on the sauer xu formula for the error in multivariate polynomial interpolation use of a new notion of multivariate divided difference leads to a short proof of a formula by sauer and xu for the error in interpolation by polynomials of total degree n in d variables at a correct point set

constructing nested bases approximations from the entries of non local operators in this article a method for constructing nested bases approximations to large scale fully populated discretizations of integral operators is introduced the scheme uses only few of the matrix entries for approximating the whole matrix in this sense it is similar to the adaptive cross approximation method however its computational complexity is improved

recent progress in unconstrained nonlinear optimization without derivatives we present an introduction to a new class of derivative free methods for unconstrained optimization we start by discussing the motivation for such methods and why they are in high demand by practitioners we then review the past developments in this field before introducing the features that characterize the newer algorithms in the context of a trust region framework we focus on techniques that ensure a suitable geometric quality of the considered models we then outline the class of algorithms based on these techniques as well as their respective merits we finally conclude the paper with a discussion of open questions and perspectives

asymptotic stability of rarefaction wave for the navier stokes equations for a compressible fluid in the half space this paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the navier stokes equations in a compressible fluid in the eulerian coordinate in the half space this is the second one of our series of papers on this subject in this paper firstly we classify completely the time asymptotic states according to some parameters that is the spatial asymptotic states and boundary conditions for this initial boundary value problem and some pictures for the classification of time asymptotic states are drawn in the state space in order to prove the stability of the rarefaction wave we use the solution to burgers equation to construct a suitably smooth approximation of the rarefaction wave and establish some time decay estimates in lp norm for the smoothed rarefaction wave we then employ the l2 energy method to prove that the rarefaction wave is non linearly stable under a small perturbation as time goes to infinity

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

a locally parametrized reduced order model for the linear frequency domain approach to time accurate computational fluid dynamics for transonic flows governed by the time accurate navier stokes equations small approximately periodic perturbations can be calculated accurately by transition to the frequency domain and truncating the fourier expansion after the first harmonic this is referred to as the linear frequency domain lfd method in this paper a parametric trajectory of reduced order models roms for the lfd solver is presented to this end several local projection based roms which are essentially specified by suitable low order subspaces are computed by the method of proper orthogonal decomposition pod in an offline stage the claimed trajectory is obtained locally by interpolating the given local subspaces considered as sample points in the grassmann manifold it is shown that the manifold interpolation technique is subject to certain restrictions moreover it turns out that the application of computing accurate roms for the lfd solver requires a special choice of underlying inner product necessitating a non eu

a leibniz formula for multivariate divided differences the leibniz formula for the divided difference of a product and opitz s formula for the divided difference table of a function as the result of evaluating that function at a certain matrix are shown to be special cases of a formula available for the coefficients with respect to any basis of an ideal or hermite polynomial interpolant in any number of variables

averaging of directional derivatives in vertices of nonobtuse regular triangulations for a shape regular triangulation mathcal t h without obtuse angles of a bounded polygonal domain omega subset re 2 let mathcal l h be the space of continuous functions linear on the triangles from mathcal t h and h the interpolation operator from c overline omega to mathcal l h this paper is devoted to the following classical problem find a second order approximation of the derivative partial u partial z a in a direction z of a function u in c 3 overline omega in a vertex a in the form of a linear combination of the constant directional derivatives partial pi h u partial z on the triangles surrounding a an effective procedure for such an approximation is presented its error is proved to be of the size o h 2 an operator mbox w h mathcal l h longrightarrow mathcal l h times mathcal l h relating a second order approximation wh h u of nabla u to every u in c 3 overline omega is constructed and shown to be a so called recovery operator the accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically

optimal sampling patterns for zernike polynomials a pattern of interpolation nodes on the disk is studied for which the interpolation problem is theoretically unisolvent and which renders a minimal numerical condition for the collocation matrix when the standard basis of zernike polynomials is used it is shown that these nodes have an excellent performance also from several alternative points of view providing a numerically stable surface reconstruction starting from both the elevation and the slope data sampling at these nodes allows for a more precise recovery of the coefficients in the zernike expansion of a wavefront or of an optical surface

l 2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method we consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension and we investigate the behavior of bounded l2 perturbations of shock wave solutions to the riemann problem using the relative entropy method we show that up to a time dependent translation of the shock the l2 norm of a perturbed solution relative to the shock wave is bounded above by the l2 norm of the initial perturbation

decay rate for perturbations of stationary discrete shocks for convex scalar conservation laws this paper is to study the decay rate for perturbations of stationary discrete shocks for the lax friedrichs scheme approximating the scalar conservation laws by means of an elementary weighted energy method if the summation of the initial perturbation over j is small and decays at the algebraic rate as j then the solution approaches the stationary discrete shock profiles at the corresponding rate as n this rate seems to be almost optimal compared with the rate in the continuous counterpart proofs are given by applying the weighted energy integration method to the integrated scheme of the original one the selection of the time dependent discrete weight function plays a crucial role in this procedure

on the complexity of optimization over the standard simplex we review complexity results for minimizing polynomials over the standard simplex and unit hypercube in addition we derive new results on the computational complexity of approximating the minimum of some classes of functions including lipschitz continuous functions on the standard simplex the main tools used in the analysis are bernstein approximation and lagrange interpolation on the simplex combined with an earlier result by de klerk et al a ptas for the minimization of polynomials of fixed degree over the simplex theoretical computer science 361 2 3 2006 210 225

acceleration of perturbation based electric field integral equations using fast fourier transform in this communication the computation of the perturbation based electric field integral equation of the form r n 1 n 0 1 2 ldots is accelerated by using fast fourier transform fft technique as an effective solution of the low frequency problem the perturbation method employs the taylor expansion of the scalar green s function in free space however multiple impedance matrices have to be solved at different frequency orders and the computational cost becomes extremely high especially for large scale problems since the perturbed kernels still satisfy toeplitz property on the uniform cartesian grid the fft based on lagrange interpolation can be well incorporated to accelerate the multiple matrix vector products because of the nonsingularity property of high order kernels when n geq 1 we do not need to do any near field amendment finally the efficiency of the proposed method is validated in an iterative solver with numerical examples

stochastic polynomial interpolation for uncertainty quantification with computer experiments multivariate polynomials are increasingly being used to construct emulators of computer models for uncertainty quantification for deterministic computer codes interpolating polynomial metamodels should be used instead of noninterpolating ones for logical consistency and prediction accuracy however available methods for constructing interpolating polynomials only provide point predictions there is no known method that can provide probabilistic statements about the interpolation error furthermore there are few alternatives to grid designs and sparse grids for constructing multivariate interpolating polynomials a significant disadvantage of these designs is the large gaps between allowable design sizes this article proposes a stochastic interpolating polynomial sip that seeks to overcome the problems discussed above a bayesian approach in which interpolation uncertainty is quantified probabilistically through the posterior distribution of the output is employed this allows assessment of the effe

stability of travelling waves for non convex scalar viscous conservation laws travelling waves of a viscous conservation law so called viscous profiles are shown to be stable in polynomialy weighted l spaces there is no assumption of convexity on the nonlinear term and thus earlier results of il in and ole nik sattinger and kawashima and matsumura are generalized the method uses the semigroup of the linearized equation with solutions of the full problem expressed by the variation of constants formula estimates are derived for the semigroup through a new technique for estimating the resolvent 1993 john wiley sons inc

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

on bivariate hermite interpolation with minimal degree polynomials a newton type approach is used to deal with bivariate polynomial hermite interpolation problems when the data are distributed in the intersections of two families of straight lines as a generalization of regular grids the interpolation operator is degree reducing and the interpolation space is a minimal degree space integral remainder formulas are given for the lagrange case then extended to the hermite case and finally used to obtain error estimates

scalar conservation laws with discontinuous flux function i the viscous profile condition the equation whereh is heaviside s step function appears for example in continuous sedimentation of solid particles in a liquid in two phase flow in traffic flow analysis and in ion etching the discontinuity of the flux function atx 0 causes a discontinuity of a solution which is not uniquely determined by the initial data the equation can be written as a triangular 2 2 non strictly hyperbolic system this augmentation is non unique and a natural definition is given by means of viscous profiles by a viscous profile we mean a stationary solution ofu t f x u xx wheref is a smooth approximation of the discontinuous flux i e h is smoothed in terms of the 2 2 system the discontinuity atx 0 is either a regular lax an under or overcompressive a marginal under or overcompressive or a degenerate shock wave in some cases depending onf andg there is a unique viscous profile e g undercompressive and regular lax waves and in some cases there are infinitely many e g overcompressive waves the main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution atx 0 and the viscous profile condition

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

interpolation lattices in several variables principal lattices are classical simplicial configurations of nodes suitable for multivariate polynomial interpolation in n dimensions a principal lattice can be described as the set of intersection points of n 1 pencils of parallel hyperplanes using a projective point of view lee and phillips extended this situation to n 1 linear pencils of hyperplanes in two recent papers two of us have introduced generalized principal lattices in the plane using cubic pencils in this paper we analyze the problem in n dimensions considering polynomial exponential and trigonometric pencils which can be combined in different ways to obtain generalized principal lattices we also consider the case of coincident pencils an error formula for generalized principal lattices is discussed

numerical testing of the stability of viscous shock waves a new theoretical evans function condition is used as the basis of a numerical test of viscous shock wave stability accuracy of the method is demonstrated through comparison against exact solutions a convergence study and evaluation of approximate error equations robustness is demonstrated by applying the method to waves for which no current analytic results apply highly nonlinear waves from the cubic model and strong shocks from gas dynamics an interesting aspect of the analysis is the need to incorporate features from the analytic evans function theory for purposes of numerical stability for example we find it necessary for numerical accuracy to solve odes on the space of wedge products

polynomial interpolation of minimal degree minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which lagrange interpolation with respect to the given points is uniquely solvable and degree reducing this is a generalization of the concept of least interpolation introduced by de boor and ron this paper investigates the behavior of lagrange interpolation with respect to these spaces giving a newton interpolation method and a remainder formula for the error of interpolation moreover a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view

newton cotes cubature rules over d 1 pencil lattices in this paper newton cotes cubature rules are extended to d 1 pencil lattices over simplices and simplicial partitions the closed form of the cubature rules as well as the error term are determined further the basic cubature rules can be combined with an adaptive algorithm over simplicial partitions the key point of the algorithm is a subdivision step that refines a d 1 pencil lattice over a simplex to its subsimplices if the number of function evaluations is crucial the additional freedom provided by d 1 pencil lattices may be used to decrease it significantly

condor a new parallel constrained extension of powell s uobyqa algorithm experimental results and comparison with the dfo algorithm this paper presents an algorithmic extension of powell s uobyqa algorithm unconstrained optimization by quadratical approximation we start by summarizing the original algorithm of powell and by presenting it in a more comprehensible form thereafter we report comparative numerical results between uobyqa dfo and a parallel constrained extension of uobyqa that will be called in the paper condor constrained non linear direct parallel optimization using trust region method for high computing load function the experimental results are very encouraging and validate the approach they open wide possibilities in the field of noisy and high computing load objective functions optimization from 2 min to several days like for instance industrial shape optimization based on computation fluid dynamic codes or partial differential equations solvers finally we present a new easily comprehensible and fully stand alone implementation in c of the parallel algorithm

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

multipoint taylor formulae the main result of this paper is an abstract version of the kowalewski ciarlet wagschal multipoint taylor formula for representing the pointwise error in multivariate lagrange interpolation several applications of this result are given in the paper the most important of these is the construction of a multipoint taylor error formula for a general finite element together with the corresponding l p error bounds another application is the construction of a family of error formulae for linear interpolation indexed by real measures of unit mass which includes some recently obtained formulae it is also shown how the problem of constructing an error formula for lagrange interpolation from a d invariant space of polynomials with the property that it involves only derivatives which annihilate the interpolating space can be reduced to the problem of finding such a formula for a simpler one point interpolation map

interpolation by ridge polynomials and its application in neural networks we consider multivariate interpolation at arbitrary points and provide effective interpolation formulas by using simple ridge polynomials which essentially possess the nature of univariate polynomials in computation and show that ridge polynomials with least degree can be found for interpolation purpose which a computational algorithm is provided correspondingly we then apply the interpolation results to design a computational algorithm for training feedforward neural networks

asymptotic stability of traveling wave solutions of systems for one dimensional gas motion the asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics 1 the solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet for some 0 ast provided that the initial disturbance is small and of integral zero and in addition decays at an algebraic rate for x 2 the traveling wave solution with nishida and smoller s condition of the system of a viscous heat conductive ideal gas is asymptotically stable provided the initial disturbance is small and of integral zero 3 the traveling wave solution with weak shock profile of the broadwell model system of the boltzmann equation is asymptotically stable provided the initial disturbance is small and its hydrodynamical moments are of integral zero each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one the property of integral zero of the initial disturbance plays a crucial role in this procedure

spectral stability of small shock waves this paper is the first in a series of articles that study the eigenvalue problem for small amplitude shock waves in systems of conservation laws with viscosity or relaxation the papers show in various contexts that in the zero amplitude limit appropriately scaled versions of the associated evans bundles converge to suspensions of evans bundles for fixed shock waves in related reduced systems with lower dimensional state space in this article the new geometric framework establishing this connection is introduced in the simplest context that of shock waves associated with a simple genuinely nonlinear mode in systems with identity viscosity in one space dimension

a modified fractional step method for the accurate approximation of detonation waves the numerical approximation of combustion processes may lead to numerical difficulties which are caused by different time scales of the transport part and the reactive part of the model equations here we consider a modified fractional step method that overcomes this difficulty on standard test problems and allows the use of a mesh width and time step determined by the nonreactive part without precisely resolving the very small reaction zone high resolution godunov methods are employed and the structure of the riemann solution is used to determine where burning should occur in each time step the modification is implemented in the software package clawpack numerical results for 1d and 2d detonation waves are shown including a detonation wave diffracting around a corner

comparison of fast boundary element methods on parametric surfaces we compare fast black box boundary element methods on parametric surfaces in r3 these are the adaptive cross approximation the multipole method based on interpolation and the wavelet galerkin scheme the surface representation by a piecewise smooth parameterization is in contrast to the common approximation of surfaces by panels nonetheless parametric surface representations are easily accessible from computer aided design cad and are recently topic of the studies in isogeometric analysis especially we can apply two dimensional interpolation in the multipole method a main feature of this approach is that the cluster bases and the respective moment matrices are independent of the geometry this results in a superior compression of the far field compared to other cluster methods

multivariate divided differences with simple knots in d dimensions d geq1 the order ga mathord in nd divided difference of a function f from ifmmode rr d else rr d fi to mathbb r associated with a pp n unisolvent set a subset ifmmode rr d else rr d fi is the coefficient of x ifmmode alpha else alpha fi of the interpolating polynomial of the points a f a a mathord in a it is a linear combination of pointwise values of f in a this is a generalization of the usual dd obtained for d 1 and ifmmode alpha else alpha fi n with which we compare the main properties

multivariate polynomial interpolation in newton forms techniques of univariate newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms the newton basis format with divided diffe

stability of bright solitary wave solutions to perturbed nonlinear schrodinger equations abstract the propagation of pulses in ideal nonlinear optical fibers without loss is governed by the nonlinear schrodinger equation nls when considering realistic fibers one must examine perturbed nls equations with the particular perturbation depending on the physical situation that is being modeled a common example is the complex ginzburg landau equation cgl which is a dissipative perturbation it is known that some of the stable bright solitons of the nls survive a dissipative perturbation such as the cgl given that a wave persists it is then important to determine its stability with respect to the perturbed nls a major difficulty in analyzing the stability of solitary waves upon adding dissipative terms is that eigenvalues may bifurcate out of the essential spectrum since the essential spectrum of the nls is located on the imaginary axis such eigenvalues may lead to an unstable wave in fact eigenvalues can pop out of the essential spectrum even if the unperturbed problem has no eigenvalue embedded in the essential spectrum here we present a technique which can be used to track these bifurcating eigenvalues as a consequence we are able to locate the spectrum for bright solitary wave solutions to various perturbed nonlinear schrodinger equations and determine precise conditions on parameters for which the waves are stable in addition we show that a particular perturbation the parametrically forced nls equation supports stable multi bump solitary waves the technique for tracking eigenvalues which bifurcate from the essential spectrum is very general and should therefore be applicable to a larger class of problems than those presented here

directions for computing truncated multivariate taylor series efficient recurrence relations for computing arbitrary order taylor coefficients for any univariate function can be directly applied to a function of n variables by fixing a direction in r n after a sequence of directions the multivariate taylor coefficients or partial derivatives can be reconstructed or interpolated the sequence of univariate calculations is more efficient than multivariate methods although previous work indicates a space cost for this savings and significant cost for the reconstruction we completely eliminate this space cost and develop a much more efficient algorithm to perform the reconstruction by appropriate choice of directions the reconstruction reduces to a sequence of lagrange polynomial interpolation problems in r n 1 for which a divided difference algorithm computes the coefficients of a newton form another algorithm collects like terms from the newton form and returns the desired multivariate coefficients

scalar conservation laws with discontinuous flux function ii on the stability of the viscous profiles the equation frac partial u partial t frac partial partial x left h x f u left 1 h x right g u right 0 whereh is heaviside s step function appears for example in continuous sedimentation of solid particles in a liquid in two phase flow in traffic flow analysis and in ion etching the discontinuity of the flux function atx 0 causes a discontinuity of a solution which is not uniquely determined by the initial data by a viscous profile of this discontinuity we mean a stationary solution ofu t f x eu xx wheref is a smooth approximation of the discontinuous flux i e h is smoothed we present some results on the stability of the viscous profiles which means that a small disturbance tends to zero uniformly ast this is done by weighted energy methods where the weight depending onf andg plays a crucial role

on the history of multivariate polynomial interpolation multivariate polynomial interpolation is a basic and fundamental subject in approximation theory and numerical analysis which has received and continues receiving not deep but constant attention in this short survey we review its development in the first 75 years of this century including a pioneering paper by kronecker in the 19th century

multivariate hermite interpolation by algebraic polynomials a survey abstract this is a survey of that theory of multivariate lagrange and hermite interpolation by algebraic polynomials which has been developed in the past 20 years its purpose is not to be encyclopedic but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples it takes classical hermite interpolation as a starting point but then successively broadens the assumptions so that finally interpolation of arbitrary functionals and the theory of singularities from algebraic geometry is discussed