity in the weight update equation in least-mean-squares (LMS) adaptation is investigated for a white Gaussian data model. Nonlinear difference equations are derived for the eight first and second moments, which include the effect of an error function (erf) saturation-type nonlinearity on the error sequence driving the algorithm. A nonlinear difference equation for the mean norm is explicitly solved using a differential equation approximation and integration by quadratures. The steady-state second-moment weight behavior is evaluated exactly for the erf nonlinearity. Using the above results, the tradeoff between the extent of error saturation, steady-state excess mean-square error, and rate of algorithm convergence is studied. The tradeoff shows that (1) starting with a sign detector, the convergence rate is increased by nearly a factor of two for each additional bit, and (2) as the number of bits is increased further, the additional bit by very little in convergence speed, asymptotically approaching the behavior of the linear algorithm. >

Inverse kinematic algorithms for redundant systems

Abstract: An iterative method of computing the solution of the inverse kinematic problem is developed for redundant systems using the transpose of the Jacobian matrix instead of the pseudoinverse. The solutions may be optimized on a criterion function or on physical constraints, such as obstacle avoidance. Stability and convergence of the method are shown. Although its convergence rate is only about half that of Newton's method, the advantage of the method is that it remains easily tractable close to the singular configurations of the manipulator. A hybrid method combining the Jacobian transpose and Newton's methods is proposed. Results of the application of the method on a 10-link manipulator in 2-D space are shown. >

The p‐version of the finite element method for domains with corners and for infinite domains

Abstract: A special approach to deal with elliptic problems with singularities is introduced. It is shown that this approach, to be called an auxiliary mapping technique, in the framework of the p-version of the finite element method yields an exponential rate of convergence. It is also shown that this technique can deal with elliptic problems on unbounded domains in R2 as well. (AMS(MOS) subject classifications: Primary, 65N30, 65N15.)

The recursive reduced-order numerical solution of the singularly perturbed matrix differential Riccati equation

Abstract: Under stability-observability conditions imposed on a singularly perturbed system, an efficient numerical method for solving the corresponding matrix differential Riccati equation is obtained in terms of the reduced-order problems. The order reduction is achieved via the use of the Chang transformation applied to the Hamiltonian matrix of a singularly perturbed linear-quadratic control problem. An efficient numerical recursive algorithm with a quadratic rate of convergence is developed for solving the algebraic equations comprising the Chang transformation. >

Convergence of the random vortex method in two dimensions

Abstract: A theoretical framework for analyzing the random vortex method is presented. It extends and modifies the analysis of the inviscid vortex method in a natural and unified manner. The rate of convergence of the random vortex method in two dimensions is obtained by analyzing the consistency error and justifying the stability estimate. The sampling error introduced by the random motions of finitely many vortices is the dominant component of the consistency error in terms of order. It is estimated by applying Bennett's inequality. MATHEMATICAL SCIENCES INSTITUTE, CORNELL UNIVERSITY, ITHACA, NEW YORK 14850 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 05:39:41 UTC All use subject to http://about.jstor.org/terms

Unified multilevel adaptive finite element methods for elliptic problems

Abstract: Many elliptic partial differential equations can be solved numerically with near optimal efficiency through the uses of adaptive refinement and multigrid solution techniques It is our goal to develop a more unified approach to the combined process of adaptive refinement and multigrid solution which can be used with high order finite elements The basic step of the refinement process is the bisection of a pair of triangles, which corresponds to the addition of one or more basis functions to the approximation space An approximation of the resulting change in the solution is used as an error indicator to determine which triangles to divide The multigrid iteration uses a red-black Gauss-Seidel relaxation in which the black relaxations are used only locally The grid transfers use the change between the nodal and hierarchical bases This multigrid iteration requires only O(N) operations, even for highly nonuniform grids, and is defined for any finite element space The full multigrid method is an optimal blending of the processes of adaptive refinement and multigrid iteration So as to minimize the number of operations required, the duration of the refinement phase is based on increasing the dimension of the approximation space by some fixed factor which is determined to be the largest possible for the given error-reducing power of the multigrid iteration The result is an algorithm which (i) uses only O(N) operations with a reasonable constant of proportionality, (ii) solves the discrete system to the accuracy of the discretization error, (iii) is able to achieve the optimal order of convergence of the discretization error in the presence of singularities Numerical experiments confirm this for linear, quadratic and cubic elements It is believed that the method can also be applied to more practical problems involving systems of PDE's, time dependence, and three spatial dimensions

Product integration-collocation methods for noncompact integral operator equations

Abstract: We discuss the numerical solution of a class of second-kind integral equations in which the integral operator is not compact Such equations arise, for example, when boundary integral methods are applied to potential problems in a two-dimensional domain with corners in the boundary We are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes, provided some simple modifications are made to the underlying basis functions These are sufficient to ensure stability, but do not damage the rate of convergence Numerical experiments show that such modifications are necessary in certain circumstances

Trapping and cascading of eigenvalues in the large coupling limit

Abstract: We consider eigenvaluesEλ of the HamiltonianHλ=−Δ+V+λW,W compactly supported, in the λ→∞ limit. ForW≧0 we find monotonic convergence ofEλ to the eigenvalues of a limiting operatorH∞ (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for 1-dimensional systems. In 1-dimensional systems withW≦0, or withW changing sign, we do not find convergence. Instead, we find a cascade phenomenon, in which, as λ→∞, each eigenvalueEλ stays near a Dirichlet eigenvalue for a long interval (of lengthO(\(\sqrt \lambda \))) of the scaling range, quickly drops to the next lower Dirichlet eigenvalue, stays there for a long interval, drops again, and so on. As a result, for most large values of λ the discrete spectrum ofHλ is close to that ofE∞, but when λ reaches a transition region, the entire spectrum quickly shifts down by one. We also explore the behavior of several explicit models, as λ→∞.

Analytic quadratic integration over the two-dimensional Brillouin zone

Abstract: A new method is described to evaluate analytically integrals of quadratically interpolated functions over the two-dimensional Brillouin zone. The method is not geometric in nature like the quadratic method of Methfessel and co-workers, but is algebraic. It allows quadratic interpolation not only for the dispersion relation epsilon n(k), but for property functions fn(k) as well. Comparisons are made between the analytic quadratic integration and the commonly used analytic linear integration by calculating tight-binding Brillouin zone integrals with the same number of k-points for both methods. It is shown that convergence behaviour and convergence rate are far better for the analytic quadratic integration than for the analytic linear integration. Roughly, analytic quadratic integration can achieve the same accuracy as analytic linear integration with only about the square root of the total number of k-points needed.

On implementation of computational algorithms for optimal design 1: Preliminary investigation

Abstract: In this two part paper, the problem of implementation of computational algorithms for design optimization into a computer software is discussed. A recently developed algorithm that generates and incorporates approximate second order information about the problem is selected for detailed analyses and discussions. It is shown that numerical behaviour of the algorithm is influenced by variation of the key parameters and procedures. The concept of numerical experiments is introduced, and certain variations of the algorithm and parameters are selected and their influence on its performance is studied. It is shown that the numerical rate of convergence can be substantially improved with proper procedures and values of the parameters. The first part of the paper describes some preliminary analyses and investigations. The second part describes further numerical analyses and detailed procedures for evaluation of performance of various variations of an algorithm or different computer codes. The basic conclusion from the study is that robust and efficient implementation of algorithms requires expert knowledge and considerable numerical experimentation. A wide range of small scale and large scale problems of varying difficulty must be solved to evaluate performance of an algorithm. The study suggests development of knowledge-based systems for practical design optimization.