preconditioned minimal residual methods for chebyshev spectral calculations abstract the problem of preconditioning the pseudospectral chebyshev approximation of an elliptic operator is considered the numerical sensitivity to variations of the coefficients of the operator are investigated for two classes of preconditioning matrices one arising from finite differences the other from finite elements the preconditioned system is solved by a conjugate gradient type method and by a dufort frankel method with dynamical parameters the methods are compared on some test problems with the richardson method 13 and with the minimal residual richardson method 21

finite element preconditioning for spectral multigrid methods abstract the good performance of finite element preconditioning for the algebraic spectral systems is demonstrated different types of multigrid strategies are compared the most efficient strategy consists of a pure richardson relaxation preconditioned by one v cycle of a finite element multigrid method the good smoothing properties of line relaxation become obvious the computational work for the various methods is estimated

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

higher order properties of gmm and generalized empirical likelihood estimators in an effort to improve the small sample properties of generalized method of moments gmm estimators a number of alternative estimators have been suggested these include empirical likelihood el continuous updating and exponential tilting estimators we show that these estimators share a common structure being members of a class of generalized empirical likelihood gel estimators we use this structure to compare their higher order asymptotic properties we find that gel has no asymptotic bias due to correlation of the moment functions with their jacobian eliminating an important source of bias for gmm in models with endogeneity we also find that el has no asymptotic bias from estimating the optimal weight matrix eliminating a further important source of bias for gmm in panel data models we give bias corrected gmm and gel estimators we also show that bias corrected el inherits the higher order property of maximum likelihood that it is higher order asymptotically efficient relative to the other bias corrected estimators

a posteriori steered p robust multigrid with optimal step sizes and adaptive number of smoothing steps we develop a multigrid solver steered by an a posteriori estimator of the algebraic error we adopt the context of a second order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree p 1 our solver employs zero pre and one post smoothing by the overlapping schwarz block jacobi method and features an optimal choice of the step sizes in the smoothing correction on each level by line search this leads to a simple pythagorean formula of the algebraic error in the next step in terms of the current error and level wise and patch wise error reductions we show the two following results and their equivalence the solver contracts the algebraic error independently of the polynomial degree p and the estimator represents a two sided p robust bound on the algebraic error the p robustness results are obtained by carefully applying the results of schoberl et al ima j numer anal 28 2008 pp 1 24 for one mesh combined with a multilevel stable decomposition for piecewise affine polynomials of xu et al multiscale nonlinear and adaptive approximation springer berlin 2009 pp 599 659 we consider quasi uniform or graded bisection simplicial meshes and prove mild dependence on the number of mesh levels for minimal h 1 regularity and complete independence for h 2 regularity we also present a simple and effective way for the solver to adaptively choose the number of post smoothing steps necessary at each individual level yielding a yet improved error reduction numerical tests confirm p robustness and show the benefits of the adaptive number of smoothing steps

empirical likelihood and general estimating equations for some time so called empirical likelihoods have been used heuristically for purposes of nonparametric estimation owen showed that empirical likelihood ratio statistics for various parameters theta f of an unknown distribution f have limiting chi square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods our objective in this paper is twofold first to link estimating functions or equations and empirical likelihood second to develop methods of combining information about parameters we do this by assuming that information about f and theta is available in the form of unbiased estimating functions empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood efficiency results for estimates of both theta and f are obtained the methods are illustrated on several problems and areas for future investigation are noted

model free forward screening via cumulative divergence feature screening plays an important role in the analysis of ultrahigh dimensional data due to complicated model structure and high noise level existing screening methods often suffer from model

extending the scope of empirical likelihood this article extends the scope of empirical likelihood methodology ill three directions to allow for plug in estimates of nuisance parameters in estimating equations slower than root n rates of convergence and settings in which there are a relatively large number of estimating equations compared to the sample size calibrating empirical likelihood confidence regions with plug in is sometimes intractable due to the complexity of the asymptotics so we introduce a bootstrap approximation that call be used in such situations we provide a range of examples from survival analysis and nonparametric statistics to illustrate the main results

defect correction for the advection diffusion equation abstract a defect correction method for the advection diffusion equation is presented the discretization is performed by 2nd order finite difference schemes schemes where the 2nd order upstream scheme is combined with the standard central scheme for preconditioning the usual first order upstream scheme is employed the defect correction iteration is used for relaxation inside a multigrid procedure it is shown that the smoothing analysis yields rather pessimistic results with rates of about 0 7 0 8 in the practical computation the discretization error is reached in 2 v cycles numerical results are presented which demonstrate the high efficiency of our treatment

trace pursuit a general framework for model free variable selection abstractwe propose trace pursuit for model free variable selection under the sufficient dimension reduction paradigm two distinct algorithms are proposed stepwise trace pursuit and forward trace pursuit stepwise trace pursuit achieves selection consistency with fixed p forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality the screening consistency property of forward trace pursuit based on sliced inverse regression is established finite sample performances of trace pursuit and other model free variable selection methods are compared through numerical studies supplementary materials for this article are available online

splitting techniques for the pseudospectral approximation of the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is considered using the uzawa algorithm the spectral system is decoupled into a helmholtz equation and an equation with a pseudo laplace operator it is proven that the eigenvalues of the pseudo laplacian are real and negative except one zero eigenvalue belonging to the constant mode the pseudospectral method avoids spurious modes since the pressure is approximated with lower order two degrees lower polynomials than the velocity only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used for discretization for the iterative solution of the spectral system an effective preconditioner is necessary here effective finite difference preconditioners for the spectral pseudo laplacian are presented

confidence regions for entries of a large precision matrix we consider the statistical inference for high dimensional precision matrices specifically we propose a data driven procedure for constructing a class of simultaneous confidence regions for a subset of the entries of a large precision matrix the confidence regions can be applied to test for specific structures of a precision matrix and to recover its nonzero components we first construct an estimator for the precision matrix via penalized node wise regression we then develop the gaussian approximation to approximate the distribution of the maximum difference between the estimated and the true precision coefficients a computationally feasible parametric bootstrap algorithm is developed to implement the proposed procedure the theoretical justification is established under the setting which allows temporal dependence among observations therefore the proposed procedure is applicable to both independent and identically distributed data and time series data numerical results with both simulated and real data confirm the good performance of the proposed method

marginal empirical likelihood and sure independence feature screening we study a marginal empirical likelihood approach in scenarios when the number of variables grows exponentially with the sample size the marginal empirical likelihood ratios as functions of the parameters of interest are systematically examined and we find that the marginal empirical likelihood ratio evaluated at zero can be used to differentiate whether an explanatory variable is contributing to a response variable or not based on this finding we propose a unified feature screening procedure for linear models and the generalized linear models different from most existing feature screening approaches that rely on the magnitudes of some marginal estimators to identify true signals the proposed screening approach is capable of further incorporating the level of uncertainties of such estimators such a merit inherits the self studentization property of the empirical likelihood approach and extends the insights of existing feature screening methods moreover we show that our screening approach is less restrictive to distributional assumptions and can be conveniently adapted to be applied in a broad range of scenarios such as models specified using general moment conditions our theoretical results and extensive numerical examples by simulations and data analysis demonstrate the merits of the marginal empirical likelihood approach

sure independence screening in generalized linear models with np dimensionality ultrahigh dimensional variable selection plays an increasingly important role in contemporary scientific discoveries and statistical research among others fan and lv 2008 propose an independent screening framework by ranking the marginal correlations they showed that the correlation ranking procedure possesses a sure independence screening property within the context of the linear model with gaussian covariates and responses in this paper we propose a more general version of the independent learning with ranking the maximum marginal likelihood estimates or the maximum marginal likelihood itself in generalized linear models we show that the proposed methods with fan and lv 2008 as a very special case also possess the sure screening property with vanishing false selection rate the conditions under which that the independence learning possesses a sure screening is surprisingly simple this justifies the applicability of such a simple method in a wide spectrum we quantify explicitly the extent to which the dimensionality can be reduced by independence screening which depends on the interactions of the covariance matrix of covariates and true parameters simulation studies are used to illustrate the utility of the proposed approaches in addition we supported in part by grant nsf grants dms 0714554 and dms 0704337 the bulk of the work was conducted when rui song was a postdoctoral research fellow at princeton university the authors would like to thank the associate editor and two referees for their constructive comments that improve the presentation and the results of the paper ams 2000 subject classifications primary 68q32 62j12 secondary 62e99 60f10

local independence feature screening for nonparametric and semiparametric models by marginal empirical likelihood we consider an independence feature screening technique for identifying explanatory variables that locally contribute to the response variable in high dimensional regression analysis without requiring a specific parametric form of the underlying data model our approach accommodates a wide spectrum of nonparametric and semiparametric model families to detect the local contributions of explanatory variables our approach constructs empirical likelihood locally in conjunction with marginal nonparametric regressions since our approach actually requires no estimation it is advantageous in scenarios such as the single index models where even specification and identification of a marginal model is an issue by automatically incorporating the level of variation of the nonparametric regression and directly assessing the strength of data evidence supporting local contribution from each explanatory variable our approach provides a unique perspective for solving feature screening problems theoretical analysis shows that our approach can handle data dimensionality growing exponentially with the sample size with extensive theoretical illustrations and numerical examples we show that the local independence screening approach performs promisingly

spectral collocation on triangular elements we consider the poisson problem on a segment of the unit disc and on triangles on the segment we transform the poisson problem by means of polar coordinates in these new coordinates we have a problem in a rectangle which can easily be mapped onto the square here standard chebyshev collocation techniques can be applied then the segment is mapped onto a triangle where the same spectral scheme may be used by numerical tests we observed the expected high spectral accuracy due to the corner singularity a singular behaviour of the solution can be expected here we improved the accuracy by auxiliary mapping techniques further it is shown that finite difference preconditioning can be successfully applied in order to construct an efficient iterative solver finally a domain decomposition technique is applied to the patching of a rectangular and a triangular element

improved spectral multigrid methods for periodic elliptic problems abstract the spectral multigrid method for periodic elliptic problems is examined several modifications are introduced including a midpoint pseudospectral discretization which eliminates the need for filtering the highest fourier mode and new relaxation schemes for isotropic and anisotropic problems numerical results are presented demonstrating substantial increases in efficiency and accuracy over previous methods

factor profiled sure independence screening we propose a method of factor profiled sure independence screening for ultrahigh dimensional variable selection the objective of this method is to identify nonzero components consistently from a sparse coefficient vector the new method assumes that the correlation structure of the high dimensional data can be well represented by a set of low dimensional latent factors which can be estimated consistently by eigenvalue eigenvector decomposition the estimated latent factors should then be profiled out from both the response and the predictors such an operation referred to as factor profiling produces uncorrelated predictors therefore sure independence screening can be applied subsequently and the resulting screening result is consistent for model selection a major advantage that standard sure independence screening does not share we refer to the new method as factor profiled sure independence screening numerical studies confirm its outstanding performance copyright 2012 oxford university press

spectral collocation methods and polar coordinate singularities this paper considers the numerical solution of elliptic differential equations on the unit disk using polar coordinates the disk is mapped onto a rectangle the resulting transformed problem is solved by a method related to collocation since the origin is a coordinate singularity some natural trial functions are singular there and a special technique is applied to use zero as a collocation point for poisson and helmholtz equations a fast algorithm with an operation count of o n2 log n is presented numerical results show the different stability and convergence properties of the algorithms

a fast spectral element solver combining static condensation and multigrid techniques we propose a spectral element multigrid method for the two dimensional helmholtz equation discretized on regular grids combining p multigrid with static condensation the method achieves nearly linear complexity with an order independent convergence rate for solving the condensed equations for smoothing we consider two groups of edge based relaxation schemes the best of which attains a multigrid convergence rate of 0 014 to 0 028 numerical experiments have been carried out that demonstrate the robustness of the approach for orders up to 32 and a total of 109 degrees of freedom in comparison with a fast finite difference solver the latter is clearly outperformed already for errors of one percent or lower

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

cramer type moderate deviations for studentized two sample u statistics with applications author s chang j shao qm zhou wx abstract 2016 institute of mathematical statistics two sample u statistics are widely used in a broad range of applications including those in the fields of biostatistics and econometrics in this paper we establish sharp cramer type moderate deviation theorems for studentized two sample u statistics in a general framework including the two sample t statistic and studentized mann whitney test statistic as prototypical examples in particular a refined moderate deviation theorem with second order accuracy is established for the two sample t statistic these results extend the applicability of the existing statistical methodologies from the one sample t statistic to more general nonlinear statistics applications to two sample large scale multiple testing problems with false discovery rate control and the regularized bootstrap method are also discussed

splitting techniques for the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is presented using the uzawa algorithm the spectral system is decoupled into helmholtz equations for the velocity components and an equation with the pseudo laplacian for the pressure both the velocity components and the pressure are approximated by polynomials of the same degree n compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well posed the resulting discrete velocity field is identically divergence free only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used here we further compare our treatment with an inhomogeneous neumann condition for the pressure this scheme was proposed in g karniadakis m israeli and s a orszag j comp phys 97 1991 pp 414 443 the comparison shows that the neumann condition yields somewhat better approximations for the pressure in the time discretization a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure it is numerically shown that a stable third order method in time can be achieved in the case of periodic boundary conditions this result is confirmed by fourier analysis

line relaxation for spectral multigrid methods abstract spectral multigrid methods for elliptic problems with dirichlet and periodic boundary conditions are examined the multigrid convergence is improved by certain line relaxation techniques which are suggested by a smoothing analysis furthermore several choices of relaxation parameters are compared numerical results are presented showing the gain in efficiency and accuracy over previous methods

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

penalized high dimensional empirical likelihood we propose penalized empirical likelihood for parameter estimation and variable selection for problems with diverging numbers of parameters our results are demonstrated for estimating the mean vector in multivariate analysis and regression coefficients in linear models by using an appropriate penalty function we showthat penalized empirical likelihood has the oracle property that is with probability tending to 1 penalized empirical likelihood identifies the true model and estimates the nonzero coefficients as efficiently as if the sparsity of the true model was known in advance the advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated by testing hypotheses and constructing confidence regions numerical simulations confirm our theoretical findings

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

defect correction for convection dominated flow a defect correction scheme with the first order upwind preconditioner is considered by fourier analysis the preconditioning properties for the second order upwind scheme the central scheme and spectral methods are examined since the eigenvalues of the preconditioned operator are complex the generalized minimal residual iteration is used for the iterative solution this procedure is applied to the boussinesq flow problem in vorticity streamfunction formulation numerical results are presented for increasing rayleigh numbers

a 3d spectral multigrid method we consider a 3d spectral multigrid method for 3d elliptic problems with arbitrary anisotropies the smoothing method is a preconditioned richardson iteration where a pseudospectral discretization is used on the right hand side and a finite difference discretization on the left hand side in each step of this iteration a finite difference problem is solved approximately by plane relaxation sweeps the plane relaxation requires the solution of 2d finite difference problems these problems are solved approximately using 2d multigrid cycles

asymptotic properties of bridge estimators in sparse high dimensional regression models we study the asymptotic properties of bridge estimators in sparse high dimensional linear regression models when the number of covariates may increase to infinity with the sample size we are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero we show that under appropriate conditions bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance thus bridge estimators have an oracle property in the sense of fan and li j amer statist assoc 96 2001 1348 1360 and fan and peng ann statist 32 2004 928 961 in general the oracle property holds only if the number of covariates is smaller than the sample size however under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size

numerical solution of nonlinear volterrafredholmhammerstein integral equations in two dimensional spaces based on block pulse functions in this paper the numerical technique based on block pulse functions bpfs has been developed to approximate the solutions of nonlinear volterrafredholmhammerstein integral equations in two dimensional spaces these functions are orthogonal and have compact support on 0 1 the proposed method reduces the integral equations to a system of nonlinear algebraic equations that can be easily solved by any numerical method also the convergence of the proposed approach is discussed furthermore in order to show the accuracy and reliability of the above mentioned algorithm the new approach is verified through some numerical examples a numerical technique based on block pulse functions is proposed two dimensional nonlinear volterrafredholmhammerstein integral equations are firstly numerically solved the proposed method is to transform the volterrafredholm integral equations into a system of algebraic equations the main characteristic behind the algorithm is operator theory