boosting the margin a new explanation for the effectiveness of voting methods one of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large and often is observed to decrease even after the training error reaches zero in this paper we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label we show that techniques used in the analysis of vapnik s support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error we also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples finally we compare our explanation to those based on the bias variance decomposition

an adaptive mesh refinement method for solution of the transported pdf equation this paper presents the results of an investigation into a possible alternative to monte carlo methods for solving the transported probability density function pdf equation for scalars compositions the method uses a finite volume approach combined with adaptive mesh refinement amr in a multi dimensional compositional space comparisons are made between the new method and monte carlo solutions for analytical test cases involving the reaction of two or three chemical species these tests demonstrate the potential of the new method in terms of both accuracy and run time additional test cases involving various models for molecular mixing were also conducted with similar conclusions copyright 2009 john wiley sons ltd

end point corrected trapezoidal quadrature rules for singular functions abstract a group of quadrature formulae for end point singular functions is presented generalizing classical end point corrected trapezoidal quadrature rules the actual values of the end point corrections are obtained for each singularity as a solution of a system of linear algebraic equations the algorithm is applicable to a wide class of monotonic singularities and does not require that an analytical expression for the singularity be known only the knowledge of its first several moments and the ability to evaluate it on the interval of integration are needed keywords convergence numerical analysis numerical integration author

eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

active reranking for web image search image search reranking methods usually fail to capture the user s intention when the query term is ambiguous therefore reranking with user interactions or active reranking is highly demanded to effectively improve the search performance the essential problem in active reranking is how to target the user s intention to complete this goal this paper presents a structural information based sample selection strategy to reduce the user s labeling efforts furthermore to localize the user s intention in the visual feature space a novel local global discriminative dimension reduction algorithm is proposed in this algorithm a submanifold is learned by transferring the local geometry and the discriminative information from the labelled images to the whole global image database experiments on both synthetic datasets and a real web image search dataset demonstrate the effectiveness of the proposed active reranking scheme including both the structural information based active sample selection strategy and the local global discriminative dimension reduction algorithm

accuracy analysis of an adaptive mesh refinement method using benchmarks of 2 d steady incompressible lid driven cavity flows and coarser meshes lid driven cavity flows have been widely investigated and accurate results have been achieved as benchmarks for testing the accuracy of computational methods this paper verifies the accuracy of an adaptive mesh refinement method numerically using 2 d steady incompressible lid driven cavity flows and coarser meshes the accuracy is shown by verifying that the centres of vortices given in the benchmarks are located in the refined grids of refined meshes for reynolds numbers 100 1000 and 2500 using coarser meshes the adaptive mesh refinement method performs mesh refinement based on the numerical solutions of navier stokes equations solved by a finite volume method with a well known simple algorithm for pressure velocity coupling the accuracy of the refined meshes is shown by comparing the profiles of horizontal and vertical components of velocity fields with the corresponding components of the benchmarks together and drawing closed streamlines the adaptive mesh refinement method verified in this paper can be applied to find the accurate numerical solutions of any mathematical models containing continuity equations for incompressible fluid steady state fluid flows or mass and heat transfer

an efficient high order nystrom scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface this text proposes a fast rapidly convergent nystrom method for the solution of the lippmann schwinger integral equation that mathematically models the scattering of time harmonic acoustic waves by inhomogeneous obstacles while allowing the material properties to jump across the interface the method works with overlapping coordinate charts as a description of the given scatterer in particular it employs partitions of unity to simplify the implementation of high order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations to deal with the discontinuous material interface in a high order manner a specialized quadrature is used in the boundary region the approach further utilizes an fft based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of o n log n for an n point discretization a detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance are presented in this paper

high order corrected trapezoidal quadrature rules for the coulomb potential in three dimensions we construct correction coefficients for high order trapezoidal quadrature rules to evaluate three dimensional singular integrals of the form j v dv x y z x2 y2 z2dxdydz where the domain d is a cube containing the point of singularity 0 0 0 and v is a c function defined on 3 the procedure employed here is a generalization to 3 d of the method of central corrections for logarithmic singularities 1 in one dimension and 2 in two dimensions as in one and two dimensions the correction coefficients for high order trapezoidal rules for j v are independent of the number of sampling points used to discretize the cube d when v is compactly supported in d the approximation is the trapezoidal rule plus a local weighted sum of the values of v around the point of singularity these quadrature rules provide an efficient stable and accurate way of approximating j v we demonstrate the performance of these quadratures of orders up to 17 for highly oscillatory functions v these type of integrals appear in scattering calculations in 3 d

adaptive mesh and algorithm refinement using direct simulation monte carlo adaptive mesh and algorithm refinement amar embeds a particle method within a continuum method at the finest level of an adaptive mesh refinement amr hierarchy the coupling between the particle region and the overlaying continuum grid is algorithmically equivalent to that between the fine and coarse levels of amr direct simulation monte carlo dsmc is used as the particle algorithm embedded within a godunov type compressible navier stokes solver several examples are presented and compared with purely continuum calculations

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

general tensor discriminant analysis and gabor features for gait recognition traditional image representations are not suited to conventional classification methods such as the linear discriminant analysis lda because of the undersample problem usp the dimensionality of the feature space is much higher than the number of training samples motivated by the successes of the two dimensional lda 2dlda for face recognition we develop a general tensor discriminant analysis gtda as a preprocessing step for lda the benefits of gtda compared with existing preprocessing methods such as the principal components analysis pca and 2dlda include the following 1 the usp is reduced in subsequent classification by for example lda 2 the discriminative information in the training tensors is preserved and 3 gtda provides stable recognition rates because the alternating projection optimization algorithm to obtain a solution of gtda converges whereas that of 2dlda does not we use human gait recognition to validate the proposed gtda the averaged gait images are utilized for gait representation given the popularity of gabor function based image decompositions for image understanding and object recognition we develop three different gabor function based image representations 1 gabord is the sum of gabor filter responses over directions 2 gabors is the sum of gabor filter responses over scales and 3 gaborsd is the sum of gabor filter responses over scales and directions the gabord gabors and gaborsd representations are applied to the problem of recognizing people from their averaged gait images a large number of experiments were carried out to evaluate the effectiveness recognition rate of gait recognition based on first obtaining a gabor gabord gabors or gaborsd image representation then using gdta to extract features and finally using lda for classification the proposed methods achieved good performance for gait recognition based on image sequences from the university of south florida usf humanid database experimental comparisons are made with nine state of the art classification methods in gait recognition

a conservative three dimensional eulerian method for coupled solid fluid shock capturing a new method is presented for the explicit eulerian finite difference computation of shock capturing problems involving multiple resolved material phases in three dimensions we solve separately for each phase the equations of fluid dynamics or solid mechanics using as interface boundary conditions artificially extended representations of the individual phases for fluids we use a new 3d spatially unsplit implementation of the piecewise parabolic ppm method of colella and woodward for solids we use the 3d spatially unsplit eulerian solid mechanics method of miller and colella vacuum and perfectly incompressible obstacles may also be employed as phases a separate problem is the time evolution of material interfaces which are represented by planar segments constructed with a volume of fluid method the volume fractions are advanced in time using a second order 3d spatially unsplit advection routine with a velocity field determined by solution of interface normal two phase riemann problems from the riemann problem solutions we also determine cross interface momentum and energy fluxes the volume fractions in mixed cells may be arbitrarily small which would ordinarily make the courant friedrichs lewy time step stability limit arbitrarily small as well we overcome this limitation using the mass redistribution formalism to conservatively redistribute generalized mass in the neighborhood of the split cells we present an application of this method to an explosion contained in a metal can a reactive fluid approximating pbx 9404 is encased within an elastic plastic solid approximating copper surrounded by vacuum our implementation is in parallel and with adaptive mesh refinement

distance metric learning by minimal distance maximization classic linear dimensionality reduction ldr methods such as principal component analysis pca and linear discriminant analysis lda are known not to be robust against outliers following a systematic analysis of the multi class ldr problem in a unified framework we propose a new algorithm called minimal distance maximization mdm to address the non robustness issue the principle behind mdm is to maximize the minimal between class distance in the output space mdm is formulated as a semi definite program sdp and its dual problem reveals a close connection to weighted ldr methods a soft version of mdm in which lda is subsumed as a special case is also developed to deal with overlapping centroids finally we drop the homoscedastic gaussian assumption made in mdm by extending it in a non parametric way along with a gradient based convex approximation algorithm to significantly reduce the complexity of the original sdp the effectiveness of our proposed methods are validated on two uci datasets and two face datasets

generalized iterative relief for supervised distance metric learning the relief algorithm is a popular approach for feature weighting many extensions of the relief algorithm are developed and i relief is one of the famous extensions in this paper i relief is generalized for supervised distance metric learning to yield a mahananobis distance function the proposed approach is justified by showing that the objective function of the generalized i relief is closely related to the expected leave one out nearest neighbor classification rate in addition the relationships among the generalized i relief the neighbourhood components analysis and graph embedding are also pointed out experimental results on various data sets all demonstrate the superiority of the proposed approach

hybrid block amr in cartesian and curvilinear coordinates mhd applications abstract we present a novel hybrid block adaptive scheme for use in solving sets of near conservation laws in general orthogonal coordinate systems the adaptive mesh refinement amr scheme is block based i e individual grids have a pre fixed number of grid cells and is implemented for any dimensionality d its hybrid character relaxes the common approach where a block that needs refinement triggers 2 d subblocks when grids are refined by a factor of 2 this introduces incomplete families in the grid hierarchy but approaches the optimal fit to developing flow features inherent in the original patch based amr strategy our hybrid block amr approach is compared with a patch based amr one both exploiting openmp parallelism the implementation is able to handle general curvilinear coordinates for which restriction and prolongation formulae are presented along with boundary treatments at singular boundaries demonstrative examples cover hydro and magnetohydrodynamic mhd applications including tests on a 2d polar grid a 2 5d spherical and a 3d cylindrical configuration the applications cover classical up to relativistic mhd simulations of particular relevance for astrophysical magnetized jet and stellar wind studies

hamilton jacobi equations on a manifold and applications to grid generation or refinement in this paper we consider hamilton jacobi equations on a manifold typically on the graph of some previously computed function z x y and we show how the corresponding level set method allows us to generate and or to refine a mesh in regions where this function z has large derivatives such as it is the method needs to be strongly improved and accelerated but the principle is awfully natural and in principle the method is fully automatic similar ideas are also useful in image processing in particular for the active contours method

a boosting approach for supervised mahalanobis distance metric learning determining a proper distance metric is often a crucial step for machine learning in this paper a boosting algorithm is proposed to learn a mahalanobis distance metric similar to most boosting algorithms the proposed algorithm improves a loss function iteratively in particular the loss function is defined in terms of hypothesis margins and a metric matrix base learner specific to the boosting framework is also proposed experimental results show that the proposed approach can yield effective mahalanobis distance metrics for a variety of data sets and demonstrate the feasibility of the proposed approach

radiation diffusion for multi fluid eulerian hydrodynamics with adaptive mesh refinement block structured meshes provide the ability to concentrate grid points and computational effort in interesting regions of a flow field without sacrificing the efficiency and low memory requirements of a regular grid we describe an algorithm for simulating radiation diffusion on such a mesh coupled to multi fluid gasdynamics conservation laws are enforced by using locally conservative difference schemes along with explicit synchronization operations between different levels of refinement in unsteady calculations each refinement level is advanced at its own optimal timestep particular attention is given to the appropriate coupling between the fluid energy and the radiation field the behavior of the discretization at sharp interfaces and the form of synchronization between levels required for energy conservation in the diffusion process two and three dimensional examples are presented including parallel calculations performed on an ibm sp 2

local discriminative distance metrics ensemble learning the ultimate goal of distance metric learning is to incorporate abundant discriminative information to keep all data samples in the same class close and those from different classes separated local distance metric methods can preserve discriminative information by considering the neighborhood influence in this paper we propose a new local discriminative distance metrics lddm algorithm to learn multiple distance metrics from each training sample a focal sample and in the vicinity of that focal sample focal vicinity to optimize local compactness and local separability those locally learned distance metrics are used to build local classifiers which are aligned in a probabilistic framework via ensemble learning theoretical analysis proves the convergence rate bound the generalization bound of the local distance metrics and the final ensemble classifier we extensively evaluate lddm using synthetic datasets and large benchmark uci datasets

learning a mahalanobis distance metric for data clustering and classification distance metric is a key issue in many machine learning algorithms this paper considers a general problem of learning from pairwise constraints in the form of must links and cannot links as one kind of side information a must link indicates the pair of the two data points must be in a same class while a cannot link indicates that the two data points must be in two different classes given must link and cannot link information our goal is to learn a mahalanobis distance metric under this metric we hope the distances of point pairs in must links are as small as possible and those of point pairs in cannot links are as large as possible this task is formulated as a constrained optimization problem in which the global optimum can be obtained effectively and efficiently finally some applications in data clustering interactive natural image segmentation and face pose estimation are given in this paper experimental results illustrate the effectiveness of our algorithm

locally adaptive metric nearest neighbor classification nearest neighbor classification assumes locally constant class conditional probabilities this assumption becomes invalid in high dimensions with finite samples due to the curse of dimensionality severe bias can be introduced under these conditions when using the nearest neighbor rule we propose a locally adaptive nearest neighbor classification method to try to minimize bias we use a chi squared distance analysis to compute a flexible metric for producing neighborhoods that are highly adaptive to query locations neighborhoods are elongated along less relevant feature dimensions and constricted along most influential ones as a result the class conditional probabilities are smoother in the modified neighborhoods whereby better classification performance can be achieved the efficacy of our method is validated and compared against other techniques using both simulated and real world data

a tensor train accelerated solver for integral equations in complex geometries abstract we present a framework using the quantized tensor train qtt decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions we describe how the qtt decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations for a broad range of problems computational and storage costs of the inversion scheme are extremely modest o log n and once the inverse is computed it can be applied in o n log n we analyze the qtt ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as fmm and hss we prove that the qtt ranks are bounded for translation invariant systems and argue that this behavior extends to non translation invariant volume and boundary integrals for volume integrals the qtt decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers we present results demonstrating the remarkable performance of the qtt based solver when applied to both translation and non translation invariant volume integrals in 3d for boundary integral equations we demonstrate that using a qtt decomposition to construct preconditioners for a krylov subspace method leads to an efficient and robust solver with a small memory footprint we test the qtt preconditioners in the iterative solution of an exterior elliptic boundary value problem laplace formulated as a boundary integral equation in complex multiply connected geometries

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

discriminant adaptive nearest neighbor classification nearest neighbour classification expects the class conditional probabilities to be locally constant and suffers from bias in high dimensions we propose a locally adaptive form of nearest neighbour classification to try to ameliorate this curse of dimensionality we use a local linear discriminant analysis to estimate an effective metric for computing neighbourhoods we determine the local decision boundaries from centroid information and then shrink neighbourhoods in directions orthogonal to these local decision boundaries and elongate them parallel to the boundaries thereafter any neighbourhood based classifier can be employed using the modified neighbourhoods the posterior probabilities tend to be more homogeneous in the modified neighbourhoods we also propose a method for global dimension reduction that combines local dimension information in a number of examples the methods demonstrate the potential for substantial improvements over nearest neighbour classification

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

a parallel fast multipole accelerated integral equation scheme for 3d stokes equations in this paper we discuss a numerical scheme for the stokes equations in three dimensions it uses an integral equation formulation and is accelerated by the new version of fast multipole method first introduced by greengard and rokhlin in 1997 acta numerica 1997 6 229 269 the code is parallelized to solve problems of extremely large size the resulting numerical solver can be applied to stokes flows in complex geometry and also serves as a building block for solving the navier stokes equations of low to moderate reynold s numbers copyright 2006 john wiley sons ltd

high order quadratures for integral operators with singular kernels a numerical integration method that has rapid convergence for integrands with known singularities is presented based on endpoint corrections to the trapezoidal rule the quadratures are suited for the discretization of a variety of integral equations encountered in mathematical physics the quadratures are based on a technique introduced by rokhlin 1990 the present modification controls the growth of the quadrature weights and permits higher order rules in practice several numerical examples are included

a fast direct algorithm for the solution of the laplace equation on regions with fractal boundaries an algorithm is presented for the rapid direct solution of the laplace equation on regions with fractal boundaries in a typical application the numerical simulation has to be on a very large scale involving at least tens of thousands of equations with as many unknowns in order to obtain any meaningful results attempts to use conventional techniques have encountered insurmountable difficulties due to excessive cpu time requirements of the computations involved indeed conventional direct algorithms for the solution of linear systems require order o n3 operations for the solution of an n n problem while classical iterative methods require order o n2 operations with the constant strongly dependent on the problem in question in either case the computational expense is prohibitive for large scale problems the direct algorithm of the present paper requires o n operations with a constant dependent only on the geometry of the boundary making it considerably more practical for large scale problems encountered in the computation of harmonic measure of fractals complex iteration theory potential theory and growth phenomena such as crystallization electrodeposition viscous fingering and diffusion limited aggregation

a high order godunov method for multiple condensed phases presented in this work is a nonlinear adaptive nonreflecting boundary condition nrbc that when compared with classical local nrbcs further reduces the wave reflection error at far fields in the numerical simulation of wave dominated problems a nonlinear procedure can considerably improve the accuracy of nrbcs even if the problem itself is linear as in the case of the 2d wave equation in fact the first and second order nrbcs of engquist and majda can be modified by adding proper weights to the coefficients of the difference equations the weights are adaptively determined in such a way that the angle of complete absorption is locally coincident with the outgoing wave direction the theoretical reflection coefficients and the numerical experiments show that a considerable reduction of the numerical error due to wave reflections is achieved by applying the present adaptive algorithm

an adaptive level set approach for incompressible two phase flows we present a numerical method using the level set approach for solving incompressible two phase flow with surface tension in the level set approach the free surface is represented as the zero level set of a smooth function this has the effect of replacing the advection of density which has steep gradients at the free surface with the advection of the level set function which is smooth in addition the free surface can merge or break up with no special treatment we maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects in this work we couple the level set scheme to an adaptive projection method for the incompressible navier stokes equations in order to achieve higher resolution of the free surface with a minimum of additional expense we present two dimensional axisymmetric and fully three dimensional results of air bubble and water drop computations

new resolution strategy for multiscale reaction waves using time operator splitting space adaptive multiresolution and dedicated high order implicit explicit time integrators we tackle the numerical simulation of reaction diffusion equations modeling multi scale reaction waves this type of problem induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts spatially very localized in this paper we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts the paper considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is restricted by neither the fastest scales in the source term nor by stability constraints of the diffusive steps but only by the physics of the phenomenon we aim thus at solving complete models including all time and space scales within a prescribed accuracy considering large simulation domains with conventional computing resources the efficiency is evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for two dimensional spiral waves and three dimensional scroll waves as an illustration future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are discussed

distance learning in discriminative vector quantization discriminative vector quantization schemes such as learning vector quantization lvq and extensions thereof offer efficient and intuitive classifiers based on the representation of classes by prototypes the original methods however rely on the euclidean distance corresponding to the assumption that the data can be represented by isotropic clusters for this reason extensions of the methods to more general metric structures have been proposed such as relevance adaptation in generalized lvq glvq and matrix learning in glvq in these approaches metric parameters are learned based on the given classification task such that a data driven distance measure is found in this letter we consider full matrix adaptation in advanced lvq schemes in particular we introduce matrix learning to a recent statistical formalization of lvq robust soft lvq and we compare the results on several artificial and real life data sets to matrix learning in glvq a derivation of lvq like learning based on a heuristic cost function in all cases matrix adaptation allows a significant improvement of the classification accuracy interestingly however the principled behavior of the models with respect to prototype locations and extracted matrix dimensions shows several characteristic differences depending on the data sets

an adaptive mesh refinement algorithm for the radiative transport equation the discrete ordinates form of the radiative transport equation rte is spatially discretized and solved using an adaptive mesh refinement amr algorithm this technique permits local grid refinement to minimize spatial discretization error of the rte an error estimator is applied to define regions for local grid refinement overlapping refined grids are recursively placed in these regions and the rte is then solved over the entire domain the procedure continues until the spatial discretization error has been reduced to a sufficient level the following aspects of the algorithm are discussed error estimation grid generation communication between refined levels and solution sequencing this initial formulation employs the step scheme and is valid for absorbing and isotropically scattering media in two dimensional enclosures the utility of the algorithm is tested by comparing the convergence characteristics and accuracy to those of the standard single grid algorithm for two simple benchmark problems the amr algorithm maintains the convergence characteristics of the standard single grid algorithm but it does not provide any efficiency gains due to a lack of disparate spatial scales in a third more localized problem however the amr algorithm demonstrates significant memory and cpu time reductions

high order corrected trapezoidal quadrature rules for singular functions a class of quadrature formulae is presented applicable to both nonsingular and singular functions generalizing the classical endpoint corrected trapezoidal quadrature rules while the latter rules are usually derived by means of the euler maclaurin formula their generalizations are obtained as solutions of certain systems of linear algebraic equations a procedure is developed for the construction of very high order quadrature rules applicable to functions with a priori specified singularities and relaxing the requirements on the distribution of nodes the scheme applies not only to nonsingular functions but also to a wide class of functions with monotonic singularities numerical experiments are presented demonstrating the practical usefulness of the new class of quadratures tables of quadrature weights are included for singularities of the form s x x lambda for a variety of values of lambda and s x log x

a time accurate explicit multi scale technique for gas dynamics abstract we present a new time accurate algorithm for the explicit numerical integration of the compressible euler equations of gas dynamics this technique is based on the discrete event simulation des methodology for nonlinear flux conservative pdes y a omelchenko h karimabadi self adaptive time integration of flux conservative equations with sources j comput phys 216 1 2006 179 194 des enables adaptive distribution of cpu resources in accordance with local time scales of the underlying numerical solution it distinctly stands apart from multiple local time stepping algorithms in that it requires neither selecting a global synchronization time step nor pre determining a sequence of time integration operations for individual parts of a heterogeneous numerical system in this paper we extend the des methodology in three important directions i we apply des to a system of coupled gas dynamics equations discretized via a central upwind scheme a kurganov e tadmor new high resolution central schemes for nonlinear conservation laws and convection diffusion equations j comput phys 160 2000 241 282 a kurganov s noelle g petrova semidiscrete central upwind schemes for hyperbolic conservation laws and hamilton jacobi equations siam j sci comput 23 3 2001 707 740 ii we introduce a new preemptive event processing pep technique which automatically enforces synchronous execution of events with sufficiently close update times iii we significantly improve the accuracy of the previous algorithm y a omelchenko h karimabadi self adaptive time integration of flux conservative equations with sources j comput phys 216 1 2006 179 194 by applying locally second order in time flux conserving corrections to the solution obtained with the forward euler scheme the performance of the new technique is demonstrated in a series of one dimensional gas dynamics test problems by comparing numerical solutions obtained in event driven and equivalent time stepping simulations

a direct forcing embedded boundary method with adaptive mesh refinement for fluid structure interaction problems in the present work we developed a structured adaptive mesh refinement s amr strategy for fluid structure interaction problems in laminar and turbulent incompressible flows the computational grid consists of a number of nested grid blocks at different refinement levels the coarsest grid blocks always cover the entire computational domain and local refinement is achieved by the bisection of selected blocks in every coordinate direction the grid topology and data structure is managed using the paramesh toolkit the filtered navier stokes equations for incompressible flow are advanced in time using an explicit second order projection scheme where all spatial derivatives are approximated using second order central differences on a staggered grid for transitional and turbulent flow regimes the large eddy simulation les approach is used where special attention is paid on the discontinuities introduced by the local refinement for all the fluid structure interaction problems reported in this study the complete set of equations governing the dynamics of the flow and the structure are simultaneously advanced in time using a predictor corrector strategy an embedded boundary method is utilized to enforce the boundary conditions on a complex moving body which is not aligned with the grid lines several examples of increasing complexity are given to demonstrate the robustness and accuracy of the proposed formulation

adaptive solution techniques for simulating underwater explosions and implosions adaptive solution techniques are presented for simulating underwater explosions and implosions the liquid is assumed to be an adiabatic fluid and the solution in the gas is assumed to be uniform in space the solution in water is integrated in time using a semi implicit time discretization of the adiabatic euler equations results are presented either using a non conservative semi implicit algorithm or a conservative semi implicit algorithm a semi implicit algorithm allows one to compute with relatively large time steps compared to an explicit method the interface solver is based on the coupled level set and volume of fluid method clsvof m sussman a second order coupled level set and volume of fluid method for computing growth and collapse of vapor bubbles j comput phys 187 2003 110 136 m sussman e g puckett a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows j comput phys 162 2000 301 337 several underwater explosion and implosion test cases are presented to show the performances of our proposed techniques

a high order fast algorithm for scattering calculation in two dimensions abstract we present a high order fast iterative solver for the direct scattering calculation for the helmholtz equation in two dimensions our algorithm solves the scattering problem formulated as the lippmann schwinger integral equation for compactly supported smoothly vanishing scatterers there are two main components to this algorithm first the integral equation is discretized with quadratures based on high order corrected trapezoidal rules for the logarithmic singularity present in the kernel of the integral equation second on the uniform mesh required for the trapezoidal rule we rewrite the discretized integral operator as a composition of two linear operators a discrete convolution followed by a diagonal multiplication therefore the application of these operators to an arbitrary vector required by an iterative method for the solution of the discretized linear system will cost n 2 log n for a n by n mesh with the help of fft we will demonstrate the performance of the algorithm for scatterers of complex structures and at large wave numbers for numerical implementations gmres iterations will be used and corrected trapezoidal rules up to order 20 will be tested

dimensionality reduction of multimodal labeled data by local fisher discriminant analysis reducing the dimensionality of data without losing intrinsic information is an important preprocessing step in high dimensional data analysis fisher discriminant analysis fda is a traditional technique for supervised dimensionality reduction but it tends to give undesired results if samples in a class are multimodal an unsupervised dimensionality reduction method called locality preserving projection lpp can work well with multimodal data due to its locality preserving property however since lpp does not take the label information into account it is not necessarily useful in supervised learning scenarios in this paper we propose a new linear supervised dimensionality reduction method called local fisher discriminant analysis lfda which effectively combines the ideas of fda and lpp lfda has an analytic form of the embedding transformation and the solution can be easily computed just by solving a generalized eigenvalue problem we demonstrate the practical usefulness and high scalability of the lfda method in data visualization and classification tasks through extensive simulation studies we also show that lfda can be extended to non linear dimensionality reduction scenarios by applying the kernel trick

geometric mean for subspace selection subspace selection approaches are powerful tools in pattern classification and data visualization one of the most important subspace approaches is the linear dimensionality reduction step in the fisher s linear discriminant analysis flda which has been successfully employed in many fields such as biometrics bioinformatics and multimedia information management however the linear dimensionality reduction step in flda has a critical drawback for a classification task with c classes if the dimension of the projected subspace is strictly lower than c 1 the projection to a subspace tends to merge those classes which are close together in the original feature space if separate classes are sampled from gaussian distributions all with identical covariance matrices then the linear dimensionality reduction step in flda maximizes the mean value of the kullback leibler kl divergences between different classes based on this viewpoint the geometric mean for subspace selection is studied in this paper three criteria are analyzed 1 maximization of the geometric mean of the kl divergences 2 maximization of the geometric mean of the normalized kl divergences and 3 the combination of 1 and 2 preliminary experimental results based on synthetic data uci machine learning repository and handwriting digits show that the third criterion is a potential discriminative subspace selection method which significantly reduces the class separation problem in comparing with the linear dimensionality reduction step in flda and its several representative extensions

metric learning by discriminant neighborhood embedding in this paper we learn a distance metric in favor of classification task from available labeled samples multi class data points are supposed to be pulled or pushed by discriminant neighbors we define a discriminant adjacent matrix in favor of classification task and learn a map transforming input data into a new space such that intra class neighbors become even more nearby while extra class neighbors become as far away from each other as possible our method is non parametric non iterative and immune to small sample size sss problem target dimensionality of the new space is selected by spectral analysis in the proposed method experiments on real world data sets demonstrate the effectiveness of our method

a consistent approach to large eddy simulation using adaptive mesh refinement the large eddy simulation of turbulent flows is discussed with particular attention paid to the issue of commutation of differentiation and filtering multi level adaptive mesh refinement is proposed as a means of mostly avoiding commutation errors where increased grid resolution is required to capture key flow features the strategy is to employ multiple uniform grids in a nested hierarchy using a constant width filter for each grid it is shown that commutivity of fine and coarse grid filters must be enforced in order to consistently relate variables at different refinement levels methods for treating fine grid boundaries and walls are also discussed it is shown that errors associated with boundary treatments are small and localized

accuracy verification of a 2d adaptive mesh refinement method for incompressible or steady flow locating accurate centres of vortices is one of the accuracy measures for computational methods in fluid flow and the lid driven cavity flows are widely used as benchmarks this paper analyses the accuracy of an adaptive mesh refinement method using 2d steady incompressible lid driven cavity flows the adaptive mesh refinement method performs mesh refinement based on the numerical solutions of navier stokes equations solved by navier2d a vertex centred finite volume code that uses the median dual mesh to form the control volumes cvs about each vertex the accuracy of the refined meshes is demonstrated by the centres of vortices obtained in the benchmarks being contained in the twice refined cells the adaptive mesh refinement method investigated in this paper is proposed based on the qualitative theory of differential equations theoretically infinite refinements can be performed on an initial mesh practically we can stop the process of refinement based on tolerance conditions the method can be applied to find accurate numerical solutions of any mathematical models containing the continuity equation for incompressible fluid or steady state fluid flow

distance metric learning for large margin nearest neighbor classification the accuracy of k nearest neighbor knn classification depends significantly on the metric used to compute distances between different examples in this paper we show how to learn a mahalanobis distance metric for knn classification from labeled examples the mahalanobis metric can equivalently be viewed as a global linear transformation of the input space that precedes knn classification using euclidean distances in our approach the metric is trained with the goal that the k nearest neighbors always belong to the same class while examples from different classes are separated by a large margin as in support vector machines svms the margin criterion leads to a convex optimization based on the hinge loss unlike learning in svms however our approach requires no modification or extension for problems in multiway as opposed to binary classification in our framework the mahalanobis distance metric is obtained as the solution to a semidefinite program on several data sets of varying size and difficulty we find that metrics trained in this way lead to significant improvements in knn classification sometimes these results can be further improved by clustering the training examples and learning an individual metric within each cluster we show how to learn and combine these local metrics in a globally integrated manner

local rectangular refinement with application to nonreacting and reacting fluid flow problems a new solution adaptive gridding method has been developed for the solution of discretized systems of coupled nonlinear elliptic partial differential equations on rectangular domains such a method is required for the numerical solution of realistic combustion problems in which physical quantities may vary by orders of magnitude over one tenth of a millimeter at atmospheric pressure or over micrometers at higher pressures the local rectangular refinement lrr method maintains orthogonality at grid line intersections but lifts the tensor product restriction common to traditional grids producing unstructured grids governing equations are discretized throughout the domain using newly derived forms and newton s method is used to solve the resulting system on a simple test case with a known solution the lrr method and its new discretizations are found to be more accurate than gridding methods representative of those appearing previously in the literature for the more realistic problem of nonreacting driven square cavity flow the lrr solution agrees very well with previously published data when the lrr method is applied to a practical reacting flow a rich axisymmetric laminar bunsen flame with complex chemistry multicomponent transport and an optically thin radiation submodel grid spacing highly influences the inner flame s position which stabilizes only with adequate refinement the vorticity velocity formulation of the governing equations is shown to produce valid results when used in conjunction with the lrr gridding technique furthermore each lrr grid is used to form a nonuniform equivalent tensor product etp grid and also in most cases an equispaced fully refined fr grid these additional grids are supersets of the lrr grids and thus contain refinement in exactly the same regions performance comparisons between the lrr etp and fr grids indicate that the lrr method provides substantial savings in execution time and computer memory requirements without compromising solution accuracy

a new adaptive mesh refinement strategy for numerically solving evolutionary pde s a graph based implementation of quadtree meshes for dealing with adaptive mesh refinement amr in the numerical solution of evolutionary partial differential equations is discussed using finite volume methods the technique displays a plug in feature that allows replacement of a group of cells in any region of interest for another one with arbitrary refinement and with only local changes occurring in the data structure the data structure is also specially designed to minimize the number of operations needed in the amr implementation of the new scheme allows flexibility in the levels of refinement of adjacent regions moreover storage requirements and computational cost compare competitively with mesh refinement schemes based on hierarchical trees low storage is achieved for only the children nodes are stored when a refinement takes place these nodes become part of a graph structure thus motivating the denomination autonomous leaves graph alg for the new scheme neighbors can then be reached without accessing their parent nodes additionally linear system solvers based on the minimization of functionals can be easily employed alg was not conceived with any particular problem or geometry in mind and can thus be applied to the study of several phenomena some test problems are used to illustrate the effectiveness of the technique

efficient multi modal geometric mean metric learning abstract with the fast development of information acquisition there is a rapid growth of multi modality data e g text audio image and video in health care multimedia retrieval and many other applications confronted with the challenges of clustering classification or regression with multi modality information it is essential to effectively measure the distance or similarity between objects described with heterogeneous features metric learning aimed at finding a task oriented distance function is a hot topic in machine learning however most existing algorithms lack efficiency for high dimensional multi modality tasks in this work we develop an effective and efficient metric learning algorithm for multi modality data i e efficient multi modal geometric mean metric learning emgmml the proposed algorithm learns a distinctive distance metric for each view by minimizing the distance between similar pairs while maximizing the distance between dissimilar pairs to avoid overfitting the optimization objective is regularized by symmetrized logdet divergence emgmml is very efficient in that there is a closed form solution for each distance metric experimental results show that the proposed algorithm outperforms the state of the art metric learning methods in terms of both accuracy and efficiency

formulation analysis and numerical study of an optimization based conservative interpolation remap of scalar fields for arbitrary lagrangian eulerian methods we develop and study the high order conservative and monotone optimization based remap obr of a scalar conserved quantity mass between two close meshes with the same connectivity the key idea is to phrase remap as a global inequality constrained optimization problem for mass fluxes between neighboring cells the objective is to minimize the discrepancy between these fluxes and the given high order target mass fluxes subject to constraints that enforce physically motivated bounds on the associated primitive variable density in so doing we separate accuracy considerations handled by the objective functional from the enforcement of physical bounds handled by the constraints the resulting obr formulation is applicable to general unstructured heterogeneous grids under some weak requirements on grid proximity but not on the cell types we prove that the obr algorithm is linearity preserving in one two and three dimensions the paper also examines connections between the obr and the recently proposed flux corrected remap fcr liska et al 1 we show that the fcr solution coincides with the solution of a modified version of obr m obr which has the same objective but a simpler set of box constraints derived by using a worst case scenario because m obr fcr has a smaller feasible set preservation of linearity may be lost and accuracy may suffer for some grid configurations our numerical studies confirm this and show that obr delivers significant increases in robustness and accuracy preliminary efficiency studies of obr reveal that it is only a factor of 2 1 slower than fcr but admits 1 5 times larger time steps

regularization in matrix relevance learning in this paper we present a regularization technique to extend recently proposed matrix learning schemes in learning vector quantization lvq these learning algorithms extend the concept of adaptive distance measures in lvq to the use of relevance matrices in general metric learning can display a tendency towards oversimplification in the course of training an overly pronounced elimination of dimensions in feature space can have negative effects on the performance and may lead to instabilities in the training we focus on matrix learning in generalized lvq glvq extending the cost function by an appropriate regularization term prevents the unfavorable behavior and can help to improve the generalization ability the approach is first tested and illustrated in terms of artificial model data furthermore we apply the scheme to benchmark classification data sets from the uci repository of machine learning we demonstrate the usefulness of regularization also in the case of rank limited relevance matrices i e matrix learning with an implicit low dimensional representation of the data

adaptive mesh refinement applied to the scalar transported pdf equation in a turbulent jet this paper describes a novel solution method for the transported probability density function pdf equation for scalars compositions in contrast to conventional solution methods based on the monte carlo approach we use a finite volume method combined with adaptive mesh refinement amr applied in both physical and compositional space the obvious advantage of this over a uniform grid is that fine meshes are only used where the solution requires high resolution the efficiency of the method is demonstrated by a number of tests involving a turbulent jet flow with up to two scalars both reacting and non reacting we find that the amr calculation can be at a fraction of the computer cost of a uniform grid calculation with the same accuracy copyright 2010 john wiley sons ltd

an adaptive multiresolution scheme with local time stepping for evolutionary pdes we present a fully adaptive numerical scheme for evolutionary pdes in cartesian geometry based on a second order finite volume discretization a multiresolution strategy allows local grid refinement while controlling the approximation error in space for time discretization we use an explicit runge kutta scheme of second order with a scale dependent time step on the finest scale the size of the time step is imposed by the stability condition of the explicit scheme on larger scales the time step can be increased without violating the stability requirement of the explicit scheme the implementation uses a dynamic tree data structure numerical validations for test problems in one space dimension demonstrate the efficiency and accuracy of the local time stepping scheme with respect to both multiresolution scheme with global time stepping and finite volume scheme on a regular grid fully adaptive three dimensional computations for reaction diffusion equations illustrate the additional speed up of the local time stepping for a thermo diffusive flame instability

three dimensional adaptive mesh refinement for hyperbolic conservation laws a local adaptive mesh refinement algorithm for solving hyperbolic systems of conservation laws in three space dimensions is described the method is based on the use of local grid patches superimpo

mesh refinement algorithms in an unstructured solver for multiphase flow simulation using discrete particles this study developed spray adaptive mesh refinement algorithms with directional sensitivity in an unstructured solver to improve spray simulation for internal combustion engine application inadequate spatial resolution is often found to cause inaccuracies in spray simulation using the lagrangian eulerian approach due to the over estimated diffusion and inappropriate liquid gas phase coupling dynamic mesh refinement algorithms adaptive to fuel sprays and vapor gradients were developed in order to increase the grid resolution in the spray region to improve simulation accuracy the local refinement introduced the coarse fine face interface that requires advanced numerical schemes for flux calculation and grid rezoning with moving boundaries to resolve the issue in flux calculation this work implemented the refinement coarsening algorithms into a collocated solver to avoid tedious interpolations in solving the momentum equations a pressure correction method was applied to address unphysical pressure oscillations due to the collocation of pressure and velocity an edge based algorithm was used to evaluate the edge centered quantities in order to account for the contributions from all the cells around an edge at the coarse fine interface a quasi second order upwind scheme with strong monotonicity was also modified to accommodate the coarse fine interface for convective fluxes to resolve the issue related to grid rezoning rezoning was applied to the initial baseline mesh only and the new locations of the refined grids were obtained by interpolating the updated baseline mesh the time step constraints were also re evaluated to account for the change resulting from mesh refinement the present refinement algorithm was used in simulating fuel sprays in an engine combustion chamber it was found that the present approach could produce the same level of results as those using the uniformly fine mesh with substantially reduced computer time results also showed that this approach could alleviate the artifacts related to the lagrangian discrete modeling of spray drops due to insufficient spatial resolution

locally corrected multidimensional quadrature rules for singular functions accurate numerical integration of singular functions usually requires either adaptivity or product integration both interfere with fast summation techniques and thus hamper large scale computations this paper presents a method for computing highly accurate quadrature formulas for singular functions which combine well with fast summation methods given the singularity and the n nodes we first construct weights which integrate smooth functions with order k accuracy then we locally correct a small number of weights near the singularity to achieve order k accuracy on singular functions as well the method is highly efficient and runs in o nk 2d n log 2 n time and o k 2d n space we derive precise error bounds and time estimates and confirm them with numerical results which demonstrate the accuracy and efficiency of the method in large scale computations as part of our implementation we also construct a new adaptive multidimensional product gauss quadrature routine with an effective error

local algorithms for pattern recognition and dependencies estimation in previous publications bottou and vapnik 1992 vapnik 1992 we described local learning algorithms which result in performance improvements for real problems we present here the theoretical framework on which these algorithms are based first we present a new statement of certain learning problems namely the local risk minimization we review the basic results of the uniform convergence theory of learning and extend these results to local risk minimization we also extend the structural risk minimization principle for both pattern recognition problems and regression problems this extended induction principle is the basis for a new class of algorithms

high order corrected trapezoidal quadrature rules for functions with a logarithmic singularity in 2 d abstract in this report we construct correction coefficients to obtain high order trapezoidal quadrature rules to evaluate two dimensional integrals with a logarithmic singularity of the form where the domain d is a square containing the point of singularity 0 0 and v is a c function defined on the whole plane r 2 the procedure we use is a generalization to 2 d of the method of central corrections for logarithmic singularities described in 1 as in 1 d the correction coefficients are independent of the number of sampling points used to discretize the square d when v has compact support contained in d the approximation is the trapezoidal rule plus a local weighted sum of the values of v around the point of singularity these quadrature rules give an efficient stable and accurate way of approximating j v we provide the correction coefficients to obtain corrected trapezoidal quadrature rules up to order 20

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

modeling material responses by arbitrary lagrangian eulerian formulation and adaptive mesh refinement method in this paper we report an efficient numerical method combining a staggered arbitrary lagrangian eulerian ale formulation with the adaptive mesh refinement amr method for materials modeling including elastic plastic flows material failure and fragmentation predictions unlike traditional amr applied on fixed domains our investigation focuses on the application to moving and deforming meshes resulting from lagrangian motion we give details of this numerical method with a capability to simulate elastic plastic flows and predict material failure and fragmentation and our main focus of this paper is to create an efficient method which combines ale and amr methods to simulate the dynamics of material responses with deformation and failure mechanisms the interlevel operators and boundary conditions for these problems in amr meshes have been investigated and error indicators to locate material deformation and failure regions are studied the method has been applied on several test problems and the solutions of the problems obtained with the ale amr method are reported parallel performance and software design for the ale amr method are also discussed