on leja sequences some results and applications in this paper we collect and review some results concerning leja sequences on one and two dimensional domains then we present two applications and discuss some open problems on the asymptotic distribution of leja points on two dimensional domains

second and fourth painleve hierarchies and jimbo miwa linear problems the relations between the different linear problems for painleve equations is an intriguing open problem here we consider our previously given second and fourth painleve hierarchies publ res inst math sci kyoto 37 327 347 2001 and show that they could alternatively have been derived using the linear problems of jimbo and miwa that is we give a gauge transformation of our linear problems for these two hierarchies which maps those of the second and fourth painleve equations themselves onto those of jimbo and miwa

on the computation of sets of points with low lebesgue constant on the unit disk abstract in this paper of numerical nature we test the lebesgue constant of several available point sets on the disk and propose new ones that enjoy low lebesgue constant furthermore we extend some results in cuyt 2012 analyzing the case of bos arrays whose radii are nonnegative gauss gegenbauer lobatto nodes with exponent noticing that the optimal still allows to achieve point sets on with low lebesgue constant n for degrees n 30 next we introduce an algorithm that through optimization determines point sets with the best known lebesgue constants for n 25 finally we determine theoretically a point set with the best lebesgue constant for the case n 1

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

numerical integration in natural neighbour galerkin methods in this paper issues regarding numerical integration of the discrete system of equations arising from natural neighbour natural element galerkin methods are addressed the sources of error in the traditional delaunay triangle based numerical integration are investigated two alternative numerical integration schemes are analysed first a local approach in which nodal shape function supports are exactly decomposed into triangles and circle segments is shown not to give accurate enough results second a stabilized nodal quadrature scheme is shown to render high levels of accuracy while resulting specially appropriate in a natural neighbour galerkin approximation method the paper is completed with several examples showing the performance of the proposed techniques copyright 2004 john wiley sons ltd

an algorithm for computing fekete points in the triangle on the line and its tensor products fekete points are known to be the gauss lobatto quadrature points but unlike high order quadrature fekete points generalize to non tensor product domains such as the triangle thus fekete points might serve as an alternative to the gauss lobatto points for certain applications in this work we present a new algorithm to compute fekete points and give results up to degree 19 for the triangle for degree d 10 these points have the smallest lebesgue constant currently known the computations validate a conjecture of bos j approx theory 64 1991 pp 271 280 that fekete points along the boundary of the triangle are the one dimensional gauss lobatto points

on the constrained mock chebyshev least squares the algebraic polynomial interpolation on n 1 uniformly distributed nodes can be affected by the runge phenomenon also when the function f to be interpolated is analytic among all techniques that have been proposed to defeat this phenomenon there is the mock chebyshev interpolation which produces a polynomial p that interpolates f on a subset of m 1 of the given nodes whose elements mimic as well as possible the chebyshev lobatto points of order m in this work we use the simultaneous approximation theory to produce a polynomial p of degree r greater than m which still interpolates f on the m 1 mock chebyshev nodes minimizing at the same time the approximation error in a least squares sense on the other points of the sampling grid we give indications on how to select the degree r in order to obtain polynomial approximant good in the uniform norm furthermore we provide a sufficient condition under which the accuracy of the mock chebyshev interpolation in the uniform norm is improved numerical results are provided

an analysis of finite difference and finite volume formulations of conservation laws abstract finite difference and finite volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws a new coordinate free formulation of systems of conservation laws is developed which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system the analysis considers general types of equations potential euler and navier stokes three dimensional unsteady flows with time varying grids are described using a single consistent nomenclature for both formulations grid motion due to a non inertial reference frame as well as flow adaptation is covered in comparing the two formulations it is found useful to distinguish between differences in numerical methods and differences in grid definition the former plays a role for non cartesian grids and results in only cosmetic differences in the manner in which geometric terms are handled the differences in grid definition for the two formulations is found to be more important since it affects the manner in which boundary conditions zonal procedures and grid singularities are handled at computational boundaries the proper interpretation of strong and weak conservation law forms for quasi one dimensional and axisymmetric flows is brought out

on the inverse problem of calculus of variations for fourth order equations fels conditions fels m e 1996 trans amer math soc 348 5007 5029 doi 10 1090 s0002 9947 96 01720 5 ensure the existence and uniqueness of the lagrangian in the case of a fourth order equation we show that when fels conditions are satisfied the lagrangian can be derived from the jacobi last multiplier as in the case of a second order equation indeed we prove that if a lagrangian exists for an equation of any even order then it can be derived from the jacobi last multiplier two equations from a number theory paper by hall hall r r 2002 j number theory 93 235 245 doi 10 1006 jnth 2001 2719 one of the second and one of the fourth order will be used to exemplify the method the known link between jacobi last multiplier and lie symmetries is also exploited finally the lagrangians of two fourth order equations drawn from physics are determined with the same method

caustics in 1 1 integrable systems caustics arising in the asymptotic description of rapidly decaying solutions of the equations integrable via the inverse scattering transform are defined different asymptotic approaches to the description are considered the appearance of members n2 of the hierarchy of the second painleve equation as special functions of wave catastrophies is discussed the adjoining problem n2 k2 n k for the simplest example 22 2 is considered in detail

uniform weighted approximation on the square by polynomial interpolation at chebyshev nodes abstract the paper deals with de la vallee poussin type interpolation on the square at tensor product chebyshev zeros of the first kind the approximation is studied in the space of locally continuous functions with possible algebraic singularities on the boundary equipped with weighted uniform norms in particular simple necessary and sufficient conditions are proved for the uniform boundedness of the related lebesgue constants error estimates in some sobolev type spaces are also given pros and cons of such a kind of filtered interpolation are analyzed in comparison with the lagrange polynomials interpolating at the same chebyshev grid or at the equal number of padua nodes the advantages in reducing the gibbs phenomenon are shown by means of some numerical experiments

computing fekete and lebesgue points simplex square disk we have computed point sets with maximal absolute value of the vandermonde determinant fekete points or minimal lebesgue constant lebesgue points on three basic bidimensional compact sets the simplex the square and the disk using routines of the matlab optimization toolbox we have obtained some of the best bivariate interpolation sets known so far

emptiness formation probability of the six vertex model and the sixth painleve equation we show that the emptiness formation probability of the six vertex model with domain wall boundary conditions at its free fermion point is a tau function of the sixth painleve equation using this fact we derive asymptotics of the emptiness formation probability in the thermodynamic limit

hyperinterpolation on the square we show that hyperinterpolation at near minimal cubature points for the product chebyshev measure along with xu compact formula for the corresponding reproducing kernel provide a simple and powerful polynomial approximation formula in the uniform norm on the square the lebesgue constant of the hyperinterpolation operator grows like log 2 of the degree as that of quasi optimal interpolation sets recently proposed in the literature moreover we give an accurate implementation of the hyperinterpolation formula with linear cost in the number of cubature points and we compare it with interpolation formulas at the same set of points

voronoi based finite volume methods optimal voronoi meshes and pdes on the sphere we first develop and analyze a finite volume scheme for the discretization of partial differential equations pdes on the sphere the scheme uses voronoi tessellations of the sphere for a model convection diffusion problem the finite volume scheme is shown to produce first order accurate approximations with respect to a mesh dependent discrete firstderivative norm then we introduce the notion of constrained centroidal voronoi tessellations ccvts of the sphere these are special voronoi tessellation of the sphere for which the generators of the voronoi cells are also the constrained centers of mass with respect to a prescribed density function of the cells after discussing an algorithm for determining ccvt meshes on the sphere we discuss and illustrate several desirable properties possessed by these meshes in particular it is shown that ccvt meshes define very high quality uniform and non uniform meshes on the sphere finally we discuss through some computational experiments the performance of the ccvt meshes used in conjunction with the finite volume scheme for the solution of simple model pdes on the sphere the experiments show for example that the ccvt based finite volume approximations are second order accurate if errors are measured in

bivariate lagrange interpolation at the node points of lissajous curves the degenerate case in this article we study bivariate polynomial interpolation on the node points of degenerate lissajous figures these node points form chebyshev lattices of rank 1 and are generalizations of the well known padua points we show that these node points allow unique interpolation in appropriately defined spaces of polynomials and give explicit formulas for the lagrange basis polynomials further we prove mean and uniform convergence of the interpolating schemes for the uniform convergence the growth of the lebesgue constant has to be taken into consideration it turns out that this growth is of logarithmic nature

voronoi cell finite difference method for accurate electronic structure calculation of polyatomic molecules on unstructured grids we introduce a new numerical grid based method on unstructured grids in the three dimensional real space to investigate the electronic structure of polyatomic molecules the voronoi cell finite difference vfd method realizes a discrete laplacian operator based on voronoi cells and their natural neighbors featuring high adaptivity and simplicity to resolve multicenter coulomb singularity in all electron calculations of polyatomic molecules this method utilizes highly adaptive molecular grids which consist of spherical atomic grids it provides accurate and efficient solutions for the schrodinger equation and the poisson equation with the all electron coulomb potentials regardless of the coordinate system and the molecular symmetry for numerical examples we assess accuracy of the vfd method for electronic structures of one electron polyatomic systems and apply the method to the density functional theory for many electron polyatomic molecules

overview and construction of meshfree basis functions from moving least squares to entropy approximants in this paper an overview of the construction of meshfree basis functions is presented with particular emphasis on moving least squares approximants natural neighbour based polygonal interpolants and entropy approximants the use of information theoretic variational principles to derive approximation schemes is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints the maximization minimization of the shannon jaynes entropy functional relative entropy functional is used to unify the construction of globally and locally supported convex approximation schemes a java applet is used to visualize the meshfree basis functions and comparisons and links between different meshfree approximation schemes are presented copyright 2006 john wiley sons ltd

bivariate polynomial interpolation on the square at new nodal sets abstract as known the problem of choosing good nodes is a central one in polynomial interpolation while the problem is essentially solved in one dimension all good nodal sequences are asymptotically equidistributed with respect to the arc cosine metric in several variables it still represents a substantially open question in this work we consider new nodal sets for bivariate polynomial interpolation on the square first we consider fast leja points for tensor product interpolation on the other hand for interpolation in p n 2 on the square we experiment four families of points which are asymptotically equidistributed with respect to the dubiner metric which extends to higher dimension the arc cosine metric one of them nicknamed padua points gives numerically a lebesgue constant growing like log square of the degree

computing approximate fekete points by qr factorizations of vandermonde matrices we propose a numerical method implemented in matlab for computing approximate fekete points on compact multivariate domains it relies on the search of maximum volume submatrices of vandermonde matrices computed on suitable discretization meshes and uses a simple greedy algorithm based on qr factorization with column pivoting the method gives also automatically an algebraic cubature formula provided that the moments of the underlying polynomial basis are known numerical tests are presented for the interval and the square which show that approximate fekete points are well suited for polynomial interpolation and cubature

high order finite volume approximations of differential operators on nonuniform grids abstract when computing numerical solutions to partial differential equations difference operators that mimic the crucial properties of the differential operators are usually more accurate than those that do not properties such as symmetry conservation stability and the duality relationships and identities between the gradient curl and divergence operators are all important using the finite volume method we have derived local accurate reliable and efficient difference methods t divergence gradient and curl operators are defined using a discrete versions of the divergence theorem and stokes theorem these methods are especially powerful on coarse nonuniform grids and in calculations where the mesh moves to track interfaces or shocks numerical examples comparing local second and fourth order finite volume approximations to conservation laws on very rough grids are used to demonstrate the advantages of the higher order methods

discrete weighted least squares method for the poisson and biharmonic problems on domains with smooth boundary in this article a discrete weighted least squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented it is shown how to create well conditioned matrices of the resulting system of linear equations using algebraic polynomials carefully selected matching points and weight factors two simple algorithms generating suitable matching points the chebyshev matching points for standard two dimensional domains and the approximate fekete points of sommariva and vianello for general domains are described the efficiency of the presented method is demonstrated by solving the poisson and biharmonic problems with the homogeneous dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions

an isomonodromy cluster of two regular singularities we consider a linear 2 2 matrix ode with two coalescing regular singularities this coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ode in particular a zero distance limit for the ode exists the monodromy group of the limiting ode is calculated in terms of the original one this coalescing process generates a limit for the corresponding nonlinear systems of isomonodromy deformations in our main example the latter limit reads as p 6 p 5 where p n is the nth painleve equation we also discuss some general problems which arise while studying the above mentioned limits for the painleve equations

geometric weakly admissible meshes discrete least squares approximations and approximate fekete points using the concept of geometric weakly admissible meshes see 2 below together with an algorithm based on the classical qr factorization of matrices we compute efficient points for discrete multivariate least squares approximation and lagrange interpolation

asymptotics of oscillatory riemann hilbert problems a classical method of stationary phase for oscillatory integrals is generalized to oscillatory riemann hilbert problems of the kind arising in the theory of integrable nonlinear equations the proposed approach is developed for the phase with n first order stationary points and the final formulas can immediately be applied to the problem of long time behavior of the solutions of equations such as the nls kdv mkdv sine gordon and others

bivariate lagrange interpolation at the padua points the ideal theory approach the padua points are a family of points on the square 1 1 2 given by explicit formulas that admits unique lagrange interpolation by bivariate polynomials interpolation polynomials and cubature formulas based on the padua points are studied from an ideal theoretic point of view which leads to the discovery of a compact formula for the interpolation polynomials the lp convergence of the interpolation polynomials is also studied

fast calculation of scattering by 3 d inhomogeneities in uniaxial anisotropic multilayers a fast algorithm to calculate the response of uniaxial layered media on a rectilinear mesh is proposed in order to handle scattering phenomena due to inhomogeneities within them the uniaxial layered media are with their optical axes parallel to the interfaces of the layered structure the generalized poisson summation formula is used to efficiently calculate the discrete fourier transform of spatial responses of the layeredmedia from its continuous fourier spectrum using the windowing technique the integral involved in constructing the discrete fourier spectrum can be performed in a small domain without compromising the accuracy of the result furthermore a fast numerical integration method based on the recently proposed padua points is used which together with the windowing technique yields a fast solution to such scattering problems numerical simulations illustrate the accuracy and efficiency of the proposed algorithm

voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids natural neighbours are based on the voronoi diagram which partitions space into closest point regions the sibson and the laplace non sibsonian interpolants which are based on natural neighbours have shown promise within a galerkin framework for the solution of partial differential equations in this paper we focus on the laplace interpolant with a two fold objective first to unify the previous developments related to the laplace interpolant and to indicate its ties to some well known numerical methods and secondly to propose a voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids a conservation law in integral form is discretized on voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids the proposed scheme can also be viewed as a point collocation technique a detailed study on consistency is conducted and the satisfaction of the discrete maximum principle stability is established owing to symmetry of the laplace weight a symmetric positive definite stiffness matrix is realized which permits the use of efficient linear solvers on a regular rectangular or hexagonal grid the difference scheme reduces to the classical finite difference method numerical examples for the poisson equation with dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme copyright 2003 john wiley sons ltd

bivariate lagrange interpolation at the padua points the so called padua points give a simple geometric and explicit construction of bivariate polynomial interpolation in the square moreover the associated lebesgue constant has minimal order o

bivariate lagrange interpolation at the padua points computational aspects the so called padua points give a simple geometric and explicit construction of bivariate polynomial interpolation in the square moreover the associated lebesgue constant has minimal order of growth o log 2 n here we show four families of padua points for interpolation at any even or odd degree n and we present a stable and efficient implementation of the corresponding lagrange interpolation formula based on the representation in a suitable orthogonal basis we also discuss extension of non polynomial padua like interpolation to other domains such as triangles and ellipses we give complexity and error estimates and several numerical tests

on certain multivariate vandermonde determinants whose variables separate abstract we prove that for almost square tensor product grids and certain sets of bivariate polynomials the vandermonde determinant can be factored into a product of univariate vandermonde determinants this result generalizes the conjecture 3 lemma 1 as a special case we apply the result to padua and padua like points

natural neighbour galerkin methods summary natural neighbour co ordinates sibson co ordinates is a well known interpolation scheme for multivariate data tting and smoothing the numerical implementation of natural neighbour co ordinates in a galerkin method is known as the natural element method nem in the natural element method natural neighbour co ordinates are used to construct the trial and test functions recent studies on nem have shown that natural neighbour co ordinates which are based on the voronoi tessellation of a set of nodes are an appealing choice to construct meshless interpolants for the solution of partial dierential equations in belikov et al computational mathematics and mathematical physics 1997 37 1 9 15 a new interpolation scheme non sibsonian interpolation based on natural neighbours was proposed in the present paper the nonsibsonian interpolation scheme is reviewed and its performance in a galerkin method for the solution of elliptic partial dierential equations that arise in linear elasticity is studied a methodology to couple nite elements to nem is also described two signicant advantages of the non sibson interpolant over the sibson interpolant are revealed and numerically veried the computational eciency of the non sibson algorithm in 2 dimensions which is expected to carry over to 3 dimensions and the ability to exactly impose essential boundary conditions on the boundaries of convex and non convex domains copyright 2001 john wiley sons ltd

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

bivariate lagrange interpolation at the node points of non degenerate lissajous curves motivated by an application in magnetic particle imaging we study bivariate lagrange interpolation at the node points of lissajous curves the resulting theory is a generalization of the polynomial interpolation theory developed for a node set known as padua points with appropriately defined polynomial spaces we will show that the node points of non degenerate lissajous curves allow unique interpolation and can be used for quadrature rules in the bivariate setting an explicit formula for the lagrange polynomials allows to compute the interpolating polynomial with a simple algorithmic scheme compared to the already established schemes of the padua and xu points the numerical results for the proposed scheme show similar approximation errors and a similar growth of the lebesgue constant

gaussian cubature and bivariate polynomial interpolation gaussian cubature is used to study bivariate polynomial interpolation based on the common zeros of quasi orthogonal polynomials

tensor product gauss lobatto points are fekete points for the cube tensor products of gauss lobatto quadrature points are frequently used as collocation points in spectral element methods unfortunately it is not known if gauss lobatto points exist in non tensor product domains like the simplex in this work we show that the n dimensional tensor product of gauss lobatto quadrature points are also fekete points this suggests a way to generalize spectral methods based on gauss lobatto points to non tensor product domains since fekete points are known to exist and have been computed in the triangle and tetrahedron in one dimension this result was proved by fejer in 1932 but the extension to higher dimensions in non trivial

least squares polynomial approximation on weakly admissible meshes disk and triangle we construct symmetric polar wams weakly admissible meshes with low cardinality for least squares polynomial approximation on the disk these are then mapped to an arbitrary triangle numerical tests show that the growth of the least squares projection uniform norm is much slower than the theoretical bound and even slower than that of the lebesgue constant of the best known interpolation points for the triangle as opposed to good interpolation points such meshes are straightforward to compute for any degree the construction can be extended to polygons by triangulation

computational method for general multicenter electronic structure calculations here a three dimensional fully numerical i e chemical basis set free method p f batcho phys rev a 57 6 1998 is formulated and applied to the calculation of the electronic structure of general multicenter hamiltonian systems the numerical method is presented and applied to the solution of schr odinger type operators where a given number of nuclei point singularities is present in the potential field the numerical method combines the rapid exponential convergence rates of modern spectral methods with the multiresolution flexibility of finite element methods and can be viewed as an extension of the spectral element method the approximation of cusps in the wave function and the formulation of multicenter nuclei singularities are efficiently dealt with by the combination of a coordinate transformation and a piecewise variational spectral approximation the complete system can be efficiently inverted by established iterative methods for elliptical partial differential equations an application of the method is presented for atomic diatomic and triatomic systems and comparisons are made to the literature when possible in particular local density approximations are studied within the context of kohn sham density functional theory and are presented for selected subsets of atomic and diatomic molecules as well as the ozone molecule

electronic structure calculations using self adaptive multiscale voronoi basis functions the multiscale self adaptivity of voronoi basis functions is currently proving to be useful for the simulation of complex fluid systems involving structures on a number of distinct lengthscales in this paper we explore the possibility of extending the use of such multiscale basis functions to the framework of density functional theoretic electronic structure computations