approximate likelihood for large irregularly spaced spatial data likelihood approaches for large irregularly spaced spatial datasets are often very difficult if not infeasible to implement due to computational limitations even when we can assume normality exact calculations of the likelihood for a gaussian spatial process observed at n locations requires o n3 operations we present a version of whittle s approximation to the gaussian log likelihood for spatial regular lattices with missing values and for irregularly spaced datasets this method requires o nlog2n operations and does not involve calculating determinants we present simulations and theoretical results to show the benefits and the performance of the spatial likelihood approximation method presented here for spatial irregularly spaced datasets and lattices with missing values we apply these methods to estimate the spatial structure of sea surface temperatures using satellite data with missing values

edge effects and efficient parameter estimation for stationary random fields summary we consider the estimation of the parameters of a stationary random field on d dimensional lattice by minimizing the classical whittle approximation to the gaussian log likelihood if the usual biased sample covariances are used the estimate is efficient only in one dimension to remove this edge effect we introduce data tapers and show that the resulting modified estimate is efficient also in two and three dimensions this avoids the use of the unbiased sample covariances which are in general not positive definite

non uniform convergence of finite volume schemes for riemann problems of ideal magnetohydrodynamics this paper presents riemann test problems for ideal magnetohydrodynamics mhd finite volume schemes the test problems place emphasis on the hyperbolic irregularities of the ideal mhd system namely the occurrence of intermediate shocks and non unique solutions we investigate numerical solutions for the test problems obtained by several commonly used methods roe hlle central scheme all methods turned out to show a non uniform convergence behavior which may be paraphrased as pseudo convergence initially the methods show convergence towards a wrong solution with irregular wave patterns only after heavy grid refinement the methods switch to converge to the true solution this behavior is most pronounced whenever the solution should be unique but its phase space trajectory lies in the vicinity of that of a non unique solution we show detailed grid convergence studies and empirical error analysis the results may be related to similar results for a 2 2 model system and to time dependent investigations of other authors the non uniform convergence is expected to be present for any diffusive finite volume method and question the reliability of coarse grid ideal mhd solutions also in higher space dimensions

on the use of discrete laplace operator for preconditioning kernel matrices this paper presents a preconditioning strategy applied to certain types of kernel matrices that are increasingly ill conditioned the ill conditioning of these matrices is tied to the unbounded variation of the fourier transform of the kernel function hence the basic idea is to differentiate the kernel in order to suppress the variation the idea resembles some existing preconditioning methods for toeplitz matrices where the theory heavily relies on the underlying fixed generating function the theory does not apply to the case of a fixed domain with increasingly fine discretizations because the generating function depends on the grid size for this case we prove equal distribution results on the spectrum of the resulting matrices furthermore the proposed preconditioning technique also applies to non toeplitz matrices thus eliminating the reliance on a regular grid structure of the points the preconditioning strategy can be used to accelerate an iterative solver for solving linear systems with res

spectral methods for nonstationary spatial processes summary we propose a nonstationary periodogram and various parametric approaches for estimating the spectral density of a nonstationary spatial process we also study the asymptotic properties of the proposed estimators via shrinking asymptotics assuming the distance between neighbouring observations tends to zero as the size of the observation region grows without bound with this type of asymptotic model we can uniquely determine the spectral density avoiding the aliasing problem we also present a new class of nonstationary processes based on a convolution of local stationary processes this model has the advantage that the model is simultaneously defined everywhere unlike moving window approaches but it retains the attractive property that locally in small regions it behaves like a stationary spatial process applications include the spatial analysis and modelling of air pollution data provided by the us environmental protection agency

parameter estimation for a stationary process on a d dimensional lattice summary we study asymptotic properties of various estimation procedures for a general stationary regular process on a d dimensional lattice differences between d 1 time series and d 2 spatial processes are pointed out we suppose that the process is observed on a set pn with cn points which tends to infinity with the same speed in all directions the relative edge effect is of order cn lid increasing with d this effect is without statistical importance if d 1 but is important if d 2 we give a lcn consistent asymptotically normal estimator of the underlying parameter the procedure being constructed by a modification of whittle s approximation to the log likelihood in the gaussian case this procedure is asymptotically efficient

distribution results on the algebra generated by toeplitz sequences a finite dimensional approach abstract let a be an l 1 symbol defined on q d q with d 1 and let us consider the multi indexed sequence of toeplitz matrices t n a with n n d it is well known that t n a is spectrally distributed as a in the sense of the singular values in this paper we prove that a sequence as t n a is spectrally distributed in the sense of the singular values as the measurable function a for a l 1 and and ranging in any finite set of values

weights of twisted exponential sums let k be a finite field of characteristic p l a prime number different from p psi k to overline bf q l ast a nontrivial additive character and chi k ast n to overline bf q l ast a character on k ast n then defines an artin schreier sheaf mathcal l psi on the affine line bf a k 1 and defines a kummer sheaf mathcal k chi on the n dimensional torus bf t k n let f in k x 1 x 1 1 ldots x n x n 1 be a laurent polynomial it defines a k morphism f bf t k n to bf a k 1 in this paper we calculate the weights of h c i bf t bar k n mathcal k chi otimes f ast mathcal l psi under some non degeneracy conditions on f our results can be used to estimate sums of the form sum x 1 ldots x n in k ast chi 1 f 1 x 1 ldots x n cdots chi m f m x 1 ldots x n psi f x 1 ldots x n where chi 1 ldots chi m k ast to bf c ast are multiplicative characters psi k to bf c ast is a nontrivial additive character and f1 fm f are laurent polynomials

low dissipation centred schemes for hyperbolic equations in conservative and non conservative form abstract we propose a one parameter family of low dissipation centred numerical schemes for solving hyperbolic equations in conservative or non conservative form using finite volume or discontinuous galerkin finite element methods the new schemes spring out from the multi dimensional force method and are determined by a single parameter 1 given an increasing sequence of real numbers 1 1 2 k there corresponds a sequence of numerical schemes with stability restriction associated to a decreasing sequence of courant numbers 1 c 1 c 2 c k 0 and a decreasing sequence of corresponding numerical viscosity functions d 1 d 2 d k for a given courant number c k 1 there is a real number k 0 and a corresponding stable scheme with minimal numerical viscosity d k the proposed schemes suit very well the family of high order discontinuous galerkin finite element methods and the recently proposed class of ader tr schemes whose orders of accuracy define decreasing sequences of courant numbers as the order of accuracy increases the centred methods of this paper are stable in 2d and 3d in the frame of simultaneous updating formulae unlike other centred methods such as 1d force which are not furthermore the schemes are highly accurate for slowly moving waves which is precisely the kind of waves that traditional centred methods smear disastrously additional features of the proposed schemes include ease of implementation and applicability to any hyperbolic system either in conservative or non conservative form here the proposed schemes are analysed and computationally assessed through a suite of test problems for a linear model system for the euler equations in one and two space dimensions and for the baer nunziato equations for compressible two phase flow

minimum norm quadratic estimation of spatial variograms abstract the estimation of spatial variograms a measure of spatial correlation is a critical problem in the implementation of kriging a method for interpolating random fields we consider the use of minimum norm quadratic estimators of the variogram when it is specified up to a finite number of linear parameters we investigate the asymptotic behavior of such estimators for gaussian processes as the number of observations within some bounded region increases the basic conclusion is that we can estimate consistently those functions of the parameters and only those functions that have a nonnegligible impact asymptotically on the kriging procedure in general the behavior of the variogram over relatively short distances is the only aspect of the variogram that is asymptotically important as an example consider a gaussian process z on the real line with unknown constant mean and where is known as the semivariogram of the process and is a finite vector of unknown parameters suppose for 0

exponential sums and newton polyhedra cohomology and estimates the basic objects of this study are exponential sums on a variety defined over a finite field fq q pa p char fq as we have remarked in some earlier articles 1 2 we find it more natural to begin with exponential sums on the torus gm n extend via the usual toric decomposition of an to exponential sums on affine n space and finally proceed via a standard character argument 4 to exponential sums on an affine variety defined over fq while this is the natural order of the work what we do in fact in the first part of this article is to combine the first two steps and deal with exponential sums on varieties v of the form v gm r x as r s n let f e fq xl xn xl xr be an arbitrary regular function on v then f is a sum of monomials and as such has a well defined newton polyhedron a ff at infinity this is the convex closure in rn of the lattice points which occur as exponents of the terms of f together with the origin we have indicated in our previous work 1 2 the description of some of the invariants of the associated l function in terms of properties of this polyhedron for example in 1 we showed how bounds for the degree and total degree of the l function associated with a general exponential sum on v can be expressed in terms of the volumes of a f and the intersections of a f and the various coordinate spaces in the present article assuming f is nondegenerate and commode with respect to af f we show these estimates are sharp our methods are p adic and are based on the work of dwork 11 12 our main accomplishment from which our other results follow is the extension of dwork s cohomology theory from smooth projective hypersurfaces in characteristic p to a general class of exponential sums given f regular on v we construct a complex of p adic banach spaces on which frobenius acts the alternating product of characteristic polynomials of frobenius describes the associated l function in fact when f is nondegenerate and commode with respect to a f the complex is acyclic in dimensions other than 0 and the characteristic

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

locally toeplitz sequences spectral properties and applications abstract the notion of locally toeplitz sequence of matrices is introduced which extends the notion of toeplitz sequence of matrices the singular value distribution and in the hermitian case the eigenvalue distribution is completely characterized for locally toeplitz sequences and their sums obtaining weighted szeg formulas which extend well known results due to tyrtyshnikov concerning toeplitz matrices indeed any toeplitz sequence t n f where f is bounded is proved to be a locally toeplitz sequence moreover sufficient conditions are given for the product of two locally toeplitz sequences to be also locally toeplitz by combining these theoretic results we are able to explicitly compute the asymptotic spectral distribution of a large class of matrices arising in the applications including the algebra generated by toeplitz sequences and virtually all matrices resulting from the discretization of a unidimensional differential operator with non constant coefficients finally a large number of examples is discussed

stability of the notion of approximating class of sequences and applications given an approximating class of sequences b n m n m for a n n we prove that b n m n m x being the pseudo inverse of moore penrose is an approximating class of sequences for a n n where a n n is a sparsely vanishing sequence of matrices a n of size d n with d k d q for k q k email protected n as a consequence we extend distributional spectral results on the algebra generated by toeplitz sequences by including the pseudo inversion operation in the case where the sequences that are pseudo inverted are distributed as sparsely vanishing symbols applications to preconditioning and a potential use in image signal restoration problems are presented

a hybrid riemann solver for large hyperbolic systems of conservation laws this paper presents a riemann solver for nonlinear hyperbolic systems of conservation laws based on a krylov subspace approximation of the upwinding dissipation vector in the general case the solver does not require any detailed information of the eigensystem except an estimate of the global maximal propagation speed it uses successive flux function evaluations to obtain a numerical flux which is almost equivalent to that of a godunov scheme with complete upwinding the new krylov riemann solver is particularly efficient when used for large systems with many nonlinear equations such that typically no explicit expression for the eigensystem is available also no numerical procedures are necessary to compute the eigensystem numerical examples demonstrate the excellent performance of the solver with respect to other solvers

bilinear forms with kloosterman sums and applications we prove nontrivial bounds for general bilinear forms in hyper kloosterman sums when the sizes of both variables may be below the range controlled by fourier analytic methods polya vinogradov range we then derive applications to the second moment of cusp forms twisted by characters modulo primes and to the distribution in arithmetic progressions to large moduli of certain eisenstein hecke coefficients on gl 3 our main tools are new bounds for certain complete sums in three variables over finite fields proved using methods from algebraic geometry especially l adic cohomology and the riemann hypothesis

computing f a b via least squares polynomial approximations given a certain function f various methods have been proposed in the past for addressing the important problem of computing the matrix vector product f a b without explicitly computing the matrix f a such methods were typically developed for a specific function f a common case being that of the exponential this paper discusses a procedure based on least squares polynomials that can in principle be applied to any continuous function f the idea is to start by approximating the function by a spline of a desired accuracy then a particular definition of the function inner product is invoked that facilitates the computation of the least squares polynomial to this spline function since the function is approximated by a polynomial the matrix a is referenced only through a matrix vector multiplication in addition the choice of the inner product makes it possible to avoid numerical integration as an important application we consider the case when f t sqrt t and a is a sparse symmetric positive definite matrix which arises in sampling from a gaussian process distribution the covariance matrix of the distribution is defined by using a covariance function that has a compact support at a very large number of sites that are on a regular or irregular grid we derive error bounds and show extensive numerical results to illustrate the effectiveness of the proposed technique

hybrid entropy stable hll type riemann solvers for hyperbolic conservation laws it is known that hll type schemes are more dissipative than schemes based on characteristic decompositions however hll type methods offer greater flexibility to large systems of hyperbolic conservation laws because the eigenstructure of the flux jacobian is not needed we demonstrate in the present work that several hll type riemann solvers are provably entropy stable further we provide convex combinations of standard dissipation terms to create hybrid hll type methods that have less dissipation while retaining entropy stability the decrease in dissipation is demonstrated for the ideal mhd equations with a numerical example

the glt class as a generalized fourier analysis and applications recently the class of generalized locally toeplitz glt sequences has been introduced as a generalization both of classical toeplitz sequences and of variable coefficient differential operators and for every sequence of the class it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function the symbol that can be easily identified this generalizes the notion of a symbol for differential operators discrete and continuous or for toeplitz sequences for which it is identified through the fourier coefficients and is related to the classical fourier analysis the glt class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol sparsely vanishing symbol a symbol which vanishes at most in a set of zero lebesgue measure furthermore we show that the glt class virtually includes any finite difference or finite element discretization of pdes and based on this we demonstrate that our results on glt sequences can be used in a pde setting in various directions 1 as a generalized fourier analysis for the study of iterative and semi iterative methods when dealing with variable coefficients non rectangular domains non uniform gridding or triangulations 2 in order to provide a tool for the stability analysis of pde numerical schemes e g a necessary von neumann criterium for variable coefficient systems of pdes is obtained uniformly with respect to the boundary conditions 3 for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case etc the final part of the paper deals indeed with problems 1 3 and other possible directions in which the glt analysis can be conveniently employed

bounds on the efficiency of linear predictions using an incorrect covariance function suppose z is a random process defined on a bounded set r r d with finite second moments consider the behavior of linear predictions based on z t 1 z t n where t 1 t 2 is a dense sequence of points in r a general method is described for obtaining rates of convergence when the covariance function is misspecified but compatible with the correct covariance function when z is gaussian these bounds are related to the entropy distance between the measures for the random field under the actual and presumed covariance functions explicit bounds are given when r 0 1 and z is stationary with spectral density of the form f a 2 2 p where is a known positive integer and a is the parameter that is misspecified

l adic gkz hypergeometric sheaves and exponential sums abstract to a torus action on a complex vector space gelfand kapranov and zelevinsky introduce a system of differential equations called the gkz hypergeometric system its solutions are gkz hypergeometric functions we study the l adic counterpart of the gkz hypergeometric system which we call the l adic gkz hypergeometric sheaf it is an object in the derived category of l adic sheaves on the affine space over a finite field traces of frobenius on stalks of this object at rational points of the affine space define the hypergeometric functions over the finite field introduced by gelfand and graev we prove that the l adic gkz hypergeometric sheaf is perverse calculate its rank and prove that it is irreducible under the non resonance condition we also study the weight filtration of the gkz hypergeometric sheaf determine its lisse locus and apply our result to the study of weights of twisted exponential sums

characterizing surface smoothness via estimation of effective fractal dimension the fractal dimension d of stationary gaussian surfaces may be expressed very simply in terms of behaviour of the covariance function near the origin indeed only the covariance of line transect samples is required and that fact makes practical estimation of d relatively straightforward the case of non gaussian surfaces is more poorly understood but we might define effective fractal dimension in terms of the covariance function as though the surface were gaussian in the present paper we suggest simple practical methods for estimating effective fractal dimension based on the variogram like techniques proposed recently by taylor and taylor ours are founded on log linear regression

fast band toeplitz preconditioners for hermitian toeplitz systems this paper considers the solutions of hermitian toeplitz systems where the toeplitz matrices are generated by nonnegative functions f the preconditioned conjugate gradient method with well known circulant preconditioners fails in the case when f has zeros this paper employs toeplitz matrices of fixed bandwidth as preconditioners their generating functions g are trigonometric polynomials of fixed degree and are determined by minimizing the maximum relative error f g f infty it is shown that the condition number of systems preconditioned by the band toeplitz matrices are o 1 for f with or without zeros when f is positive the preconditioned systems converge at the same rate as other well known circulant preconditioned systems an a priori bound of the number of iterations required for convergence is also given

boundary conditions for regularized 13 moment equations for micro channel flows boundary conditions are the major obstacle in simulations based on advanced continuum models of rarefied and micro flows of gases in this paper we present a theory how to combine the regularized 13 moment equations derived from boltzmann s equation with boundary conditions obtained from maxwell s kinetic accommodation model while for the linear case these kinetic boundary conditions suffice we need additional conditions in the non linear case these are provided by the bulk solutions obtained after properly transforming the equations while keeping their asymptotic accuracy with respect to boltzmann s equation after finding a suitable set of boundary conditions and equations a numerical method for generic shear flow problems is formulated several test simulations demonstrate the stable and oscillation free performance of the new approach

fixed domain asymptotics for spatial periodograms abstract the periodogram for a spatial process observed on a lattice is often used to estimate the spectral density the bases for such estimators are two asymptotic properties that periodograms commonly possess 1 the periodogram at a particular frequency is approximately unbiased for the spectral density and 2 the correlation of the periodogram at distinct frequencies is approximately zero for spatial data it is often appropriate to use fixed domain asymptotics in which the observations get increasingly dense in some fixed region as their number increases using fixed domain asymptotics this article shows that standard asymptotic results for periodograms do not apply and that using the periodogram of the raw data can yield highly misleading results but by appropriately filtering the data before computing the periodogram it is possible to obtain results similar to the standard asymptotic results for spatial periodograms