multigrid methods for combined finite difference and fourier problems abstract multigrid methods for combined finite difference and fourier pseudospectral problems are considered applications to two and three dimensional elliptic systems are given suitable relaxation schemes for isotropic and anisotropic problems are presented numerical results demonstrate the efficiency of these methods

chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

the admissibility of the kaplan meier and other maximum likelihood estimators in the presence of censoring for the nonparametric estimation of a survival function when censoring is present the kaplan meier estimator is often used the admissibility of this estimator and other related maximum likelihood estimators is demonstrated this is done by reducing the problem to one involving just the multinomial distribution and then using the stepwise bayes technique to prove admissibility 1 introduction the admissibility of various well known nonparametric estimators was recently demonstrated in meeden ghosh and vardeman 1985 the argument proving admissibility involved two essential steps the first was to note that to prove admissibility for the nonparametric problem it was enough to prove admissibility when the family of possible distributions was just assumed to be some multinomial family of distributions rather than the large nonparametric family the second step then used the stepwise bayes technique to prove admissibility for the multinomial problem here we will use the same argument to prove the admissibility of the kaplan meier estimator for the nonparametric estimation of the survival function when censoring is present in fact the admissibility of a whole class of maximum likelihood estimators which includes the kaplan meier estimator is demonstrated the only additional complication is that the censoring mechanism must be modeled and then taken into account this description and modeling of the censoring is carried out in section 3 in section 4 the stepwise bayes technique is used to demonstrate the admissibility of the maximum likelihood estimator for the multinomial problem in section 5 the admissibility results for the nonparametric problem are given if we were just proving the admissibility of the kaplan meier estimator this paper would be considerably shorter although the proofs for the kaplan meier estimator and the more general case are essentially the same the general case requires considerably more notation for someone who is just interested in the kaplan meier estimator the argument can be simplified using the notation of

spectral methods for initial boundary value problems an alternative approach mea this paper presents a new approach to spectral methods for initial boundary value problems a filtered version of the partial differential equation and the initial and boundary conditions at an overdetermined set of points are collocated as an approximate solution the function is chosen that belongs to an appropriate finite dimensional space and minimizes a weighted average of the residuals at these points it is proved that the approximate solution converges to the actual solution at a spectral rate of accuracy in both space and time the proof is based on a priori energy estimates that have been proved for such systems although this method is restricted here to hyperbolic initial boundary value problems and chebyshev polynomials it generalizes to general initial boundary value problems boundary value problems and gegenbauer polynomials

a bayesian bootstrap for censored data a bayesian bootstrap for a censored data model is introduced its small sample distributional properties are discussed and found to be similar to efron s bootstrap for censored data in the absence of censoring the bayesian bootstrap for censored data reduces to rubin s bayesian bootstrap for complete data a first order large sample theory is developed this theory shows that both censored data bootstraps are consistent bootstraps for approximating the sampling distribution of the kaplan meier estimator it also shows that both bootstraps are consistent bootstraps for approximating a posterior distribution of the survival function with respect to each member of the class of conjugate beta neutral process priors

spectral methods for periodic initial value problems with nonsmooth data in this paper we consider hyperbolic initial value problems subject to periodic boundary conditions with nonsmooth data we show that if we filter the data and solve the problem by the galerkin collocation method recently proposed by us then we can recover pointwise values with spectral accuracy provided that the actual solution is piecewise smooth for this we have to perform a local smoothing of the computed solution

an l 1 3 algorithm for ideal class group and regulator computation in certain number fields we analyse the complexity of the computation of the class group structure regulator and a system of fundamental units of a certain class of number fields our approach differs from buchmann s who proved a complexity bound of l 1 2 o 1 when the discriminant tends to infinity with fixed degree we achieve a subexponential complexity in o l 1 3 o 1 when both the discriminant and the degree of the extension tend to infinity by using techniques due to enge and gaudry in the context of algebraic curves over finite fields

bayesian methods for censored categorical data abstract bayesian methods are given for finite category sampling when some of the observations suffer missing category distinctions dickey s 1983 generalization of the dirichlet family of prior distributions is found to be closed under such censored sampling the posterior moments and predictive probabilities are proportional to ratios of b c carlson s multiple hypergeometric functions closed form expressions are developed for the case of nested reported sets when bayesian estimates can be computed easily from relative frequencies effective computational methods are also given in the general case an example involving surveys of death penalty attitudes is used throughout to illustrate the theory a simple special case of categorical missing data is a two way contingency table with cross classified count data xij i 1 r j 1 c together with supplementary trials counted only in the margin distinguishing the rows yi i 1 r there could also be further supplementary trials report

a complete class theorem for statistical problems with finite sample spaces this paper contains a complete class theorem theorem 3 2 which applies to most statistical estimation problems having a finite sample space this theorem also applies to many other statistical problems with finite sample spaces the description of this complete class involves a stepwise algorithm at each step of the process it is necessary to construct the bayes procedures in a suitably modified version of the original problem the complete class is a minimal complete class if the loss function is strictly convex some examples are given to illustrate the application of this complete class theorem among these is a new result concerning the estimation of the parameters of a multinomial distribution under a normalized quadratic loss function see example 4 5

global properties of pseudospectral methods abstract we apply polynomial interpolation methods both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations the derivative matrix for a general sequence of collocation points is explicitly constructed we explore the effect of several factors on the performance of these methods we show that global methods cannot be interpreted in terms of local methods for example the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary this difference does not depend on the density of the collocation points near the boundaries hence intuition based on finite difference methods can lead to false results

relaxation schemes for spectral multigrid methods the effectiveness of relaxation schemes for solving the systems of algebraic equations which arise from spectral discretizations of elliptic equations is examined iterative methods are an attractive alternative to direct methods because fourier transform techniques enable the discrete matrix vector products to be computed almost as efficiently as for corresponding but sparse finite difference discretizations preconditioning is found to be essential for acceptable rates of convergence preconditioners based on second order finite difference methods are used a comparison is made of the performance of different relaxation methods on model problems with a variety of conditions specified around the boundary the investigations show that iterations based on incomplete lu decompositions provide the most efficient methods for solving these algebraic systems

on computing isomorphisms of equation orders a number geometric method for computing isomorphisms of algebraic number fields respectively z orders of such fields is developed its main advantage is its easy implementation and moderate computation time

spectral methods for hyperbolic initial boundary value problems on parallel computers abstract in this paper a parallel spectral algorithm is developed for hyperbolic initial boundary value problems in one space dimension the galerkin collocation method which is spectrally accurate in both space and time is parallelized by using domain decomposition this procedure leads to a minimization problem in which there is coupling at inter domain boundaries we construct a decoupled preconditioner which can be used to iteratively solve the minimization problem symmetric formulation of the problem which is needed to compute the residual for the normal equations is discussed the methodology outlined for computing the normal equations applies equally well to computation of the residual for the p and h p versions of the finite element method there is therefore no need to compute the mass and stiffness matrices to obtain the residual as is normally done this leads to a great saving in time and memory particularly for solving nonlinear problems using the p and h p versions of the finite element method the method we discuss in this paper generalizes to hyperbolic initial boundary value problems in multidimensions too provided the computational boundaries we have introduced are noncharacteristic and the system is symmetrizable finally we show that for the case of analytic coefficients and data satisfying all the required compatibility conditions so that the solution is analytic the numerical solution is exponentially accurate in n where n is proportional to the number of subdomains and the number of degrees of freedom in each element

extending the family of bayesian bootstraps and exchangeable urn schemes summary a bayesian bootstrap for a finite population with censored observations is introduced it is shown to reduce to the finite population bayesian bootstrap if there is no censoring and to reduce to the censored data bayesian bootstrap for a large population a class of general urn schemes for simulating exchangeable sequences of variables is introduced which is connected to the bootstrap method

dependency of units in number fields we develop a method for validating the indepencence of units in algebraic number fields in case that a given system of units has a dependency we compute a certificate for this

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

chebyshev expansion methods for the solution of elliptic partial differential equations abstract in a recent paper haidvogel and zang j comput phys 30 1979 167 described a method based on an expansion in chebyshev polynomials for the solution of poisson s equation in a rectangle they also indicated how the method might be extended to more general equations in this note we give a comparison of the method with the global element method delves and hall j inst math appl 23 1979 223 and give numerical results for those problems treated by haidvogel and zang

spectral multigrid methods for elliptic equations ii a detailed description of spectral multigrid methods is provided this includes the interpolation and coarse grid operators for both periodic and dirichlet problems the spectral methods for periodic problems use fourier series and those for dirichlet problems are based upon chebyshev polynomials an improved preconditioning for dirichlet problems is given numerical examples and practical advice are included

konvergenzaussagen fur kollokationsverfahren bei elliptischen randwertaufgaben we examine the convergence of collocation with polynomials for elliptic problems the general theory is imbedded in thel p theory from which known results of existence and regularity are adopted convergence results are deduced in the cases of periodic dirichlet and mixed boundary conditions it becomes obvious that the order of convergence essentially depends on the smoothness of the solution