eigenvalue convergence on perturbed lipschitz domains for elliptic systems with mixed general decompositions of the boundary abstract we consider eigenvalues of an elliptic operator l u x j a i j u x i where u u 1 u m t is a vector valued function and the coefficients a i j are m m matrices whose elements a i j are bounded and symmetric we perturb our domain 0 by adding a set of small measure t e to form the domain e we prescribe mixed boundary conditions on quite general decompositions of the boundary and look at the behavior of the eigenvalues of e as t e shrinks to zero we look at systems which satisfy either a strong ellipticity condition a legendre hadamard condition or in particular the system of linear elasticity

maximum likelihood estimation of functionals of discrete distributions we consider the problem of estimating functionals of discrete distributions and focus on a tight up to universal multiplicative constants for each specific functional nonasymptotic analysis of the worst case squared error risk of widely used estimators we apply concentration inequalities to analyze the random fluctuation of these estimators around their expectations and the theory of approximation using positive linear operators to analyze the deviation of their expectations from the true functional namely their bias we explicitly characterize the worst case squared error risk incurred by the maximum likelihood estimator mle in estimating the shannon entropy h p sum i 1 s p i ln p i and the power sum f alpha p sum i 1 s p i alpha alpha 0 up to universal multiplicative constants for each fixed functional for any alphabet size s leq infty and sample size n for which the risk may vanish as a corollary for shannon entropy estimation we show that it is necessary and sufficient to have n gg s observations for the mle to be consistent in addition we establish that it is necessary and sufficient to consider n gg s 1 alpha samples for the mle to consistently estimate f alpha p 0 the minimax rate optimal estimators for both problems require s ln s and s 1 alpha ln s samples which implies that the mle has a strictly sub optimal sample complexity when 1 we show that the worst case squared error rate of convergence for the mle is n 2 alpha 1 for infinite alphabet size while the minimax squared error rate is n ln n 2 alpha 1 when alpha geq 3 2 the mle achieves the minimax optimal rate n 1 regardless of the alphabet size as an application of the general theory we analyze the dirichlet prior smoothing techniques for shannon entropy estimation in this context one approach is to plug in the dirichlet prior smoothed distribution into the entropy functional while the other one is to calculate the bayes estimator for entropy under the dirichlet prior for squared error which is the conditional expectation we show that in general such estimators do not improve over the maximum likelihood estimator no matter how we tune the parameters in the dirichlet prior this approach cannot achieve the minimax rates in entropy estimation the performance of the minimax rate optimal estimator with n samples is essentially at least as good as that of dirichlet smoothed entropy estimators with n ln n samples

probability distributions of natural frequencies of uncertain dynamic systems of random eigenvalues commonly encountered in dynamic systems the method involves a hierarchical decomposition of a multivariate function in terms of variables with increasing dimensions a broad range of orthonormal polynomial bases consistent with the probability measure for a fourier polynomial expansion of component functions and an innovative dimension reduction integration for calculating the expansion coefficients the new decomposition does not require sample points yet it generates a convergent sequence of lower variate estimates of theprobabilitydistributionsofeigensolutions numericalresults includingfrequencydistributionsofapiezoelectric transducer indicate that the decomposition method developed provides accurate convergent and computationally efficient estimates of the tail probabilistic characteristics of eigenvalues

spectral convergence and nonlinear dynamics of reaction diffusion equations under perturbations of the domain abstract in this paper we obtain the continuity of attractors for semilinear parabolic problems with neumann boundary conditions relatively to perturbations of the domain we show that if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the neumann laplacian is granted then we obtain the upper semicontinuity of the attractors if moreover every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors we also give necessary and sufficient conditions for the spectral convergence of neumann problems under perturbations of the domain

robust parameter design with computer experiments using orthonormal polynomials robust parameter design with computer experiments is becoming increasingly important for product design existing methodologies for this problem are mostly for finding optimal control factor settings however in some cases the objective of the experimenter may be to understand how the noise and control factors contribute to variation in the response the functional analysis of variance anova and variance decompositions of the response in addition to the mean and variance models help achieve this objective estimation of these quantities is not easy and few methods are able to quantity the estimation uncertainty in this article we show that the use of an orthonormal polynomial model of the simulator leads to simple formulas for functional anova and variance decompositions and the mean and variance models we show that estimation uncertainty can be taken into account in a simple way by first fitting a gaussian process model to experiment data and then approximating it with the orthonormal polynomia

on the sharp effect of attaching a thin handle on the spectral rate of convergence abstract consider two domains connected by a thin tube it can be shown that the resolvent of the dirichlet laplacian is continuous with respect to the channel section parameter this in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation under an explicit nondegeneracy condition we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain as an application we prove that again under an explicit nondegeneracy condition the case of resonant domains features polynomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions

local tail bounds for functions of independent random variables it is shown that functions defined on 0 1 r 1 n satisfying certain conditions of bounded differences that guarantee sub gaussian tail behavior also satisfy a much stronger local sub gaussian property for self bounding and configuration functions we derive analogous locally subexponential behavior the key tool is talagrand s ann probab 22 1994 1576 1587 variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on 0 1 r 1 n for r 2

the effect of domain squeezing upon the dynamics of reaction diffusion equations abstract let be an arbitrary smooth bounded domain in r m r n and e 0 be arbitrary write x y for a generic point of r m r n squeeze by the factor e in the y direction to obtain the squeezed domain e x ey x y consider the following reaction diffusion equation on e formula here e is the exterior normal vector field on e and f r r is a nonlinearity satisfying some growth and dissipativeness conditions assuring such that ee generates a semiflow on h1 e with a global attractor a e we prove that in some strong sense the equations ee have a limiting equationu au f u e0 as e 0 this limiting equation is an abstract semilinear parabolic equation which defines a semiflow on a closed linear subspace of h1 we show that has a global attractor a 0 and the family of attractors a e e 0 is upper semicontinuous at e 0 if m n 1 and satisfies some natural additional assumptions then the limiting equation e0 is equivalent to a parabolic boundary value problem defined on a finite graph the results of this paper extend previous results obtained by hale and raugel for domains which are ordinate sets of a positive function

monotonic quantile regression with bernstein polynomials for stochastic simulation quantile regression is an important tool to determine the quality level of service product and operation systems via stochastic simulation it is frequently known that the quantiles of the output distribution are monotonic functions of certain inputs to the simulation model because there is typically high variability in estimation of tail quantiles it can be valuable to incorporate this information in quantile modeling however the existing literature on monotone quantile regression with multiple inputs is sparse in this article we propose a class of monotonic regression models which consists of functional analysis of variance fanova decomposition components modeled with bernstein polynomial bases for estimating quantiles as a function of multiple inputs the polynomial degrees of the bases for the model and the fanova components included in the model are selected by a greedy algorithm real examples demonstrate the advantages of incorporating the monotonicity assumption in quantile regression a

jackknifing l statistics with smooth weight functions abstract the behavior of linear combinations of order statistics l statistics under jackknifing is discussed the asymptotic behavior of a jackknifed l statistic is described and the pseudovalue based variance estimate is shown to be consistent under moderate smoothness and a trimming condition on the weight function the results extend easily to functions of l statistics and thus answer a question posed in miller 1974 monte carlo results support small sample applicability of the large sample results for the construction of approximate confidence intervals

random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

neumann spectral problem in a domain with very corrugated boundary let rn be a bounded domain we perturb it to a domain e attaching a family of small protuberances with room and passage like geometry e 0 is a small parameter peculiar spectral properties of neumann problems in so perturbed domains were observed for the first time by r courant and d hilbert we study the case when the number of protuberances tends to infinity as e 0 and they are e periodically distributed along a part of our goal is to describe the behavior of the spectrum of the operator ae e 1 e where e is the neumann laplacian in e and the positive function e is equal to 1 in we prove that the spectrum of ae converges as e 0 to the spectrum of a certain boundary value problem for the neumann laplacian in with boundary conditions containing the spectral parameter in a nonlinear manner its eigenvalues may accumulate to a finite point

scrambled net variance for integrals of smooth functions hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter one version randomized t m s nets has the property that the integral estimates are unbiased and that the variance is o 1 n for any square integrable integrand stronger assumptions on the integrand allow one to find rates of convergence this paper shows that for smooth integrands over s dimensions the variance is of order n 3 log n 3 1 compared to n 1 for ordinary monte carlo thus the integration errors are of order n 3 2 log n s 1 2 in probability this compares favorably with the rate n 1 log n s 1 for unrandomized t m s nets

a law of large numbers for nearest neighbour statistics in practical data analysis methods based on proximity near neighbour relationships between sample points are important because these relations can be computed in time n log n as the number of points n associated with such methods are a class of random variables defined to be functions of a given point and its nearest neighbours in the sample if the sample points are independent and identically distributed the associated random variables will also be identically distributed but not independent despite this we show that random variables of this type satisfy a strong law of large numbers in the sense that their sample means converge to their expected values almost surely as the number of sample points n

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

estimating mean dimensionality of analysis of variance decompositions analysis of variance anova is now often applied to functions defined on the unit cube where it serves as a tool for the exploratory analysis of functions the mean dimension of a function defined as a natural weighted combination of its anova mean squares provides one measure of how hard or easy it is to integrate the function by quasi monte carlo sampling this article presents some new identities relating the mean dimension and some analogously defined higher moments to the variance importance measures of i m sobol as a result we are able to measure the mean dimension of certain functions arising in computational finance we produce an unbiased and nonnegative estimate of the variance contribution of the highest order interaction that avoids the cancellation problems of previous estimates in an application to extreme value theory we find that among other things the minimum of d independent u 0 1 random variables has a mean dimension of 2 d 1 d 3

the properties of the cross match estimate and split sampling by noting the connection with k sample u statistics we find a simple decomposition of the variance of the cross match estimate which can be regarded as a generalization of efron and stein we apply the decomposition in assessing efficiencies of several plans of using the weighted samples from an importance scheme the applications of the formula to multiple imputations lead to a method of crossing jointly imputed data to gain more accuracy

uncertainty quantification of high dimensional complex systems by multiplicative polynomial dimensional decompositions summary the central theme of this paper is multiplicative polynomial dimensional decomposition pdd methods for solving high dimensional stochastic problems when a stochastic response is dominantly of multiplicative nature the standard pdd approximation predicated on additive function decomposition may not provide sufficiently accurate probabilistic solutions of a complex system to circumvent this problem two multiplicative versions of pdd referred to as factorized pdd and logarithmic pdd were developed both versions involve a hierarchical multiplicative decomposition of a multivariate function a broad range of orthonormal polynomial bases for fourier polynomial expansions of component functions and a dimension reduction or sampling technique for estimating the expansion coefficients three numerical problems involving mathematical functions or uncertain dynamic systems were solved to corroborate how and when a multiplicative pdd is more efficient or accurate than the additive pdd the results show that indeed both the factorized and logarithmic pdd approximations can effectively exploit the hidden multiplicative structure of a stochastic response when it exists since a multiplicative pdd recycles the same component functions of the additive pdd no additional cost is incurred finally the random eigensolutions of a sport utility vehicle comprising 40 random variables were evaluated demonstrating the ability of the new methods to solve industrial scale problems copyright 2013 john wiley sons ltd received 31 january 2012 revised 6 september 2012 accepted 20 september 2012

classification of homogeneous data with large alphabets given training sequences generated by two distinct but unknown distributions on a common alphabet we study the problem of determining whether a third sequence was generated according to the first or second distribution to model sources such as natural language for which the underlying distributions are difficult to learn from realistic amounts of data we allow the alphabet size to grow and therefore the probability distributions to change with the block length our primary focus is the situation in which the underlying probabilities are all of the same order and in this regime we show that consistent classification is possible if and only if the alphabet grows subquadratically with the block length we also show that some commonly used statistical tests are suboptimal in that they are consistent only if the alphabet grows sublinearly

random points and lattice points in convex bodies assume k subset of r d is a convex body and x is a large finite subset of k how many convex polytopes are there whose vertices belong to x is there a typical shape of such polytopes how well does the maximal such polytope which is actually the convex hull of x approximate k we are interested in these questions mainly in two cases the first is when x is a random sample of n uniform independent points from k in this case motivation comes from sylvester s famous four point problem and from the theory of random polytopes the second case is when x k boolean and z d where z d is the lattice of integer points in r d and the questions come from integer programming and geometry of numbers surprisingly or not so surprisingly the answers in the two cases are rather similar

breaking the curse of nonregularity with subagging inference of the mean outcome under optimal treatment regimes precision medicine is an emerging medical approach that allows physicians to select the treatment options based on individual patient information the goal of precision medicine is to identify the optimal treatment regime otr that yields the most favorable clinical outcome prior to adopting any otr in clinical practice it is crucial to know the impact of implementing such a policy although considerable research has been devoted to estimating the otr in the literature less attention has been paid to statistical inference of the otr challenges arise in the nonregular cases where the otr is not uniquely defined to deal with nonregularity we develop a novel inference method for the mean outcome under an otr the optimal value function based on subsample aggregating subagging the proposed method can be applied to multi stage studies where treatments are sequentially assigned over time bootstrap aggregating bagging and subagging have been recognized as effective variance reduction techniques to improve unstable estimators or classifiers buhlmann and yu 2002 however it remains unknown whether these approaches can yield valid inference results we show the proposed confidence interval ci for the optimal value function achieves nominal coverage in addition due to the variance reduction effect of subagging our method enjoys certain statistical optimality specifically we show that the mean squared error of the proposed value estimator is strictly smaller than that based on the simple sample splitting estimator in the nonregular cases moreover under certain conditions the length of our proposed ci is shown to be on average shorter than cis constructed based on the existing state of the art method luedtke and van der laan 2016 and the oracle method which works as well as if an otr were known extensive numerical studies are conducted to back up our theoretical findings

domain perturbation for elliptic equations subject to robin boundary conditions abstract we prove domain perturbation theorems for linear and nonlinear elliptic equations under robin boundary conditions the theory allows very singular perturbation of domains in particular it includes cutting holes parts degenerating to a set of measure zero such as the dumbbell problem or wildly oscillating boundaries in the last case we show that the limiting problem is the dirichlet problem

global sensitivity analysis for repeated measures studies with informative drop out a semi parametric approach in practice both testable and untestable assumptions are generally required to draw inference about the mean outcome measured at the final scheduled visit in a repeated measures study with drop out scharfstein et al 2014 proposed a sensitivity analysis methodology to determine the robustness of conclusions within a class of untestable assumptions in their approach the untestable and testable assumptions were guaranteed to be compatible their testable assumptions were based on a fully parametric model for the distribution of the observable data while convenient these parametric assumptions have proven especially restrictive in empirical research here we relax their distributional assumptions and provide a more flexible semi parametric approach we illustrate our proposal in the context of a randomized trial for evaluating a treatment of schizoaffective disorder

quadrature error bounds with applications to lattice rules reproducing kernel hilbert spaces are used to derive error bounds and worst case integrands for a large family of quadrature rules in the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature however the theory developed here does not require periodicity and is not restricted to lattice rules an analysis of variance anova decomposition is employed in defining the inner product it is shown that imbedded rules are superior when integrating functions with large high order anova effects

strong data processing inequalities and phi sobolev inequalities for discrete channels the noisiness of a channel can be measured by comparing suitable functionals of the input and output distributions for instance the worst case ratio of output relative entropy to input relative entropy for all possible pairs of input distributions is bounded from above by unity by the data processing theorem however for a fixed reference input distribution this quantity may be strictly smaller than one giving the so called strong data processing inequalities sdpis the same considerations apply to an arbitrary phi divergence this paper presents a systematic study of optimal constants in the sdpis for discrete channels including their variational characterizations upper and lower bounds structural results for channels on product probability spaces and the relationship between the sdpis and the so called phi sobolev inequalities another class of inequalities that can be used to quantify the noisiness of a channel by controlling entropy like functionals of the input distribution by suitable measures of input output correlation several applications to information theory discrete probability and statistical physics are discussed

symmetric statistics poisson point processes and multiple wiener integrals the asymptotic behaviour of symmetric statistics of arbitrary order is studied as an application we describe all limit distributions of square integrable u statistics we use as a tool a randomization of the sample size a sample of poisson size n lambda with en lambda lambda can be interpreted as a poisson point process with intensity lambda and randomized symmetric statistics are its functionals as lambda rightarrow infty the probability distribution of these functionals tend to the distribution of multiple wiener integrals this can be considered as a stronger form of the following well known fact properly normalized a poisson point process with intensity lambda approaches a gaussian random measure as lambda rightarrow infty

generalized entropy power inequalities and monotonicity properties of information new families of fisher information and entropy power inequalities for sums of independent random variables are presented these inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables for an arbitrary collection of subsets as a consequence a simple proof of the monotonicity of information in central limit theorems is obtained both in the setting of independent and identically distributed i i d summands as well as in the more general setting of independent summands with variance standardized sums

geometric lower bounds for the spectrum of elliptic pdes with dirichlet conditions in part an extension of the lower bound lemma of boggio is given for the weak forms of certain elliptic operators which are in general nonlinear and have partially dirichlet and partially neumann boundary conditions its consequences and those of an adapted hardy inequality for the location of the bottom of the spectrum are explored in corollaries wherein a variety of assumptions are placed on the shape of the dirichlet and neumann boundaries

facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term abstract we study crystalline driven curvature flow with spatially nonuniform driving force term we assume special monotonicity properties of the driving term which are motivated by our previous work on berg s effect we consider special initial data which we call bent rectangles we prove existence of solutions for a generic forcing term as well as generic subclass of bent rectangles we show the initially flat facets may begin to bend provided loosely speaking they are too large moreover depending on the initial configuration we notice instantaneous loss of regularity of the moving curve

an application of the efron stein inequality in density estimation the inequality originates from studies of the jackknife estimate of variance several interesting proofs see e g vitale 1984 and applications see e g steele 1981 1982 highlight the depth and usefulness of the efron stein inequality the purpose of this short note is to point out an application in density estimation we consider in particular a density estimate fn of a density f on rd with the following properties

hoeffding anova decompositions for symmetric statistics of exchangeable observations consider a possibly infinite exchangeable sequence x xn 1 n 0 xn c consists of the first n instants of a generalized polya urn sequence for every choice of and c the hoeffding anova decomposition of a symmetric and square integrable statistic t xn c is explicitly computed in terms of linear combinations of well chosen conditional expectations of t our formulae generalize and unify the classic results of hoeffding ann math statist 19 1948 293 325 for i i d variables zhao and chen acta math appl sinica 6 1990 263 272 and bloznelis and gotze ann statist 29 2001 353 365 and ann probab 30 2002 1238 1265 for finite population statistics applications are given to construct infinite weak urn sequences and to characterize the covariance of symmetric statistics of generalized urn sequences

adaptive sparse polynomial dimensional decomposition methods for high dimensional stochastic computing this article presents two novel adaptive sparse polynomial dimensional decomposition pdd methods for solving high dimensional uncertainty quantification problems in computational science and engineering the methods entail global sensitivity analysis for retaining important pdd component functions and a full or sparse grid dimension reduction integration or quasi monte carlo simulation for estimating the pdd expansion coefficients a unified algorithm endowed with two distinct ranking schemes for grading component functions was created for their numerical implementation the fully adaptive sparse pdd method is comprehensive and rigorous leading to the second moment statistics of a stochastic response that converges to the exact solution when the tolerances vanish a partially adaptive sparse pdd method obtained through regulated adaptivity and sparsity is economical and is therefore expected to solve practical problems with numerous variables compared with past developments the adaptive sparse pdd methods do not require their truncation parameter s to be assigned a priori or arbitrarily the numerical results reveal that an adaptive sparse pdd method achieves a desired level of accuracy with considerably fewer coefficients compared with existing pdd approximations for a required accuracy in calculating the probabilistic response characteristics the new bivariate adaptive sparse pdd method is more efficient than the existing bivariately truncated pdd method by almost an order of magnitude finally stochastic dynamic analysis of a disk brake system was performed demonstrating the ability of the new methods to tackle practical engineering problems

dynamics in dumbbell domains i continuity of the set of equilibria we analyze the dynamics of a reaction diffusion equation with homogeneous neumann boundary conditions in a dumbbell domain we provide an appropriate functional setting to treat this problem and as a first step we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds

stochastic nonlinear aeroelastic analysis of a supersonic lifting surface using an adaptive spectral method an adaptive stochastic spectral projection method is deployed for the uncertainty quantification in limit cycle oscillations of an elastically mounted two dimensional lifting surface in a supersonic flow field variabilities in the structural parameters are propagated in the aeroelastic system which accounts for nonlinear restoring force and moment by means of hardening cubic springs the physical nonlinearities promote sharp and sudden flutter onset for small change of the reduced velocity in a stochastic context this behavior translates to steep solution gradients developing in the parametric space a remedy is to expand the stochastic response of the airfoil on a piecewise generalized polynomial chaos basis accurate approximation andaffordable computational costs are obtained using sensitivity based adaptivity for various types of supersonic stochastic responses depending on the selected values of the mach number on the bifurcation map sensitivity analysis via sobol indices shows how the probability density function of the peak pitch amplitude responds to combined uncertainties e g the elastic axis location torsional stiffness and flap angle we believe that this work demonstrates the capability and flexibility of the approach for more reliable predictions of realistic aeroelastic systems subject to a moderate number of uncertainties

a multivariate fast discrete walsh transform with an application to function interpolation for high dimensional problems such as approximation and integration one cannot afford to sample on a grid because of the curse of dimensionality an attractive alternative is to sample on a low discrepancy set such as an integration lattice or a digital net this article introduces a multivariate fast discrete walsh transform for data sampled on a digital net that requires only o n log n operations where n is the number of data points this algorithm and its inverse are digital analogs of multivariate fast fourier transforms this fast discrete walsh transform and its inverse may be used to approximate the walsh coefficients of a function and then construct a spline interpolant of the function this interpolant may then be used to estimate the function s effective dimension an important concept in the theory of numerical multivariate integration numerical results for various functions are presented

estimation and inference of heterogeneous treatment effects using random forests many scientific and engineering challenges ranging from personalized medicine to customized marketing recommendations require an understanding of treatment effect heterogeneity in this article we

allocation of effort in monte carlo simulation for power of permutation tests abstract consider a randomization test in which an observed statistic is compared to a null distribution constructed by first drawing a random sample from the set of all possible permutations of the data and then calculating the statistic anew for each member of the random sample if the power of such a test is investigated by simulation the computer code must have two main loops in the outer loop which has o iterations data are generated under a specific alternative hypothesis for each outer iteration there are i inner iterations in each of which the data are permuted and the statistic calculated assume that the costs of inner and outer iterations are known and the total cost of the study is fixed so that i and o are constrained by o io for fixed this article considers what the optimal choice for i is in a mean squared sense a general expression is derived for the mean squared error of the distribution of p values if the alternative hypothesis is close to the null the optimal c

anchored analysis of variance petrov galerkin projection schemes for linear stochastic structural dynamics in this paper we propose anchored functional analysis of variance petrovgalerkin aapg projection schemes originally developed in the context of parabolic stochastic partial differential equatio

a multigrid method for adaptive sparse grids sparse grids have become an important tool to reduce the number of degrees of freedom of discretizations of moderately high dimensional partial differential equations however the reduction in degrees of freedom comes at the cost of an almost dense and unconventionally structured system of linear equations to guarantee overall efficiency of the sparse grid approach special linear solvers are required we present a multigrid method that exploits the sparse grid structure to achieve an optimal runtime that scales linearly with the number of sparse grid points our approach is based on a novel decomposition of the right hand sides of the coarse grid equations that leads to a reformulation in so called auxiliary coefficients with these auxiliary coefficients the right hand sides can be represented in a nodal point basis on low dimensional full grids our proposed multigrid method directly operates in this auxiliary coefficient representation circumventing most of the computationally cumbersome sparse gr

anchored anova petrov galerkin projection schemes for parabolic stochastic partial differential equations abstract we present a numerical scheme based on the combination of hoeffding s functional analysis of variance anova decomposition with stochastic galerkin projection for solving a class of high dimensional parabolic stochastic partial differential equations spdes the central idea underpinning the proposed approach is to approximate the spde solution using a hoeffding anova decomposition whose component functions are constrained to be orthogonal with respect to an appropriate measure we show that when a dirac product measure is chosen to enforce the orthogonality constraints and the test functions are chosen appropriately the original stochastic weak formulation can be decoupled into low dimensional subproblems that can be efficiently solved in parallel using polynomial chaos based stochastic projection schemes as a result the proposed approach scales very well to spdes with large number of random variables we theoretically analyze the proposed formulation and provide a priori error estimates as a function of the spatial stochastic and temporal discretization parameters and the anova expansion order that hold under appropriate stochastic regularity assumptions numerical studies are presented for a set of time dependent stochastic diffusion problems with up to 50 random variables to demonstrate the effectiveness of the proposed approach and comparisons are made against classical polynomial chaos stochastic galerkin projection scheme gpc and the generalized spectral decomposition scheme these studies show that the proposed approach provides accuracy comparable to classical gpc galerkin projection schemes while incurring significantly lower computational cost

higher order elliptic operators on variable domains stability results and boundary oscillations for intermediate problems abstract we study the spectral behavior of higher order elliptic operators upon domain perturbation we prove general spectral stability results for dirichlet neumann and intermediate boundary conditions moreover we consider the case of the bi harmonic operator with those intermediate boundary conditions which appears in the study of hinged plates in this case we analyze the spectral behavior when the boundary of the domain is subject to a periodic oscillatory perturbation we will show that there is a critical oscillatory behavior and the limit problem depends on whether we are above below or just sitting on this critical value in particular in the critical case we identify the strange term appearing in the limiting boundary conditions by using the unfolding method from homogenization theory

perturbation of semi linear evolution equations under weak assumptions at initial time abstract we prove perturbation results for abstract semi linear evolution equations in a banach space the main feature is that only very weak assumptions are needed at initial time this allows to prove weak continuity properties and to deal with rather general domain perturbation problems for semi linear parabolic and hyperbolic boundary value problems with various boundary conditions the theory also implies the well known theory on parameter dependent equations

a polynomial dimensional decomposition for stochastic computing this article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences the method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions a broad range of orthonormal polynomial bases consistent with the probability measure for fourier polynomial expansion of component functions and an innovative dimension reduction integration for calculating the coefficients of the expansion the new decomposition method does not require sample points as in the previous version yet it generates a convergent sequence of lower variate estimates of the probabilistic characteristics of a generic stochastic response the results of five numerical examples indicate that the proposed decomposition method provides accurate convergent and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

local and global uniform convergence for elliptic problems on varying domains abstract the aim of the paper is to prove optimal results on local and global uniform convergence of solutions to elliptic equations with dirichlet boundary conditions on varying domains we assume that the limit domain be stable in the sense of keldys amer math soc transl 51 1966 1 73 we further assume that the approaching domains satisfy a necessary condition in the inside of the limit domain and only require l 2 convergence outside as a consequence uniform and l 2 convergence are the same in the trivial case of homogenisation of a perforated domain we are also able to deal with certain cracking domains

monte carlo variance of scrambled net quadrature hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter this paper studies the variance of one such hybrid scrambled nets by applying a multidimensional multiresolution wavelet analysis to the integrand the integrand is assumed to be measurable and square integrable but not necessarily of bounded variation in simple monte carlo every nonconstant term of the multiresolution contributes to the variance of the estimated integral for scrambled nets certain low dimensional and coarse terms do not contribute to the variance for any integrand in l2 the sampling variance tends to zero faster under scrambled net quadrature than under monte carlo sampling as the number of function evaluations n tends to infinity some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the monte carlo variance uniformly over all integrands f latin hypercube sampling is a special case of scrambled net quadrature

interval estimation for treatment effects using propensity score matching in causal studies without random assignment of treatment causal effects can be estimated using matched treated and control samples where matches are obtained using estimated propensity scores propensity score matching can reduce bias in treatment effect estimators in cases where the matched samples have overlapping covariate distributions despite its application in many applied problems there is no universally employed approach to interval estimation when using propensity score matching in this article we present and evaluate approaches to interval estimation when using propensity score matching

sharp convergence rate of eigenvalues in a domain with a shrinking tube abstract in this paper we consider a class of singularly perturbed domains obtained by attaching a cylindrical tube to a fixed bounded region and letting its section shrink to zero we use an almgren type monotonicity formula to evaluate the sharp convergence rate of perturbed simple eigenvalues via courant fischer min max characterization and blow up analysis for scaled eigenfunctions

about adaptive coding on countable alphabets this paper sheds light on adaptive coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non decreasing hazard rate log concave envelope distributions we prove that the auto censuring ac code is adaptive with respect to the collection of such classes the analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy and on a careful analysis of the performance of the ac coding algorithm the latter relies on nonasymptotic bounds for maxima of samples from discrete distributions with finite and nondecreasing hazard rate

euclidean semi matchings of random samples a linear programming relaxation of the minimal matching problem is studied for graphs with edge weights determined by the distances between points in a euclidean space the relaxed problem has a simple geometric interpretation that suggests the name minimal semi matching the main result is the determination of the asymptotic behavior of the length of the minimal semi matching it is analogous to the theorem of beardwood halton and hammersley 1959 on the asymptotic behavior of the traveling salesman problem associated results on the length of non random euclidean semi matchings and large deviation inequalities for random semi matchings are also given

elicitation of multiattribute value functions through high dimensional model representations monotonicity and interactions abstract this work addresses the early phases of the elicitation of multiattribute value functions proposing a practical method for assessing interactions and monotonicity we exploit the link between multiattribute value functions and the theory of high dimensional model representations the resulting elicitation method does not state any a priori assumption on an individual s preference structure we test the approach via an experiment in a riskless context in which subjects are asked to evaluate mobile phone packages that differ on three attributes

uncertainty quantification by alternative decompositions of multivariate functions this article advocates factorized and hybrid dimensional decompositions fdd hdd as alternatives to analysis of variance dimensional decomposition add for second moment statistical analysis of

dynamics in dumbbell domains ii the limiting problem in this work we continue the analysis of the asymptotic dynamics of reaction diffusion problems in a dumbbell domain started in j m arrieta a n carvalho g lozada cruz dynamics in dumbbell domains i continuity of the set of equilibria j differential equations 231 2 2006 551 597 here we study the limiting problem that is an evolution problem in a domain which consists of an open bounded and smooth set rn with a curve r0 attached to it the evolution in both parts of the domain is governed by a parabolic equation in the evolution is independent of the evolution in r0 whereas in r0 the evolution depends on the evolution in through the continuity condition of the solution at the junction points we analyze in detail the linear elliptic and parabolic problem the generation of linear and nonlinear semigroups the existence and structure of attractors

control variate remedies other than common random numbers control variates is the most promising variance reduction technique in terms of its potential for widespread use control variates is applicable in single or multiple response simulation it does not require altering the simulation run in any way and any stochastic simulation contains potential control variates a rich theory of control variates has been developed in recent years most of this theory assumes a specific probabilistic structure for the simulation output process usually joint normality of the response and the control variates when these assumptions are not satisfied desirable properties of the estimator such as unbiasedness may be lost a number of remedies for violations of the assumptions have been proposed including jackknifing and splitting however there has been no systematic analytical and empirical evaluation of these remedies this paper presents such an evaluation including evaluation of the small sample statistical properties of the proposed remedies

complete convergence of short paths and karp s algorithm for the tsp let xi 1 i 0 sum infty n 1 p left t n sqrt n beta gt epsilon right lt infty this result is essential in justifying karp s algorithm for the traveling salesman problem under the independent model and it settles a question posed by b w weide

estimating the unseen improved estimators for entropy and other properties we show that a class of statistical properties of distributions which includes such practically relevant properties as entropy the number of distinct elements and distance metrics between pairs of distributions can be estimated given a sublinear sized sample specifically given a sample consisting of independent draws from any distribution over at most k distinct elements these properties can be estimated accurately using a sample of size o k log k for these estimation tasks this performance is optimal to constant factors complementing these theoretical results we also demonstrate that our estimators perform exceptionally well in practice for a variety of estimation tasks on a variety of natural distributions for a wide range of parameters the key step in our approach is to first use the sample to characterize the unseen portion of the distribution effectively reconstructing this portion of the distribution as accurately as if one had a logarithmic factor larger sample this goes beyond such tools as the good turing frequency estimation scheme which estimates the total probability mass of the unobserved portion of the distribution we seek to estimate the shape of the unobserved portion of the distribution this work can be seen as introducing a robust general and theoretically principled framework that for many practical applications essentially amplifies the sample size by a logarithmic factor we expect that it may be fruitfully used as a component within larger machine learning and statistical analysis systems

poisson process fock space representation chaos expansion and covariance inequalities we consider a poisson process on an arbitrary measurable space with an arbitrary sigma finite intensity measure we establish an explicit fock space representation of square integrable functions of as a consequence we identify explicitly in terms of iterated difference operators the integrands in the wiener ito chaos expansion we apply these results to extend well known variance inequalities for homogeneous poisson processes on the line to the general poisson case the poincare inequality is a special case further applications are covariance identities for poisson processes on strictly ordered spaces and harris fkg inequalities for monotone functions of

quenched voronoi percolation we prove that the probability of crossing a large square in quenched voronoi percolation converges to 1 2 at criticality confirming a conjecture of benjamini kalai and schramm from 1999 the main new tools are a quenched version of the box crossing property for voronoi percolation at criticality and an efron stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences as a corollary of the proof we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive

spectral stability of general non negative self adjoint operators with applications to neumann type operators we prove a stability theorem for the eigenvalues of general non negative self adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous neumann boundary conditions upon domain perturbation

walsh spaces containing smooth functions and quasi monte carlo rules of arbitrary high order we define a walsh space which contains all functions whose partial mixed derivatives up to order delta ge 1 exist and have finite variation in particular for a suitable choice of parameters this implies that certain sobolev spaces are contained in these walsh spaces for this walsh space we then show that quasi monte carlo rules based on digital t alpha s sequences achieve the optimal rate of convergence of the worst case error for numerical integration this rate of convergence is also optimal for the subspace of smooth functions explicit constructions of digital t alpha s sequences are given hence providing explicit quasi monte carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions

long common subsequences and the proximity of two random strings let x 1 x 2 cdots x n and x 1 x 2 cdots x n be two strings from an alphabet mathcal a and let l n denote their longest common subsequence the probabilistic behavior of l n is studied under various probability models for the x s and x s

a new method of normal approximation we introduce a new version of stein s method that reduces a large class of normal approximation problems to variance bounding exercises thus making a connection between central limit theorems and concentration of measure unlike skorokhod embeddings the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily as an application we derive a general clt for functions that are obtained as combinations of many local contributions where the definition of local itself depends on the data several examples are given including the solution to a nearest neighbor clt problem posed by p bickel

extended stochastic finite element method for the numerical simulation of heterogeneous materials with random material interfaces an extended stochastic finite element method has been recently proposed for the numerical solution of partial differential equations defined on random domains this method is based on a marriage between the extended finite element method and spectral stochastic methods in this article we propose an extension of this method for the numerical simulation of random multi phased materials the random geometry of material interfaces is described implicitly by using random level set functions a fixed deterministic finite element mesh which is not conforming to the random interfaces is then introduced in order to approximate the geometry and the solution classical spectral stochastic finite element approximation spaces are not able to capture the irregularities of the solution field with respect to spatial and stochastic variables which leads to a deterioration of the accuracy and convergence properties of the approximate solution in order to recover optimal convergence properties of the approximation we propose an extension of the partition of unity method to the spectral stochastic framework this technique allows the enrichment of approximation spaces with suitable functions based on an a priori knowledge of the irregularities in the solution numerical examples illustrate the efficiency of the proposed method and demonstrate the relevance of the enrichment procedure copyright 2010 john wiley sons ltd

orthogonal decomposition of finite population statistics and its applications to distributional asymptotics we study orthogonal decomposition of symmetric statistics based on samples drawn without replacement from finite populations several applications to finite population statistics are given we establish one term edge worth expansions for general asymptotically normal symmetric statistics prove an efron stein inequality and the consistency of the jackknife estimator of variance our expansions provide second order a s approximations to wu s jackknife histogram

a high dimensional wilks phenomenon a theorem by wilks asserts that in smooth parametric density estimation the difference between the maximum likelihood and the likelihood of the sampling distribution converges toward a chi square distribution where the number of degrees of freedom coincides with the model dimension this observation is at the core of some goodness of fit testing procedures and of some classical model selection methods this paper describes a non asymptotic version of the wilks phenomenon in bounded contrast optimization procedures using concentration inequalities for general functions of independent random variables it proves that in bounded contrast minimization as for example in statistical learning theory the difference between the empirical risk of the minimizer of the true risk in the model and the minimum of the empirical risk the excess empirical risk satisfies a bernstein like inequality where the variance term reflects the dimension of the model and the scale term reflects the noise conditions from a mathematical statistics viewpoint the significance of this result comes from the recent observation that when using model selection via penalization the excess empirical risk represents a minimum penalty if non asymptotic guarantees concerning prediction error are to be provided from the perspective of empirical process theory this paper describes a concentration inequality for the supremum of a bounded non centered actually non positive empirical process combining the now classical analysis of m estimation building on talagrand s inequality for suprema of empirical processes and versatile moment inequalities for functions of independent random variables this paper develops a genuine bernstein like inequality that seems beyond the reach of traditional tools

on the effects of dimension reduction techniques on some high dimensional problems in finance many problems in finance can be formulated as high dimensional integrals which are often attacked by quasi monte carlo qmc algorithms to enhance qmc algorithms dimension reduction techniques such as the brownian bridge bb and principal component analysis pca are used to reduce the effective dimension this paper explores in depth the effects of these techniques on the dimension structure of some typical high dimensional problems from finance the pricing of path dependent options and bond valuation according to term structure models by deriving explicit expressions for the underlying integrands and the associated weights that control the relative importance of different variables and by investigating the variance ratios the effective dimensions the mean dimension and their limiting behavior as the nominal dimension tends to infinity we show theoretically and empirically how and to what extent the bb and pca algorithms change the dimension structure including the degree of additivity of the underlying functions they change the functions to be strongly weighted and substantially reduce the effective dimensions and the mean dimension and they enhance the degree of additivity which is particularly important for qmc moreover the resulting functions are of low effective dimension not only in the superposition sense but also in the truncation sense the variance ratios the effective dimensions and the mean dimension associated with these techniques are very insensitive to the nominal dimension they are essentially constant which highlights the possibility of removing the curse of dimensionality when dimension reduction techniques are used in combination with qmc a counterexample is also shown for which the bb and pca may increase the effective dimension the investigation provides further insight into the effects of dimension reduction techniques

unimodal regression using bernstein schoenberg splines and penalties summary research in the field of nonparametric shape constrained regression has been intensive however only few publications explicitly deal with unimodality although there is need for such methods in applications for example in dose response analysis in this article we propose unimodal spline regression methods that make use of bernstein schoenberg splines and their shape preservation property to achieve unimodal and smooth solutions we use penalized splines and extend the penalized spline approach toward penalizing against general parametric functions instead of using just difference penalties for tuning parameter selection under a unimodality constraint a restricted maximum likelihood and an alternative bayesian approach for unimodal regression are developed we compare the proposed methodologies to other common approaches in a simulation study and apply it to a dose response data set all results suggest that the unimodality constraint or the combination of unimodality and a penalty can substantially improve estimation of the functional relationship