optimal isoparametric finite elements and error estimates for domains involving curved boundaries we describe a practical procedure for the triangulation of arbitrary n dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy along similar lines we prove general error estimates for the finite element solution of second order elliptic problems various kinds of nonhomogeneous boundary conditions can be dealt with such as dirichlet neumann and robin boundary conditions

advances in metric embedding theory metric embedding plays an important role in a vast range of application areas such as computer vision computational biology machine learning networking statistics and mathematical psychology to name a few the mathematical theory of metric embedding is well studied in both pure and applied analysis and has more recently been a source of interest for computer scientists as well most of this work is focused on the development of bi lipschitz mappings between metric spaces in this paper we present new concepts in metric embeddings as well as new embedding methods for metric spaces we focus on finite metric spaces however some of the concepts and methods are applicable in other settings as well one of the main cornerstones in finite metric embedding theory is a celebrated theorem of bourgain which states that every finite metric space on n points embeds in euclidean space with o logn distortion bourgain s result is best possible when considering the worst case distortion over all pairs of points in the metric space yet it is natural to ask can an embedding do much better in terms of the average distortion indeed in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs in this paper we provide an embedding with constant average distortion for arbitrary metric spaces while maintaining the same worst case bound provided by bourgain s theorem in fact our embedding possesses a much stronger property we define the lq distortion of a uniformly distributed pair of points our embedding achieves the best possible lq distortion for all 1 q simultaneously the results are based on novel embedding methods which improve on previous methods in another important aspect the dimension of the host space the dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it however no previous result improved the bound on the dimension which can be derived from bourgain s embedding our embedding methods achieve better dimension and in fact shed new light on another fundamental question in metric embedding which is whether the embedding dimension of a metric space is related to its intrinsic dimension i e whether the dimension in which it can be embedded in some real normed space is related to the intrinsic dimension which is reflected by the inherent geometry of the space measured by the space s doubling dimension the existence of such an embedding was conjectured by assouad 4 and was later posed as an open problem in several papers our embeddings give the first positive result of this type showing any finite metric space obtains a low distortion and constant average distortion embedding in euclidean space in dimension proportional to its doubling dimension underlying our results is a novel embedding method probabilistic metric decomposition techniques have played a central role in the field of finite metric embedding in recent years here we introduce a novel notion of probabilistic metric decompositions which comes particularly natural in the context of embedding our new methodology provides a unified approach to all known results on embedding of arbitrary finite metric spaces moreover as described above with some additional ideas they allow to get far stronger results the results presented in this paper5have been the basis for further developments both within the field of metric embedding and in other areas such as graph theory distributed computing and algorithms we present a comprehensive study of the notions and concepts introduced here and provide additional extensions related results and some examples of algorithmic applications

a third order boundary value problem with jumping nonlinearities abstract a third order boundary value problem with jumping nonlinearities is studied some results obtained for second order boundary value problems with jumping nonlinearities are shown to hold also for the third order case

a lag averaged generalization of euler s method for accelerating series abstract euler s method is an efficient accelerator for an alternating series that is one whose terms a j are such that sign a j sign a j 1 however its effectiveness deteriorates rapidly as the period p of the oscillations in degree j increases to reduce this nonuniformity we introduce an algorithm which begins with the recursive computation of a triangular array of numbers the diagonal elements define a generalization of euler s method each horizontal row defines a delayed euler method which is even more effective the key element is that the array is generated by repeatedly averaging partial sums of the original series with a lag that is a distance in degree between the partial sums that are combined which optimally is half the period p of the oscillations in the series we illustrate the algorithm with the fourier series for a function with a shock wave jump discontinuity the terms a j 1 j sin jx vary from strictly alternating p 2 at x 0 to monotonic p as x the lag averaged euler scheme with and without delay is much more efficient than its classical counterpart especially near the discontinuities we present an analytical convergence theory for both the standard and generalized euler methods

on the peroidic bvp for the forced duffing equation with jumping nonlinearity on etudie le probleme suivant u cu f t u e t p p sur 0 t u t u 0 0 u t u 0 0 c est une fonction arbitraire dans l 1 0 t r c r et l application non lineaire f est a saut

a united approach to accelerating trigonometric expansions abstract the fourier series of a smooth function on a compact interval usually has slow convergence due to the fact that the periodic extension of the function has jumps at the interval endpoints for various symmetry conditions polynomial interpolation methods have been developed for performing a boundary correction the resulting variants of krylov approximants are a sum of a correction polynomial and a fourier sum of the corrected function 1 8 in this paper we review these methods and derive estimates in the maximum norm we further show that derivatives of the krylov approximants are again krylov approximants of derivatives of the considered function this enables us to give a unified treatment of the problem of simultaneous approximation

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

quality mesh generation in higher dimensions we consider the problem of triangulating a d dimensional region our mesh generation algorithm called qmg is a quadtree based algorithm that can triangulate any polyhedral region including nonconvex regions with holes furthermore our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles finally our algorithm is guaranteed never to overrefine the domain in the sense that the number of simplices produced by qmg is bounded above by a factor times the number produced by any competing algorithm where the factor depends on the aspect ratio bound satisfied by the competing algorithm the qmg algorithm has been implemented in c and is used as a mesh generator for the finite element method

combinatorial characterization of the null spaces of symmetric h matrices abstract we characterize the structure of null spaces of symmetric diagonally dominant sdd matrices and symmetric h matrices with non negative diagonal entries we show that the structure of the null space of a symmetric sdd matrix or h matrix a depends on the structure of the connected components of its underlying graph each connected component contributes at most one vector to the null space this paper provides a combinatorial characterization of the rank of each connected component and a combinatorial characterization of a null vector if one exists for sdd matrices we also present an efficient combinatorial algorithm for constructing an orthonormal basis for the null space the paper also shows a close connection between gain graphs and h matrices which extends known results regarding the connection between undirected graphs and laplacian matrices and between signed graphs and sdd matrices we show how to exploit these combinatorial algorithms to reliably solve certain singular linear systems in finite precision arithmetic

convergence acceleration of modified fourier series in one or more dimensions modified fourier series have recently been introduced as an adjustment of classical fourier series for the approximation of nonperiodic functions defined on d variate cubes such approximations offer a number of advantages including uniform convergence however like fourier series the rate of convergence is typically slow in this paper we extend eckhoff s method to the convergence acceleration of multivariate modified fourier series by suitable augmentation of the approximation basis we demonstrate how to increase the convergence rate to an arbitrary algebraic order moreover we illustrate how numerical stability of the method can be improved by utilising appropriate auxiliary functions in the univariate setting it is known that eckhoff s method exhibits an auto correction phenomenon we extend this result to the multivariate case finally we demonstrate how a significant reduction in the number of approximation coefficients can be achieved by using a hyperbolic cross index set

acceleration of algebraically converging fourier series when the coefficients have series in powers of 1 n if the coefficients in a fourier cosine series f x f email protected n 0 a ncos nx decrease as a small negative power of n then one may need millions of terms to sum the series to high accuracy we show that if the a n are known analytically and have a power series in 1 n then it is straightforward to approximate f x as a series of what we shall the lanczos krylov lk functions we describe the similar methodology for sine series general fourier series are merely the sum of a cosine series with a sine series and thus are implicitly handled too for cosine coefficients that involve only even powers of n and sine coefficients that are functions of odd powers of n the lk functions may be expressed in terms of bernoulli polynomials the lk functions for cosine coefficients involving odd powers of n and for sine coefficients in even powers of n are not known explicitly these are also known as clausen functions we provide rapidly convergent series to compute these clausen functions to high accuracy our method includes the endpoint subtraction ideas of lanczos and krylov but is more general the sum n 1 1 n 1 1 email protected cos nx where l 0 is a constant arises in phase transitions in absorbed monolayers on metal surfaces it is easily summed by our method which correctly incorporates the logarithmic singularities at x email protected

stable finite elements for problems with wild coefficients we consider solving an elliptic boundary value problem in the case that the coefficients vary by many orders of magnitude over the domain a linear finite element method is used it is shown that the standard method for solving the resulting linear equations in finite precision arithmetic can give an arbitrarily inaccurate answer because of ill conditioning in the stiffness matrix a new method for solving the linear equations is proposed this method is based on a mixed formulation and gives a numerically accurate answer independent of the variation in the coefficients the numerical error in the soution of the linear system for the new method is shown to depend on the aspect ratio of the triangulation

the calculation of trigonometric fourier coefficients a method for the numerical evaluation of trigonometric fourier coefficients is described this method is suitable for sufficiently smooth functions and is essentially a modification of the standard method based on trapezoidal rule sums using optionally the fast fourier transform

remarks and generalizations related to a recent multiplicity result of a lazer and p mckenna we will show here that this result has a natural generalization to the case that u 5 ai to state the result we consider the model problem lu au autel 2 where 0 0 and set t 0 we have a nonlinear eigenvalue problem lu yu au 4 the following proposition is known cf fueik 3 dancer 2 and also lo where the result appears in the formulation given below

on the number of solutions of two classes of sturm liouville boundary value problems abstract we employ global bifurcation theory of p h rabinowitz together with sturm comparison theorem to establish multiplicity results for some classes of sturm liouville boundary value problems

solving elliptic finite element systems in near linear time with support preconditioners we consider linear systems arising from the use of the finite element method for solving scalar linear elliptic problems our main result is that these linear systems which are symmetric and positive semidefinite are well approximated by symmetric diagonally dominant matrices our framework for defining matrix approximation is support theory significant graph theoretic work has already been developed in the support framework for preconditioners in the diagonally dominant case and in particular it is known that such systems can be solved with iterative methods in nearly linear time thus our approximation result implies that these graph theoretic techniques can also solve a class of finite element problems in nearly linear time we show that the support number bounds which control the number of iterations in the preconditioned iterative solver depend on mesh quality measures but not on the problem size or shape of the domain

perturbation theory for the ldu factorization and accurate computations for diagonally dominant matrices we present a structured perturbation theory for the ldu factorization of row diagonally dominant matrices and we use this theory to prove that a recent algorithm of ye math comp 77 264 2195 2230 2008 computes the l d and u factors of these matrices with relative errors less than 14n 3 u where u is the unit roundoff and n n is the size of the matrix the relative errors for d are componentwise and for l and u are normwise with respect the max norm a m max ij a ij these error bounds guarantee that for any diagonally dominant matrix a we can compute accurately its singular value decomposition and the solution of the linear system ax b for most vectors b independently of the magnitude of the traditional condition number of a and in o n 3 flops

algebraic mesh quality metrics quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics the theory based on the jacobian and related matrices provides a means of constructing classifying and evaluating mesh quality metrics the jacobian matrix is factored into geometrically meaningful parts a nodally invariant jacobian matrix can be defined for simplicial elements using a weight matrix derived from the jacobian matrix of an ideal reference element scale and orientation invariant algebraic mesh quality metrics are defined the singular value decomposition is used to study relationships between metrics equivalence of the element condition number and mean ratio metrics is proved the condition number is shown to measure the distance of an element to the set of degenerate elements algebraic measures for skew length ratio shape volume and orientation are defined abstractly with specific examples given two combined metrics shape volume and shape volume orientation are algebraically defined and examples of such metrics are given algebraic mesh quality metrics are extended to nonsimplicial elements a series of numerical tests verifies the theoretical properties of the metrics defined

nonautonomous boundary value problems with jumping nonlinearities soit f 0 1 r r une fonction continue continument differentiable par rapport au second argument sous l hypothese que les limites lim s f x s s x lim s f x s x x existent et sont uniformes sur 0 1 on etudie le probleme aux valeurs limites u f x u th u 0 u 1 0 ou t est un nombre reel et h une fonction continue pour f fixee on etudie le nombre de solutions

a robust method for accurately representing nonperiodic functions given fourier coefficient information when a function is smooth but not smoothly periodic with a particular period and nonetheless is represented by partial sums of a fourier series calculated using that period the well known gibbs phenomenon defeats uniform convergence of the sums and convergence is slow in recent years several workers have developed methods for recovering accurate and fast converging representations for functions in this situation these efforts have not concentrated on bounds for the operators corresponding to the methods and thus have not explicitly proven robustness in the presence of noise in this paper we present a method for which explicit bounds are established for the operator the method is in effect least squares fitting of the given fourier coefficients by the coefficients of polynomial splines with appropriate discontinuities we obtain bounds by exact calculations of projections in spline spaces using a computer algebra system we give examples of the method and two other published methods working with noisy data

convergence acceleration of fourier series by analytical and numerical application of poisson s formula a form of the poisson summation formula appropriate for the transformation of infinite fourier series is derived and shown to be applicable to slowly converging series arising in the solution of laplace s equation in a rectangular electrochemical cell the result is a rapidly convergent fourier series which is particularly useful in giving the potential close to the electrode surface results obtained from the analytical expression are found to be in close agreement with those resulting from numerical evaluation of the fourier integrals using simpson s rule and the euler transformation and direct summation by using goertzel s 1958 algorithm

generalization of the inverse polynomial reconstruction method in the resolution of the gibbs phenomenon the finite fourier representation of a function f x exhibits oscillations where the function or its derivatives are nonsmooth this is known as the gibbs phenomenon a robust and accurate reconstruction method that resolves the gibbs oscillations was proposed in a previous paper j comput appl math 161 2003 41 based on the inversion of the transformation matrix which represents the projection of a set of basis functions onto the fourier space if the function is a polynomial this inverse polynomial reconstruction method iprm is exact in this paper we develop the iprm by requiring that the proper error be orthogonal to the fourier or polynomial space the iprm is generalized to any set of basis functions the primitive basis polynomials nonclassical orthogonal polynomials and the gegenbauer polynomials are used to illustrate the wide validity of the iprm it is shown that the iprm yields a unique reconstruction irrespective of the basis set for any analytic function and yields spectral convergence the ill posedness of the transformation matrix due to the exponential growth of the condition number of the matrix is also discussed

on three classes of matrices with variants of the diagonal dominance property abstract we introduce three new classes jgd n ludd n and dwdd n of n n complex matrices with variants of the diagonal dominance property each of the three classes contains singular matrices we investigate in depth the relations between the three classes and their relations with the classes of diagonally dominant and generalized diagonally dominant matrices we also study the problem of providing necessary and sufficient conditions for the singularity of a matrix in each of the three classes jgd n ludd n and dwdd n we present an application of the theory of the classes jgd n ludd n and dwdd n to the iterative jacobi method for the solution of the linear system a x b

a unified approach for predicting sound radiation from baffled rectangular plates with arbitrary boundary conditions this paper discusses sound radiation from a baffled rectangular plate with each of its edges arbitrarily supported in the form of elastic restraints the plate displacement function is universally expressed as a 2 d fourier cosine series supplemented by several 1 d series the unknown fourier expansion coefficients are then determined by using the rayleigh ritz procedure once the vibration field is solved the displacement function is further simplified to a single standard 2 d fourier cosine series in the subsequent acoustic analysis thus the sound radiation from a rectangular plate can always be obtained from the radiation resistance matrix for an invariant set of cosine functions regardless of its actual dimensions and boundary conditions further this radiation resistance matrix unlike the traditional ones for modal functions only needs to be calculated once for all plates with the same aspect ratio in order to determine the radiation resistance matrix effectively an analytical formula is derived in the form of a power series of the non dimensional acoustic wavenumber the formula is mathematically valid and accurate for any wavenumber several numerical examples are presented to validate the formulations and show the effect of the boundary conditions on the radiation behavior of planar sources