ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

a fast iterative solver for the variable coefficient diffusion equation on a disk we present an efficient iterative method for solving the variable coefficient diffusion equation on a unit disk the equation is written in polar coordinates and is discretized by the standard centered difference approximation under the grid arrangement by shifting half radial mesh away from the origin so that the coordinate singularity can be handled naturally without pole conditions the resultant matrix is symmetric positive definite so the preconditioned conjugate gradient pcg method can be applied different preconditioners have been tested for comparison in particular a fast direct solver derived from the equation and the semi coarsening multigrid are shown to be almost scalable with the problem size and outperform other preconditioners significantly the present elliptic solver has been applied to study the vortex dynamics of the ginzburg landau equation with a variable diffusion coefficient

hubs in space popular nearest neighbors in high dimensional data different aspects of the curse of dimensionality are known to present serious challenges to various machine learning methods and tasks this paper explores a new aspect of the dimensionality curse referred to as hubness that affects the distribution of k occurrences the number of times a point appears among the k nearest neighbors of other points in a data set through theoretical and empirical analysis involving synthetic and real data sets we show that under commonly used assumptions this distribution becomes considerably skewed as dimensionality increases causing the emergence of hubs that is points with very high k occurrences which effectively represent popular nearest neighbors we examine the origins of this phenomenon showing that it is an inherent property of data distributions in high dimensional vector space discuss its interaction with dimensionality reduction and explore its influence on a wide range of machine learning tasks directly or indirectly based on measuring distances belonging to supervised semi supervised and unsupervised learning families

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

fast direct solver for the biharmonic equation on a disk and its application to incompressible flows we develop a simple and efficient fft based fast direct solver for the biharmonic equation on a disk the biharmonic equation is split into a coupled system of harmonic problems we first use the truncated fourier series expansion to derive a set of coupled singular odes then we solve those singular equations by second order finite difference discretizations using a radial grid with shifting a half mesh away from the origin we can handle the coordinate singularity easily without pole conditions the sherman morrison formula is then applied to solve the resultant linear system in a cost efficient way the computational complexity of the method consists of o mnlog 2n arithmetic operations for mxn grid points the numerical accuracy check and some applications to the incompressible navier stokes flows inside a disk are conducted

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

a high order fast direct solver for singular poisson equation we present a fourth order numerical solution method for the singular neumann boundary problem of poisson equations such problems arise in the solution process of incompressible navier stokes equations and in the time harmonic wave propagation in the frequence space with the zero wavenumber the equation is first discretized with a fourth order modified collatz difference scheme producing a singular discrete equation then an efficient singular value decomposition svd method modified from a fast poisson solver is employed to project the discrete singular equation into the orthogonal complement of the null space of the singular matrix in the complement of the null space the projected equation is uniquely solvable and its solution is proven to be a solution of the original singular discrete equation when the original equation has a solution analytical and experimental results show that this newly proposed singular equation solver is efficient while retaining the accuracy of the high order discretization c 2001 academic press

high order difference methods for heat equations in polar cylindrical coordinates abstract fourth order difference methods for the solution of poisson equations in cylindrical polar coordinates are proposed the same technique is then applied to obtain o k 2 h 4 two level unconditionally stable adi methods for the solution of the heat equation in two dimensional polar coordinates and three dimensional cylindrical coordinates numerical examples given here show that the methods developed here retain their order and accuracy everywhere including the region in the vicinity of the singularity r 0

best unbiased ensemble linearization and the quasi linear kalman ensemble generator 1 linearized representations of the stochastic groundwater flow and transport equations have been heavily used in hydrogeology e g for geostatistical inversion or generating conditional realizations the respective linearizations are commonly defined via jacobians numerical sensitivity matrices this study will show that jacobian based linearizations are biased with nonminimal error variance in the ensemble sense an alternative linearization approach will be derived from the principles of unbiasedness and minimum error variance the resulting paradigm prefers empirical cross covariances from monte carlo analyses over those from linearized error propagation and points toward methods like ensemble kalman filters enkfs unlike conditional simulation in geostatistical applications enkfs condition transient state variables rather than geostatistical parameter fields recently modifications toward geostatistical applications have been tested and used this study completes the transformation of enkfs to geostatistical conditioning tools on the basis of best unbiased ensemble linearization to distinguish it from the original enkf the new method is called the kalman ensemble generator keg the new context of best unbiased ensemble linearization provides an additional theoretical foundation to enkf like methods such as the keg like enkfs and derivates the keg is optimal for gaussian variables toward increased robustness and accuracy in non gaussian and nonlinear cases sequential updating acceptance rejection sampling successive linearization and a levenberg marquardt formalism are added state variables are updated through simulation with updated parameters always guaranteeing the physicalness of all state variables the keg combines the computational efficiency of linearized methods with the robustness of enkfs and accuracy of expensive realization based methods while drawing on the advantages of conditional simulation over conditional estimation such as adequate representation of solute dispersion as proof of concept a large scale numerical test case with 200 synthetic sets of flow and tracer data is conducted and analyzed

comparison of fickian and temporally nonlocal transport theories over many scales in an exhaustively sampled sandstone slab 1 it is not a compelling argument solely on the basis of a better fit to solute breakthrough curve btc data that a temporally nonlocal model is necessary to simulate transport in an advection dominated system one may counter that the classical advection dispersion equation ade is a valid model at some small scale and that the detailed hydraulic conductivity k data must be well represented then the nonlocality is only a result of upscaling and loss of information but is the nonlocal model demonstrably necessary at all scales we examine the experiment conducted by klise et al 2008 in which a 30 5 30 5 cm slab of relatively homogeneous cross bedded sandstone was exhaustively sampled for k the slab was sealed saturated with potassium iodide and x rayed 10 times while being flushed with fresh water the 8 649 air permeameter measurements were down and upscaled to make finer and coarser grids on which the velocity field was solved and the ade applied the optimized parameters in the ade were found to scale predictably most notably the longitudinal dispersivity which grew linearly with upscaling but at all levels of up and downscaling including the original k measurement scale of 0 33 cm the ade did not adequately represent the late time tails the temporally nonlocal time fractional ade t fade was applied and the optimized parameters and the immobile capacity did not depend on scale the better fit provided by the t fade in the late btc tails did not bring about a sacrificed fit elsewhere in the btc furthermore the optimized ade and t fade solutions do not converge at the smallest scale directly implying that the temporal nonlocality is a necessary model component we conclude that the logical inference if the ade is valid in heterogeneous material then there is tailing in the btc is not a proof that the reverse is true we provide a clear counterexample a corollary is that a mismatch between data and a discretized solution to the ade does not imply that more data will improve fits or predictive ability

a formally fourth order accurate compact scheme for 3d poisson equation in cylindrical coordinates in this paper we extend our previous work m c lai a simple compact fourth order poisson solver on polar geometry j comput phys 182 2002 337 345 to 3d cases more precisely we present a spectral finite difference scheme for poisson equation in cylindrical coordinates the scheme relies on the truncated fourier series expansion where the partial differential equations of fourier coefficients are solved by a formally fourth order accurate compact difference discretization here the formal fourth order accuracy means that the scheme is exactly fourth order accurate while the poles are excluded and is third order accurate otherwise despite the degradation of one order of accuracy due to the presence of poles the scheme handles the poles naturally thus no pole condition is needed the resulting linear system is then solved by the bi cgstab method with the preconditioner arising from the second order discretization which shows the scalability with the problem size

on the use of block effective macrodispersion for numerical simulations of transport in heterogeneous formations 1 the fundamental question we consider in this paper is how to allow flexibility in numerical grid design without discounting the dispersive action of the unmodeled variability in doing that we wish to preserve the interplay between all relevant length scales those relevant to the spatial variability as well as those created by design in this study we extend and test the concept of block scale macrodispersion introduced by rubin et al 1999 for modeling unresolved hydraulic property variations at scales smaller than the numerical grid blocks we present closed form analytical results for the block scale macrodispersion and test them numerically closed form analytical results are presented for the large time aymptotic limits and it is shown that these limits are attained very fast the conditions of applicability are investigated and we show that ergodicity with regard to block scale heterogeneity is attained surprisingly fast

self improved gaps almost everywhere for the agnostic approximation of monomials given a learning sample we focus on the hardness of finding monomials having low error inside the interval bounded below by the smallest error achieved by a monomial the best rule and bounded above by the error of the default class the poorest rule it is well known that when its lower bound is zero it is an easy task to find in linear time a monomial with zero error what we prove is that when this bound is not zero regardless of the location of the default class in 0 1 2 it becomes a huge complexity burden to beat significantly the default class in fact under some complexity theoretical assumptions it may already be hard to beat the trivial approximation ratios even when relaxing the time complexity constraint to be quasi polynomial or sub exponential our results also hold with uniform weights over the examples

a transformation free hoc scheme for incompressible viscous flows past an impulsively started circular cylinder in this paper we present a higher order compact scheme for the unsteady two dimensional 2d navier stokes equations on nonuniform polar grids specifically designed for the incompressible viscous flows past a circular cylinder the scheme is second order accurate in time and at least third order accurate in space the scheme very efficiently computes both unsteady and time marching steady state flow for a wide range of reynolds numbers re ranging from 10 to 9500 for the impulsively started cylinder the robustness of the scheme is highlighted when it accurately captures the vortex shedding for moderate re represented by the von karman street and the so called and s phenomena for higher re comparisons are made with established numerical and experimental results and excellent agreement is found in all the cases both qualitatively and quantitatively

boosted classification trees and class probability quantile estimation the standard by which binary classifiers are usually judged misclassification error assumes equal costs of misclassifying the two classes or equivalently classifying at the 1 2 quantile of the conditional class probability function p y 1 x boosted classification trees are known to perform quite well for such problems in this article we consider the use of standard off the shelf boosting for two more general problems 1 classification with unequal costs or equivalently classification at quantiles other than 1 2 and 2 estimation of the conditional class probability function p y 1 x we first examine whether the latter problem estimation of p y 1 x can be solved with logitboost and with adaboost when combined with a natural link function the answer is negative both approaches are often ineffective because they overfit p y 1 x even though they perform well as classifiers a major negative point of the present article is the disconnect between class probability estimation and classification next we consider the practice of over under sampling of the two classes we present an algorithm that uses adaboost in conjunction with over under sampling and jittering of the data jous boost this algorithm is simple yet successful and it preserves the advantage of relative protection against overfitting but for arbitrary misclassification costs and equivalently arbitrary quantile boundaries we then use collections of classifiers obtained from a grid of quantiles to form estimators of class probabilities the estimates of the class probabilities compare favorably to those obtained by a variety of methods across both simulated and real data sets

additive logistic regression a statistical view of boosting boosting is one of the most important recent developments in classification methodology boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced for many classification algorithms this simple strategy results in dramatic improvements in performance we show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles namely additive modeling and maximum likelihood for the two class problem boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum bernoulli likelihood as a criterion we develop more direct approximations and show that they exhibit nearly identical results to boosting direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most situations and far superior in some we suggest a minor modification to boosting that can reduce computation often by factors of 10 to 50 finally we apply these insights to produce an alternative formulation of boosting decision trees this approach based on best first truncated tree induction often leads to better performance and can provide interpretable descriptions of the aggregate decision rule it is also much faster computationally making it more suitable to large scale data mining applications

a bayesian approach for inverse modeling data assimilation and conditional simulation of spatial random fields 1 this paper addresses the inverse problem in spatially variable fields such as hydraulic conductivity in groundwater aquifers or rainfall intensity in hydrology common to all these problems is the existence of a complex pattern of spatial variability of the target variables and observations the multiple sources of data available for characterizing the fields the complex relations between the observed and target variables and the multiple scales and frequencies of the observations the method of anchored distributions mad that we propose here is a general bayesian method of inverse modeling of spatial random fields that addresses this complexity the central elements of mad are a modular classification of all relevant data and a new concept called anchors data types are classified by the way they relate to the target variable as either local or nonlocal and as either direct or indirect anchors are devices for localization of data they are used to convert nonlocal indirect data into local distributions of the target variables the target of the inversion is the derivation of the joint distribution of the anchors and structural parameters conditional to all measurements regardless of scale or frequency of measurement the structural parameters describe large scale trends of the target variable fields whereas the anchors capture local inhomogeneities following inversion the joint distribution of anchors and structural parameters is used for generating random fields of the target variable s that are conditioned on the nonlocal indirect data through their anchor representation we demonstrate mad through a detailed case study that assimilates point measurements of the conductivity with head measurements from natural gradient flow the resulting statistical distributions of the parameters are non gaussian similarly the moments of the estimates of the hydraulic head are non gaussian we provide an extended discussion of mad vis a vis other inversion methods including maximum likelihood and maximum a posteriori with an emphasis on the differences between mad and the pilot points method

geostatistical inference of hydraulic conductivity and dispersivities from hydraulic heads and tracer data 1 in groundwater hydraulic heads and solute arrival times depend primarily on the hydraulic conductivity field and hydraulic boundary conditions the spread of breakthrough curves in contrast depends also on longitudinal and transverse dispersion coefficients the shape of point like measured breakthrough curves can be reproduced by simulations only when appropriate dispersivities are applied the values and spatial distributions of dispersivities depend on the resolution of the underlying hydraulic conductivity field we present a geostatistical method for the joint estimation of log conductivity and log dispersivities from measurements of hydraulic heads and temporal moments of local breakthrough curves the parameter fields are considered random space variables and they are conditioned on the measurements by bayesian inference the estimated longitudinal and transverse dispersivities are to be applied in conjunction with the estimated conductivity field we apply our technique to data of a technical scale tracer experiment in the particular application the amount and quality of measured data are sufficient to infer the conductivity distribution at a spatial resolution at which the spread of locally obtained breakthrough curves is dominated by pore scale transverse dispersion

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

simulation of dispersion in heterogeneous porous formations statistics first order theories convergence of computations this paper discusses the results of numerical analysis of dispersion of passive solutes in two dimensional heterogeneous porous formations statistics of flow and transport variables the accuracy and the role of approximations implicit in existing first order theories and the convergence of computational results are investigated the results suggest that quite different rates of convergence with monte carlo runs hold for different spatial moments and that over 1000 realizations are required to stabilize second moments even for relatively mild heterogeneity sigma y 2 1 6 this has implications for the extent of the spatial domain for single realization numerical studies of the same type a comparison of the variance of plumes with the results of linear theories 0 05 sigma y 2 1 6 shows an unexpectedly broad validity field for the theoretical solution obtained from a suitable linearization of flow and transport reformulation of the same problem linearizing in tum the flow or the transport equations shows opposite deviations from the linear theory the interesting consequence is that the errors induced by linearizations in the flow or the transport equations have different signs and their effects on the moments of dispersing plumes are compensating thereby yielding consistent formulations unexpected features of the statistics of probability distributions of longitudinal and transverse velocities and travel times are also computed and discussed

numerical studies on nonlinear schrodinger equations by spectral collocation method with preconditioning in this study we use the spectral collocation method with preconditioning to solve various nonlinear schrodinger equations to reduce round off error in spectral collocation method we use preconditioning we study the numerical accuracy of the method the numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method

upscaling and dispersion for transport in heterogeneous media this paper focuses on upscaling of the transport equation for heterogeneous porous media with random flow we consider the local flow field being a stationary random field and develop an upscaling by the recently developed coarse graining method which is based on filtering procedures in fourier space the coarse graining method is used to obtain an upscaled dispersion tensor which depends on the given length scale of the upscaling we give explicit results for the scale dependent dispersion coefficient in lowest order perturbation theory for finite length scales the upscaled dispersion models the effect of the unresolved subscale flow fluctuations and for a global upscaling the upscaled value agrees with the well known macrodispersion coefficient which is however nearly approached for length scales larger than tenfold of the correlation length

on robustness properties of convex risk minimization methods for pattern recognition the paper brings together methods from two disciplines machine learning theory and robust statistics we argue that robustness is an important aspect and we show that many existing machine learning methods based on the convex risk minimization principle have besides other good properties also the advantage of being robust robustness properties of machine learning methods based on convex risk minimization are investigated for the problem of pattern recognition assumptions are given for the existence of the influence function of the classifiers and for bounds on the influence function kernel logistic regression support vector machines least squares and the adaboost loss function are treated as special cases some results on the robustness of such methods are also obtained for the sensitivity curve and the maxbias which are two other robustness criteria a sensitivity analysis of the support vector machine is given

data reduction using a discrete wavelet transform in discriminant analysis of very high dimensionality data we present a method of data reduction using a wavelet transform in discriminant analysis when the number of variables is much greater than the number of observations the method is illustrated with a prostate cancer study where the sample size is 248 and the number of variables is 48 538 generated using the proteinchip technology using a discrete wavelet transform the 48 538 data points are represented by 1271 wavelet coefficients information criteria identified 11 of the 1271 wavelet coefficients with the highest discriminatory power the linear classifier with the 11 wavelet coefficients detected prostate cancer in a separate test set with a sensitivity of 97 and specificity of 100

a simple compact fourth order poisson solver on polar geometry we present a simple and efficient compact fourth order poisson solver in polar coordinates this solver relies on the truncated fourier series expansion where the differential equations of the fourier coefficients are solved by the compact fourth order finite difference scheme by shifting a grid a half mesh away from the origin and incorporating the symmetry constraint of fourier coefficients we can easily handle coordinate singularities without pole conditions the numerical evidence confirms fourth order accuracy for the problem on an annulus and third order accuracy for the problem on a disk in addition a simple and comparably accurate approximation for the derivatives of the solution is also presented

greedy function approximation a gradient boosting machine function estimation approximation is viewed from the perspective of numerical optimization in function space rather than parameter space a connection is made between stagewise additive expansions and steepest descent minimization a general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion specific algorithms are presented for least squares least absolute deviation and huber m loss functions for regression and multiclass logistic likelihood for classification special enhancements are derived for the particular case where the individual additive components are regression trees and tools for interpreting such treeboost models are presented gradient boosting of regression trees produces competitive highly robust interpretable procedures for both regression and classification especially appropriate for mining less than clean data connections between this approach and the boosting methods of freund and shapire and friedman hastie and tibshirani are discussed

a fast parallel algorithm for the poisson equation on a disk a parallel algorithm for solving the poisson equation with either dirichlet or neumann conditions is presented the solver follows some of the principles introduced in a previous fast algorithm for evaluating singular integral transforms by daripa et al here we present recursive relations in fourier space together with fast fourier transforms which lead to a fast and accurate algorithm for solving poisson problems within a unit disk the algorithm is highly parallelizable and our implementation is virtually architecture independent theoretical estimates show good parallel scalability of the algorithm and numerical results show the accuracy of the method for problems with sharp variations on inhomogeneous term finally performance results for sequential and parallel implementations are presented c 2001 academic press

flow past an impulsively started circular cylinder using a higher order semicompact scheme in the present investigation an attempt has been made to solve the two dimensional incompressible viscous flow past an impulsively started circular cylinder for reynolds numbers ranging from 20 to 5000 using higher order semicompact scheme hosc unlike conventional higher order compact schemes the hosc scheme has been developed to handle the circular geometry of the chosen problem and the intensive algebraic manipulations have been reduced considerably by relaxing the compactness of the computational stencil for few terms but retained for most of the terms of the discretized equations for the flow past an impulsively started circular cylinder the results obtained at low and moderate reynolds numbers have been validated with the experimental and numerical observations available in the literature for high reynolds number flows the present scheme rightly captures the alpha phenomenon at re 1000 and both beta and alpha phenomena one after the other at re 5000

derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation in this paper simple polynomial interpolation is used to derive arbitrarily high order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative consisting of three second derivative nodes in the interior and two on the boundary on non uniform grids boundary and near boundary schemes of the same order as the interior are also developed using polynomial interpolation and for a general compact scheme on a non uniform grid it is shown that polynomial interpolation is more efficient than the conventional method of undetermined coefficients for finding coefficients of the scheme the high order non uniform schemes along with boundary closure of up to 14th order thus obtained are shown to be stable on a non uniform grid with appropriate stretching so that more grid points are clustered near the boundary the stability and resolution properties of the high order non uniform grid schemes are studied and the results of three numerical tests on stability and accuracy properties are also presented

compact mixed methods for convection diffusion type problems abstract this paper presents a class of fourth order compact finite difference technique for solving two dimensional convection diffusion equation the equation is recasted as a first order mixed system introducing a conservation and flux equations since flux appears explicitly in the mixed formulation we search a fourth order compact approximation of the primary solution field and flux based on taylor series expansion the proposed compact mixed formulation generalizes the work of carey and spotz g f carey w f spotz higher order compact mixed methods commun numer meth eng 13 1997 we show that their fourth order formulation corresponds to a particular case of our presented scheme and we extend their work to variable diffusion and convection coefficients some numerical experiments are performed to demonstrate the fourth order effective convergence rate

a sparse mesh for compact finite difference fourier solvers with radius dependent spectral resolution in circular domains this paper presents a new method for the resolution of elliptic and parabolic equations in circular domains it can be trivially extended to cylindrical domains the algorithm uses a mixed fourier compact finite difference method the main advantage of the method is achieved by a new concept of mesh the topology of the new grid keeps constant the aspect ratio of the cells avoiding the typical clustering for radial structured meshes at the center the reduction of the number of nodes has as a consequence the reduction in memory consumption in the case of fluid mechanics problems this technique also increases the time step for a constant courant number several examples are given in the paper which show the potential of the method

logistic regression using covariates obtained by product unit neural network models we propose a logistic regression method based on the hybridation of a linear model and product unit neural network models for binary classification in a first step we use an evolutionary algorithm to determine the basic structure of the product unit model and afterwards we apply logistic regression in the new space of the derived features this hybrid model has been applied to seven benchmark data sets and a new microbiological problem the hybrid model outperforms the linear part and the nonlinear part obtaining a good compromise between them and they perform well compared to several other learning classification techniques we obtain a binary classifier with very promising results in terms of classification accuracy and the complexity of the classifier

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed