Alternating iteration and elliptic variational inequalities

Summary Free boundary value problems, too complicated for formulation as a variational inequality, are broken up into two problems on overlapping regions. On one region the problem is treated as an ordinary boundary value problem; on the second region, the free boundary part of the problem is reduced to a variational inequality. By solving the two problems successively it is shown that under certain conditions the successive solutions converge to a single function that gives a solution of the original problem. Application to a filtration problem is given.     

Perturbation analysis of the orthogonal procrustes problem

Given two arbitrary real matricesA andB of the same size, the orthogonal Procrustes problem is to find an orthogonal matrixM such that the Frobenius norm MA – B is minimized. This paper treats the common case when the orthogonal matrixM is required to have a positive determinant. The stability of the problem is studied and supremum results for the perturbation bounds are derived.     

Discrete Newton methods and iterated defect corrections

Summary In this paper we present a general theory for discrete Newton methods, iterated defect corrections via neighbouring problems and deferred corrections based on asymptotic expansions of the discretization error.Dedicated to Professor Dr. J. Weisinger on the occasion of his sixty-fifth birthday     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Nonlinear multiscale decompositions: The approach of A. Harten

Datadependent interpolatory techniques can be used in the reconstruction step of a multiresolution scheme designed à la Harten. In this paper we carefully analyze the class of Essentially NonOscillatory (ENO) interpolatory techniques described in [11] and their potential to improve the compression capabilities of multiresolution schemes. When dealing with nonlinear multiresolution schemes the issue of stability also needs to be carefully considered.     

Parallel frontal methods for ODE's

This paper concerns parallel frontal predictor-corrector methods. Order and stability of these methods are investigated, when the corrector is solved both by the fixed point iteration method and by the Newton method.This work has been partially supported by the Italian C.N.R. within the Finalized Project Sistemi Informatici e Calcolo Parallelo.     

Vorzeichenstabile Differenzenverfahren für parabolische Anfangsrandwertaufgaben

Summary In this paper we consider certain structure conserving properties of finite difference methods for the solution of parabolic initial-boundary value problems. We are interested in conditions on the step size ratio =t/x ² in one-step methods which guarantee that the number of sign changes of the discrete approximation does not increase while proceeding from one time level to the following one. This means that difference schemes of this type possess a so-called variation-diminishing property which is known to hold for continuous diffusion equations also. It turns out that our conditions on are stronger than the classical ones which imply the maximum principle for the finite difference equations. By means of an example we show that our sign stability condition is necessary too.

     

Approximation of a nondifferentiable nonlinear problem related to MHD equilibria

Summary We present a simple method, based on a variant of the implicit function theorem, which leads to the existence of (a part of) a nontrivial solution branch of the nonlinear eigenvalue problem –u=u ⁺ in ,u=–1 on , where is a two-dimensional domain with boundary . The advantage of this method is that we can apply it for analysing the approximation of the above problem by a finite element method; the error analysis of the discrete problem appears immediately. We give also an iteration scheme which allows to solve the approximate problem.     

Para-orthogonal Laurent polynomials and associated sequences of rational functions

On the space, , of Laurent polynomials (L-polynomials) we consider a linear functional which is positive definite on (0, ) and is defined in terms of a given bisequence, { _k } _– . Two sequences of orthogonal L-polynomials, {Q _n (z) ₀ and , are constructed which span in the order {1,z ^–1,z,z ^–2,z ²,...} and {1,z,z ^–1,z ²,z ^–2,...} respectively. Associated sequences of L-polynomials {P _n (z) ₀ , and are introduced and we define rational functions , wherew is a fixed positive number. The partial fraction decomposition and integral representation of,M _n (z, w) are given and correspondence of {M _n (z, w)} is discussed. We get additional solutions to the strong Stieltjes moment problem from subsequences of {M _n (z, w)}. In particular when { _k } _– is a log-normal bisequence, {M _2n (z, w)} and {M _2n+1 (z, w)} yield such solutions.Research supported in part by the National Science Foundation under Grant DMS-9103141.     

Die mehrstellenformeln für den Laplaceoperator

Summary Difference methods for the numerical solution of linear partial differential equations may often be improved by using a weighted right hand side instead of the original right hand side of the differential equation. Difference formulas, for which that is possible, are called Mehrstellenformeln or Hermitian formulas. In this paper the Hermitian formulas for the approximation of Laplace's operator are characterized by a very simple condition. We prove, that in two-dimensional case for a Hermitian formula of ordern at leastn+3 discretization points are necessary. We give examples of such optimal formulas of arbitrary high-order.

     

Turán's extremal problem in random graphs: Forbidding odd cycles

For 0<1 and graphsG andH, we writeGH if any -proportion of the edges ofG span at least one copy ofH inG. As customary, we writeC ^k for a cycle of lengthk. We show that, for every fixed integerl1 and real >0, there exists a real constantC=C(l, ), such that almost every random graphG _{n, p} withp=p(n)Cn ^–1+1/2l satisfiesG _n,p1/2+ C ^2l+1. In particular, for any fixedl1 and >0, this result implies the existence of very sparse graphsG withG _1/2+ C ^2l+1.The first author was partially supported by NSERC. The second author was partially supported by FAPESP (Proc. 93/0603-1) and by CNP_q (Proc. 300334/93-1). The third author was partially sopported by KBN grant 2 1087 91 01.     

A note on skewcirculant preconditioners for elliptic problems

In a recent paper Chan and Chan study the use of circulant preconditioners for the solution of elliptic problems. They prove that circulant preconditioners can be chosen so that the condition number of the preconditioned system can be reduced fromO(n ² ) toO(n). In addition, using the Fast Fourier Transform, the computation of the preconditioner is highly parallelizable. To obtain their result, Chan and Chan introduce a shift /p/n ² for some >0. The aim of this paper is to consider skewcirculant preconditioners, and to show that in this case the condition number ofO(n) can easily be shown without using the somewhat unsatisfactory shift /p/n ². Furthermore, our estimates are more precise.     

Algorithmique et calculs de complexité pour un solveur de type dissections emboîtées

Résumé On considère la méthode de dissections emboîtées basée sur des théorèmes de séparation introduite par Gilbert-Tarjan et Roman utilisée pour la résolution par élimination de Gauss de grands systèmes linéaires creux. Plus précisemment, on étudie une structure de données par blocs similaire à celle proposée par George dans le cadre des graphes en grille, et on démontre les propriétés suivantes: d'une part, pour des familles de graphes à degré borné admettant unn -théorème de séparation, 1/2<1, le stockage secondaire de la structure par blocs contenant la matrice factorisée est linéaire par rapport à la taille du système; d'autre part, en rajoutant une hypothèse non restrictive sur la manière d'effectuer la séparation, la structure peut être construite en temps linéaire par une factorisation logique par blocs. Des exemples numériques illustrent ces résultats théoriques.

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Algorithmic study and complexity bounds for a nested dissection solver

Summary We consider the nested dissection method based on separator theorems introduced by Gilbert-Tarjan and Roman used for solving large sparse systems of linear equations. More precisely, we study a block storage scheme such as proposed by George for regular square grids and we prove the following results: first, for families of graphs of bounded degree withn -separator theorem, 1/2<1, the overhead storage of the block data structure for the factored matrix is linear in system dimension; on the other hand, by adding a non restrictive assumption on the separation, this structure can be constructed by a block symbolic factorization which runs in time linear in matrix dimension. Numerical experiments illustrating these theoretical results are provided.

     

Generalizations of the projection method with applications to SOR theory for hermitian positive semidefinite linear systems

Summary Ann×n complex matrixB is calledparacontracting if B₂1 and 0x[N(I-B)]Bx₂<x₂. We show that a productB=B _k B _k–1 ...B ₁ ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofB in terms of the subdominant singular values of theB _i 's and in terms of the angles between certain subspaces. Our results here extend earlier results due to Halperin and due to Smith, Solomon and Wagner. We also determine necessary and sufficient conditions forn numbers in the interval [0, 1] to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product. In the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the SOR iteration matrix associated with ann×n hermitian positive semidefinite matrixA none of whose diagonal entries vanish.The work of this author was supported in part by NSF Research Grant No. MCS-8400879     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

Round-off error analysis of iterations for large linear systems

Summary We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA _x =b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A ^*>0 andA has PropertyA. This means that the computed resultx _k approximates the exact solution with relative error of order A·A ^–1 where is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector Ax _k –b is of order A² A ^–1 and hence the remaining three iterations arenot well-behaved.This work was partly done during the author's visit at Carnegie-Mellon University and it was supported in part by the Office of Naval Research under Contract N00014-76-C-0370; NR 044-422 and by the National Science Foundation under Grant MCS75-222-55     

Efficient nonparametric estimation of distribution density in the basis of algebraic polynomials

A nonparametric estimatef ^* of an unknown distribution densityf W is called locally minimax iff it is minimax for all not too small neighborhoodsW _g ,g W, simultaneously, whereW is some dense subset ofW. Radaviius and Rudzkis proved the existence of such an estimate under some general conditions. However, the construction of the estimate is rather complicated. In this paper, a new estimate is proposed. This estimate is locally minimax under some additional assumptions which usually hold for orthobases of algebraic polynomial and is almost as simple as the linear projective estimate. Thus, it takes a form convenient for the construction of an adaptive estimator, which does not usea-priori information about the smoothness of the density. The adaptive estimation problem is briefly discussed and an unknown density fitting by Jacobi polynomials is investigated more explicitly.     

Elementarym-harmonic cardinal B-splines

We generalize the notion of B-spline to the thin plate splines and to otherd-dimensional polyharmonic splines as defined in [Duchon, [3]]; for regular nets, we give the main properties of these B-splines: Fourier transform, decay when x , stability, integration property, links between B-splines of different orders or of different dimensions and in particular link with the polynomial B-splines, approximation using B-splines... We show that, in some sense, B-splines may be considered as a regularized form of the Dirac distribution.     

Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements

Summary A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain is partitioned into two subdomains ₁ and ₂.Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in . Finally, a hybrid situation in which viscosity is dropped out only in ₁ is addressed. The latter is motivated by physical applications.In all cases, correct transmission conditions across the interface between ₁ and ₂ are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed.The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out.We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size.This work was partially supported by CIRA S.p.A. under the contract Coupling of Euler and Navier-Stokes equations in hypersonic flowsDeceased     

Adams methods for neutral functional differential equations

Summary In this paper Adams type methods for the special case of neutral functional differential equations are examined. It is shown thatk-step methods maintain orderk+1 for sufficiently small step size in a sufficiently smooth situation. However, when these methods are applied to an equation with a non-smooth solution the order of convergence is only one. Some computational considerations are given and numerical experiments are presented.