Multiscale resolution in the computation of crystalline microstructure

Summary. This paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy. It is shown that branching of different variants near interfaces of twinned martensite and austenite phases leads to reduced energies in finite element approximations. Such behavior of minimizing deformations is understood for an extended model that involves surface energies. Moreover, the closely related question of the role of different growth conditions of the employed bulk energy is discussed. By explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders. For first order laminates the estimates are optimal. Mathematics Subject Classification (2000):65K10, 65M50, 65N30, 73C50, 73S10     

Convex Envelopes of Monomials of Odd Degree

Convex envelopes of nonconvex functions are widely used to calculate lower bounds to solutions of nonlinear programming problems (NLP), particularly within the context of spatial Branch-and-Bound methods for global optimization. This paper proposes a nonlinear continuous and differentiable convex envelope for monomial terms of odd degree, x ^2k+1, where k N and the range of x includes zero. We prove that this envelope is the tightest possible. We also derive a linear relaxation from the proposed envelope, and compare both the nonlinear and linear formulations with relaxations obtained using other approaches.     

Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q ₁ ^rot , EQ ₁ ^rot and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to validate our theoretical results.     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

Hypergraphs as a mean of discovering the dependence structure of a discrete multivariate probability distribution

Most everyday reasoning and decision making is based on uncertain premises. The premises or attributes, which we must take into consideration, are random variables, therefore we often have to deal with a high dimensional multivariate random vector. A multivariate random vector can be represented graphically as a Markov network. Usually the structure of the Markov network is unknown. In this paper we construct special type of junction trees, in order to obtain good approximations of the real probability distribution. These junction trees are capable of revealing some of the conditional independences of the network. We have already introduced the concept of the t-cherry junction tree (E. Kovács and T. Szántai in Proceedings of the IFIP/IIASA//GAMM Workshop on Coping with Uncertainty, 2010), based on the t-cherry tree graph structure. This approximation uses only two and three dimensional marginal probability distributions. Now we use k-th order t-cherry trees, also called simplex multitrees to introduce the concept of the k-th order t-cherry junction tree. We prove that the k-th order t-cherry junction tree gives the best approximation among the family of k-width junction trees. Then we give a method which starting from a k-th order t-cherry junction tree constructs a (k+1)-th order t-cherry junction tree which gives at least as good approximation. In the last part we present some numerical results and some possible applications.     

Mixed Micromechanics and Continuum Damage Mechanics Approach to Transverse Cracking in [S, 90n]s Laminates

The stiffness reduction in [S, 90 _n ] _s laminates due to transverse cracking in 90-layers is analyzed using the synergistic continuum damage mechanics (SCDM) and a micromechanics approach. The material constants involved in the SCDM model are determined using the stiffness reduction data for a reference cross-ply laminate. The constraint efficiency factor, which depends on the stiffness and geometry of neighboring layers, is assumed to be proportional to the average crack opening displacement (COD). The COD as a function of the constraint effect of adjacent layers and crack spacing is described by a simple power law. The crack closure technique and Monte Carlo simulations are used to model the damage evolution: the 90-layer is divided into a large number of elements and the critical strain energy rate G _c having the Weibull distribution is randomly assigned to each element. The crack density data for a [0₂/90₄] _s cross-ply laminate are used to determine the Weibull parameters. The simulated crack density curves are combined with the CDM stiffness reduction predictions to obtain the stiffness versus strain. The methodology developed is successfully used to predict the stiffness reduction as a function of crack density in [±/90₄] _s laminates.     

EfficientL 2 approximation by splines

LetS _N ^k (t) be the linear space ofk-th order splines on [0, 1] having the simple knotst _i determined from a fixed functiont by the rulet _i=t(i/N). In this paper we introduce sequences of operators {Q _N } _N =1 fromC ^k [0, 1] toS _N ^k (t) which are computationally simple and which, asN, give essentially the best possible approximations tof and its firstk–1 derivatives, in the norm ofL ₂[0, 1]. Precisely, we show thatN ^k–1((f–Q _N f) ⁱ –dist₂(f ⁽¹⁾,S _N ^k–1 (t)))0 fori=0, 1, ...,k–1. Several numerical examples are given.The research of this author was partially supported by the National Science Foundation under Grant MCS-77-02464The research of this author was partially supported by the U.S. Army Reesearch Office under Grant No. DAHC04-75-G-0816     

Rank-one convex hulls in \mathbb{R}^{2\times2}

We study the rank-one convex hull of compact sets . We show that if K contains no two matrices whose difference has rank one, and if K contains no four matrices forming a T ₄ configuration, then the rank-one convex hull K ^rc is equal to K. Furthermore, we give a simple numerical criterion for testing for T ₄ configurations. Received: 20 August 2003, Accepted: 3 March 2004, Published online: 12 May 2004 Mathematics Subject Classification (2000): 49J45, 52A30 An erratum to this article can be found at     

On distance-regular graphs withk i =k j,II

Let be a distance-regular graph of diameterd andi-th valencyk _i. We show that ifk ₂ = k_j for 2 +j d and 2 <j, then is a polygon (k = 2) or an antipodal 2-cover (k _d = 1). We also give a short proof of Terwilliger's inequality for bipartite distance-regular graphs and a refinement of Ivanov's argument on diameter bound.     

Numerical computation of polynomial zeros by means of Aberth's method

An algorithm for computing polynomial zeros, based on Aberth's method, is presented. The starting approximations are chosen by means of a suitable application of Rouché's theorem. More precisely, an integerq 1 and a set of annuliA _i,i=1,...,q, in the complex plane, are determined together with the numberk _i of zeros of the polynomial contained in each annulusA _i. As starting approximations we choosek _i complex numbers lying on a suitable circle contained in the annulusA _i, fori=1,...,q. The computation of Newton's correction is performed in such a way that overflow situations are removed. A suitable stop condition, based on a rigorous backward rounding error analysis, guarantees that the computed approximations are the exact zeros of a nearby polynomial. This implies the backward stability of our algorithm. We provide a Fortran 77 implementation of the algorithm which is robust against overflow and allows us to deal with polynomials of any degree, not necessarily monic, whose zeros and coefficients are representable as floating point numbers. In all the tests performed with more than 1000 polynomials having degrees from 10 up to 25,600 and randomly generated coefficients, the Fortran 77 implementation of our algorithm computed approximations to all the zeros within the relative precision allowed by the classical conditioning theorems with 11.1 average iterations. In the worst case the number of iterations needed has been at most 17. Comparisons with available public domain software and with the algorithm PA16AD of Harwell are performed and show the effectiveness of our approach. A multiprecision implementation in MATHEMATICA is presented together with the results of the numerical tests performed.Work performed under the support of the ESPRIT BRA project 6846 POSSO (POlynomial System SOlving).     

Convex envelopes for edge-concave functions

Deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions. In this paper we describe the structure of the polyhedral convex envelopes of edge-concave functions over polyhedral domains using geometric arguments. An algorithm for computing the facets of the convex envelope over hyperrectangles in ³ is described. Sufficient conditions are described under which the convex envelope of a sum of edge-concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions.Author to whom all correspondence should be addressed.     

Approximation in the finite element method

Summary The rate of convergence of the finite element method depends on the order to which the solutionu can be approximated by the trial space of piecewise polynomials. We attempt to unify the many published estimates, by proving that if the trial space is complete through polynomials of degreek–1, then it contains a functionv ^h such that |u–v ^h | _s ch ^k–s|u| _k . The derivatives of orders andk are measured either in the maximum norm or in the mean-square norm, and the estimate can be made local: the error in a given element depends on the diameterh _i of that element. The proof applies to domains in any number of dimensions, and employs a uniformity assumption which avoids degenerate element shapes.This research was supported by the National Science Foundation (GP-13778).     

Numerical properties of Chern classes of certain covering manifolds

In this paper, we study numerical properties of Chern classes of certain covering manifolds. One of the main results is the following: Let ψ : X → Pⁿ be a finite covering of the n-dimensional complex projective space branched along a hypersurface with only simple normal crossings and suppose X is nonsingular. Let c_i(X) be the i-th Chern class of X. Then (i) if the canonical divisor K_X is numerically effective, then (−1)^kc_k(X) (k ≥ 2) is numerically positive, and (ii) if X is of general type, then (−1)ⁿcⁱ_l (X) cⁱ_r, (X) > 0, where i_l + … + i_r = n. Furthermore we show that the same properties hold for certain Kummer coverings.     

Strong approximations of k-th records and k-th record times by Wiener processes

Summary We develop several strong approximations for k-th records and k-th record times by Wiener processes. Our main result is that it is possible to approximate by the same Wiener process up to the almost sure rate O(log j), where j stands for the index of the observations, the k-th records, the logarithms of the k-th record times, and the logarithms of the k-th interrecord times. We also provide some limiting weak laws and obtain Berry-Esséen-type theorems for k-th record times.     

A full analysis of a new second order finite volume approximation based on a low-order scheme using general admissible spatial meshes for the unsteady one dimensional heat equation

The present work is an extension of our previous works ,  and  which dealt with first order (both in time and space) and second order time accurate (second order in time and first order in space) implicit finite volume schemes for parabolic equations. We aim in this work (and some forthcoming studies) at getting higher order (both in time and space) finite volume approximations for the exact solution of parabolic equations using the class of spatial generic meshes introduced recently in [13]. We focus in the present contribution on the one dimensional heat equation and its implicit finite volume scheme described in [3]. The implicit finite volume scheme approximating the one dimensional heat equation we consider (hereafter referred to as the basic finite volume scheme) yields linear systems to be solved successively. The matrices involved in these linear systems are tridiagonal. The finite volume approximate solution is of order h+k$h+k$, where h (resp. k  ) is the mesh size of the spatial (resp. time) discretization. We construct a new finite volume approximation of order (h+k)²${(h+k)}^2$ in several discrete norms which allows us to get approximations of order two for the exact solution and its first derivatives. This new high-order approximation can be computed using the same linear systems involved in the basic finite volume scheme while the right hand sides are corrected. The construction of these right hand sides includes the approximations of the second, third, and fourth spatial derivatives of the exact solution. The computation of the approximation of these high-order derivatives can be performed using the same matrices stated above with another two tridiagonal matrices. The manner by which this new high-order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above. These high-order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of [12] without any restrictive condition on the spatial mesh. A full analysis for the stated theoretical results as well as some numerical examples supporting the theory is presented. The results obtained in the present study are based essentially on two facts. The first fact is the use of the results provided in [3] which state the convergence order of the finite volume approximate solution in several norms. The second fact is the comparison between the stated new higher order approximations and suitable auxiliary finite volume approximations.     

Independence of Likelihood Ratio Criteria for Homogeneity of Several Populations

Let _i be an i-tb population with a probability density function f(· | _i ) with one dimensional unknown parameter _i = 1, 2, ... , k. Let n _i sample be drawn from each _i . The likelihood ratio criteria _j|(j–1) for testing hypothesis that the first j parameters are equal against alternative hypothesis that the first (j – 1) parameters are equal and the j-th parameter is different with the previous ones are defined, j = 2, 3, ... , k. The paper shows the asymptotic independence of _j|(j–1)'s up to the order 1/n under a hypothesis of equality of k parameters, where n is a number of total samples.     

On some semiconvex envelopes

We show that for a continuous function with superlinear growth, Cf = Rf if and only if Cf = Qf, where Cf, Qf and Rf being the convex, rank-one convex and quasiconvex envelopes (relaxations).     

Componentwise estimation of ordered parameters ofk (≥2) exponential populations

We consider the estimation of ordered parameters ofk ( 2) exponential distributions by improving upon the usual estimators. TheBrewsterzidek technique is used to find sufficient conditions for an estimator of _i and/or _i (i=1,...,k), to be inadmissible with respect to the MSE criterion where _i and _i are the location and scale parameters respectively of thei-th exponential population. Using these sufficient conditions improved estimators of _i and/or _i (i=1,...,k) are obtained.     

A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables

Let {X _i, 1in} be a negatively associated sequence, and let {X* _i , 1in} be a sequence of independent random variables such that X* _i and X _i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef( ⁿ _i=1 X _i)Ef( ⁿ _i=1 X* _i ) for any convex function f on R ¹ and that Ef(max_1kn ⁿ _i=k X _i)Ef(max_1kn ^k _i=1 X* _i ) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.