Statistical ensemble error bounds for homogenized microheterogeneous solids

Typically, in order to characterize the homogenized effective macroscopic response of new materials possessing random heterogeneous microstructure, a relation between averages is sought, where and where and are the stress and strain tensor fields within a statistically representative volume element (SRVE) of volume ||. The quantity, is known as the effective property, and is the elasticity tensor used in usual macroscale analyses. In order to generate homogenized responses computationally, a series of detailed boundary value representations resolving the heterogeneous microstructure, posed over the SRVEs domain, must be solved. This requires an enormous numerical effort that can overwhelm most computational facilities. A natural way of generating an approximation to the SRVEs response is by first computing the response of smaller (subrepresentative) samples, each with a different random realization of the microstructural type under investigation, and then to ensemble average the results afterwards. Compared to a direct simulation of an SRVE, testing many small samples is a computationally inexpensive process since the number of floating point operations is greatly reduced, as well as the fact that the samples responses can be computed trivially in parallel. However, there is an inherent error in this process. Clearly the populations ensemble average is not the SRVEs response. However, as shown in this work, the moments on the distribution of the population can be used to generate rigorous upper and lower error bounds on the quality of the ensemble-generated response. Two-sided bounds are given on the SRVE response in terms of the ensemble average, its standard deviation and its skewness.Received: December 11, 2001     

A note on bounds for non-linear multivalued homogenized operators

In this paper we study the behaviour of maximal monotone multivalued highly oscillatory operators. We construct Reuss-Voigt-Wiener and Hashin-Shtrikmann type bounds for the minimal sections of G-limits of multivalued operators by using variational convergence and convex analysis.     

On some sharp bounds for the homogenized p-poisson equation

We consider homogenization of a scale of p-Poisson equations in R^N. Some new bounds of the effective energy are proved and compared with the non-linear Wiener -and Hashin-Shtrikman bounds. Moreover, we point out concrete nontrivid examples where these bounds even coincide. Some new examples of “optimal” microstructures are presented.     

Improved error bounds for near-minimax approximations

Givenf εC ⁽ⁿ⁺¹⁾[?1, 1], a polynomialp _n, of degree ≤n, is said to be near-minimax if (*) $$\left\| {f - p_n } \right\|_\infty = 2^{ - n} |f^{(n + 1)} (\xi )|/(n + 1)!,$$ for some ζ ε (?1,1). For three sets of near-minimax approximations, by considering the form of the error ∥f ?p _n∥_∞ in terms of divided differences, it is shown that better upper and lower bounds can be found than those given by (*).     

Computable error bounds for nonlinear programming

This paper presents a method for obtaining computable bounds for the error in an approximate Kuhn—Tucker point of a nonlinear program. Techniques of interval analysis are employed to compute the error bounds.Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462.     

Sharp error bounds for Newton's process

Summary The method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].     

Global error bounds for gauss-gegenbauer quadrature

The object of this paper is to derive global error bounds for integrals of the form $$\int_{ - 1}^1 {(1 - x^2 )^\lambda f(x)dx,\lambda > - 1,} $$ which have been approximated by Gauss-Gegenbauer quadrature.     

Perturbation of error bounds

Mathematical Programming - Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi: 10.1137/100782206 ) and, more precisely, to...     

Guaranteed bounds on homogenized periodic media by FFT-based Galerkin method

An FFT-based method, introduced by Moulinec and Suquet [1] in 1994, is an effective alternative to conventional Finite Element Method (FEM) for numerical homogenization of periodic media. Here, we summarize the recent variational reformulation and discretizations by Vondřejc et al. [2–6], which are based on conforming Galerkin approximations with trigonometric polynomials as basis functions. This insight, naturally leading to guaranteed bounds on homogenized matrix, opens a wide area of further investigations, which are also briefly discussed here. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Backward error bounds for approximate Krylov subspaces

Let A be a matrix of order n and let be a subspace of dimension k. In this note, we determine a matrix E of minimal norm such that is a Krylov subspace of A+E.     

On error bounds for lower semicontinuous functions

We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini derivative of f. Received: April 27, 2001 / Accepted: November 6, 2001?Published online April 12, 2002     

Certified error bounds for uncertain elliptic equations

In many applications, partial differential equations depend on parameters which are only approximately known. Using tools from functional analysis and global optimization, methods are presented for obtaining certificates for rigorous and realistic error bounds on the solution of linear elliptic partial differential equations in arbitrary domains, either in an energy norm, or of key functionals of the solutions, given an approximate solution. Uncertainty in the parameters specifying the partial differential equations can be taken into account, either in a worst case setting, or given limited probabilistic information in terms of clouds.     

Global error bounds for piecewise convex polynomials

In this paper, by examining the recession properties of convex polynomials, we provide a necessary and sufficient condition for a piecewise convex polynomial to have a Hölder-type global error bound with an explicit Hölder exponent. Our result extends the corresponding results of Li (SIAM J Control Optim 33(5):1510–1529, 1995) from piecewise convex quadratic functions to piecewise convex polynomials.     

Cones and error bounds for linear iterations

Consider an ordered Banach space with a cone of positive elementsK and a norm ∥·∥. Let [a,b] denote an order-interval; under mild conditions, ifx*∈[a,b] then $$||x * - \tfrac{1}{2}(a + b)|| \leqslant \tfrac{1}{2}||b - a||.$$ This inequality is used to generate error bounds in norm, which provide on-line exit criteria, for iterations of the type $$x_r = Ax_{r - 1} + a,A = A^ + + A^ - ,$$ whereA ⁺ andA ^? are bounded linear operators, withA ⁺ K ?K andA ^? K ? ?K. Under certain conditions, the error bounds have the form $$\begin{gathered} ||x * - x_r || \leqslant ||y_r ||,y_r = (A^ + - A^ - )y_{r - 1} , \hfill \\ ||x * - x_r || \leqslant \alpha ||\nabla x_r ||, \hfill \\ ||x * - \tfrac{1}{2}(x_r + x_{r - 1} )|| \leqslant \tfrac{1}{2}||\nabla x_r ||. \hfill \\ \end{gathered} $$ These bounds can be used on iterative methods which result from proper splittings of rectangular matrices. Specific applications with respect to certain polyhedral cones are given to the classical Jacobi and Gauss-Seidel splittings.     

Sharp error bounds for complex floating-point inversion

We study the accuracy of the classic algorithm for inverting a complex number given by its real and imaginary parts as floating-point numbers. Our analyses are done in binary floating-point arithmetic, with an unbounded exponent range and in precision p; we also assume that the basic arithmetic operations (+, ?, ×, /) are rounded to nearest, so that the roundoff unit is u = 2^?p. We bound the largest relative error in the computed inverse either in the componentwise or in the normwise sense. We prove the componentwise relative error bound 3u for the complex inversion algorithm (assuming p ≥ 4), and we show that this bound is asymptotically optimal (as p → ∞) when p is even, and sharp when using one of the basic IEEE 754 binary formats with an odd precision (p = 53, 113). This componentwise bound obviously leads to the same bound 3u for the normwise relative error. However, we prove that the smaller bound 2.707131u holds (assuming p ≥ 24) for the normwise relative error, and we illustrate the sharpness of this bound for the basic IEEE 754 binary formats (p = 24,53,113) using numerical examples.     

Estimating error bounds for quaternary subdivision schemes

This paper deals with the error estimation techniques of quaternary subdivision schemes. The estimation is expressed in terms of initial control point sequences and constants. It is independent of subdivision process and parametrization therefore its evaluation is straightforward.     

Global error bounds for the extended vertical LCP

A new necessary and sufficient condition for the row -property is given. By using this new condition and a special row rearrangement, we provide two global error bounds for the extended vertical linear complementarity problem under the row -property, which extend the error bounds given in Chen and Xiang (Math. Program. 106:513–525, 2006) and Mathias and Pang (Linear Algebra Appl. 132:123–136, 1990) for the P-matrix linear complementarity problem, respectively. We show that one of the new error bounds is sharper than the other, and it can be computed easily for some special class of the row -property block matrix. Numerical examples are given to illustrate the error bounds. The work was in part supported by a Grant-in-Aid from Japan Society for the Promotion of Science, and the National Natural Science Foundation of China (10671010).