local tomography with nonsmooth attenuation ii we continue investigation of local tomography with nonsmooth attenuation which was proposed earlier by the author our main result is a new and simple characterization of the wave front of the local tomography function tilde f lambda left phi right which holds under fairly general conditions on the density function f and the attenuation coefficient

quantile estimation with latin hypercube sampling quantiles are often used to measure risk of stochastic systems we examine quantile estimators obtained using simulation with latin hypercube sampling lhs a variance reduction technique that efficiently extends stratified sampling to higher dimensions and produces negatively correlated outputs we consider single sample lhs sslhs which minimizes the variance that can be obtained from lhs and also replicated lhs rlhs we develop a consistent estimator of the asymptotic variance of the sslhs quantile estimator s central limit theorem enabling us to provide the first confidence interval ci for a quantile when applying sslhs for rlhs we construct cis using batching and sectioning on average our rlhs cis are shorter than previous rlhs cis and only slightly wider than the sslhs ci we establish the asymptotic validity of the cis by first proving that the quantile estimators satisfy bahadur representations which show that the quantile estimators can be approximated by linear transformations of e

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

on the asymptotic distribution of scrambled net quadrature recently in a series of articles owen proposed the use of scrambled t m s nets and t s sequences in high dimensional numerical integration these scrambled nets and sequences achieve the superior accuracy of equidistribution methods while allowing for the simpler error estimation techniques of monte carlo methods the main aim of this article is to use stein s method to study the asymptotic distribution of the scrambled 0 m s net integral estimate in particular it is shown that for suitably smooth integrands on the s dimensional unit hypercube the estimate has an asymptotic normal distribution

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

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

a robust non linear solid shell element based on a mixed variational formulation the paper is concerned with a geometrically non linear solid shell finite element formulation which is based on the hu washizu variational principle for the approximation of the independent displacement stress and strain fields the strain field is additively decomposed into two parts due to the fact that one part of the strain field is interpolated in the same manner as proposed by the enhanced assumed strain eas method it is denoted as eas field the other strain field is approximated with the same interpolation functions as the stress field in contrast to the eas concept the approximation spaces of the stresses and the enhanced assumed strains are not orthogonal consequently the stress field is not eliminated from the finite element equations for the displacements tri linear shape functions are considered shear locking and curvature thickness locking are treated using assumed natural strain interpolations a static condensation leads to a simple low order hexahedral solid shell element numerical tests show that the present model is very robust and allows larger load steps than an eas solid shell element

on latin hypercube sampling this paper contains a collection of results on latin hypercube sampling the first result is a berry esseen type bound for the multivariate central limit theorem of the sample mean hat mu n based on a latin hypercube sample the second establishes sufficient conditions on the convergence rate in the strong law for hat mu n finally motivated by the concept of empirical likelihood a way of constructing nonparametric confidence regions based on latin hypercube samples is proposed for vector means

local algorithms in exterior tomography exterior tomographic data are taken over lines outside a central region and such data occur in the industrial nondestructive evaluation of large objects such as rockets we explain using microlocal analysis which singularities are well reconstructed from exterior data and we explain how this phenomenon is reflected in the singular value decomposition for the exterior transform e t quinto singular value decompositions and inversion methods for the exterior radon transform and a spherical transform j math anal appl 95 1983 437 448 we extend lambda tomography to exterior data and to limited angle exterior data the algorithm is tested on industrial data from perceptics inc

coulomb interactions on planar structures inverting the square root of the laplacian we present an adaptive fast multipole method for inverting the square root of the laplacian in two dimensions solving this problem is the dominant computational cost in many applications arising in electrical engineering geophysical fluid dynamics and the study of thin films it corresponds to the evaluation of the field induced by a planar distribution of charge or vorticity our algorithm is direct and assumes only that the source distribution is discretized using an adaptive quad tree the amount of work grows linearly with the number of mesh points

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

sequential quasi monte carlo this is a written discussion of the paper sequential quasi monte carlo sampling by m gerber and n chopin following the presentation given before the royal statistical society in london on december 10th 2014

asymptotic normality of scrambled geometric net quadrature in a very recent work basu and owen found comput math 17 2017 467 496 propose the use of scrambled geometric nets in numerical integration when the domain is a product of s arbitrary spaces of dimension d having a certain partitioning constraint it was shown that for a class of smooth functions the integral estimate has variance o n 1 2 d log n s 1 for scrambled geometric nets compared to o n 1 for ordinary monte carlo the main idea of this paper is to expand on the work by loh ann statist 31 2003 1282 1324 to show that the scrambled geometric net estimate has an asymptotic normal distribution for certain smooth functions defined on products of suitable subsets of mathbb r d

transforming low discrepancy sequences from a cube to a simplex sequences of points with a low discrepancy are the basic building blocks for quasi monte carlo methods traditionally these points are generated in a unit cube to develop point sets on a simplex we will transform the low discrepancy points from the unit cube to a simplex an advantage of this approach is that most of the known results on low discrepancy sequences can be re used after introducing several transformations their efficiency as well as their quality will be evaluated we present a koksma hlawka inequality which says that under certain conditions the order of convergence using the new point set is the same as that of the original set

wavelet based multiresolution local tomography we develop an algorithm to reconstruct the wavelet coefficients of an image from the radon transform data the proposed method uses the properties of wavelets to localize the radon transform and can be used to reconstruct a local region of the cross section of a body using almost completely local data that significantly reduces the amount of exposure and computations in x ray tomography the property that distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet bases with sufficiently many vanishing moments the ramp filtered version of the scaling function as well as the wavelet function has extremely rapid decay we show that the variance of the elements of the null space is negligible in the locally reconstructed image also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm to reconstruct a local region 16 pixels in radius in a 256 spl times 256 image we require 22 of full exposure data

extrapolated fields in the formulation of the assumed strain elements part ii three dimensional problems abstract a new assumed strain quadrilateral element with accurate in plane bending behavior is presented the basic idea consists in introduction of a parallelogram domain associated with the given quadrilateral element development of the related assumed strain field for the parallelogram and subsequent extrapolation of this field to the domain of the original quadrilateral the assumed strain field for the parallelogram is constructed by identification of various modes of its deformation and then by proper modification of the strain field in some of these modes in particular the strain operator with respect to the so called in plane bending modes is modified so as to simulate the strain field resulting from the assumptions usually made in structural mechanics the modification of the strain fields leads to the assumed strain operator the proposed new assumed strain quadrilateral element exhibits remarkably accurate performance for both membrane and bending dominated problems even when severely distorted and high aspect ratio meshes are considered and when materials are nearly incompressible

transformations and hardy krause variation using a multivariable faa di bruno formula we give conditions on transformations tau 0 1 m to cal x where cal x is a closed and bounded subset of mathbb r d such that f circ tau is of bounded variation in the sense of hardy and krause for all f in c m cal x we give similar conditions for f circ tau to be smooth enough for scrambled net sampling to attain o n 3 2 epsilon accuracy some popular symmetric transformations to the simplex and sphere are shown to satisfy neither condition some other transformations due to fang and wang monogr statist appl probab 51 crc press boca raton fl 1993 satisfy the first but not the second condition we provide transformations for the simplex that makes f circ tau smooth enough to fully benefit from scrambled net sampling for all f in a class of generalized polynomials we also find sufficient conditions for conditional inversion in mathbb r 2 and for importance sampling to be of bounded variation in the sense of ha

a central limit theorem for latin hypercube sampling summary latin hypercube sampling lhs is a technique for monte carlo integration due to mckay conover and beckman m stein proved that lhs integrals have smaller variance than independent and identically distributed monte carlo integration the extent of the variance reduction depending on the extent to which the integrand is additive we extend stein s work to prove a central limit theorem variance estimation methods for nonparametric regression can be adapted to provide n 12 consistent estimates of the asymptotic variance in lhs moreover the skewness can be estimated at this rate the variance reduction may be explained in terms of certain control variates that cannot be directly measured we also show how to combine control variates with lhs finally we show how these results lead to a frequentist approach to computer experimentation

optimal quadrature for haar wavelet spaces this article considers the error of the scrambled equidistribution quadrature rules in the worst case random case and average case settings the underlying space of integrands is a hilbert space of multidimensional haar wavelet series hwav the asymptotic orders of the errors are derived for the case of the scrambled t m s nets and t s sequences these rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands hwav

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

an 8 node brick finite element an 8 node brick element based on the assumed stress hybrid formulation is described with three additional stress fields the element stiffness matrix now has the required rank of 18 and the bending response is exact with rectangular elements surprisingly the 2 2 2 gauss rule suffices for all numerical integrations to satisfy the constant stress patch test

local tomography ii standard or global tomography involves the reconstruction of a function f from line integrals local tomography in this paper involves the reconstruction of a related function lf alpha lambda f mu lambda 1 f where lambda is the square root of the positive laplacian delta this article is a sequel to the article local tomography siam j appl math 52 1992 pp 459 484 1193 1198 by faridani ritman and smith the principal new results are 1 good bounds for lambda f and lambda 1 f outside the support of f particularly when f has 0 moments up to some order 2 identification and reduction of global effects in local tomography i e identification and reduction of the dependence of lf x on the values of f at points at an intermediate distance from x 3 an algorithm for computing approximate density jumps from lambda f when f is a linear combination of characteristic functions and a smooth background several examples are given some from real x ray da

estimating functions for inhomogeneous spatial point processes with incomplete covariate data the r package spatstat provides a very flexible and useful framework for analysing spatial point patterns a fundamental feature is a procedure for fitting spatial point process models depending on covariates however in practice one often faces incomplete observation of the covariates and this leads to parameter estimation error which is difficult to quantify in this paper we introduce a monte carlo version of the estimating function used in spatstat for fitting inhomogeneous poisson processes and certain inhomogeneous cluster processes for this modified estimating function it is feasible to obtain the asymptotic distribution of the parameter estimators in the case of incomplete covariate information this allows a study of the loss of efficiency due to the missing covariate data

a geometrically non linear piezoelectric solid shell element based on a mixed multi field variational formulation this paper is concerned with a geometrically non linear solid shell element to analyse piezoelectric structures the finite element formulation is based on a variational principle of the hu washizu type and includes six independent fields displacements electric potential strains electric field mechanical stresses and dielectric displacements the element has eight nodes with four nodal degrees of freedoms three displacements and the electric potential a bilinear distribution through the thickness of the independent electric field is assumed to fulfill the electric charge conservation law in bending dominated situations exactly the presented finite shell element is able to model arbitrary curved shell structures and incorporates a 3d material law a geometrically non linear theory allows large deformations and includes stability problems linear and non linear numerical examples demonstrate the ability of the proposed model to analyse piezoelectric devices

latin hyperrectangle sampling for computer experiments latin hypercube sampling lhs is a popular method for evaluating the expectation of functions in computer experiments however when the expectation of interest is taken with respect to a nonuniform distribution the usual transformation to the probability space can cause relatively smooth functions to become extremely variable in areas of low probability consequently the equal probability cells inherent in hypercube methods often tend to sample an insufficient proportion of the total points in these areas this article introduces latin hyperrectangle sampling lhrs a generalization of lhs that allows for nonequal cell probabilities to address this problem a number of examples are given illustrating the improvement of the proposed methodology over lhs with respect to the variance of the resulting estimators extensions to orthogonal array based lhs stratified lhs and scrambled nets are also described

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

on rates of convergence for stochastic optimization problems under non independent and identically distributed sampling in this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size this is a well studied problem in case the samples are independent and identically distributed i e when standard monte carlo simulation is used here we study the case where that assumption is dropped broadly speaking our results show that under appropriate assumptions the rates of convergence for pointwise estimators under a sampling scheme carry over to the optimization case in the sense that convergence of approximating optimal solutions and optimal values to their true counterparts has the same rates as in pointwise estimation we apply our results to two well established sampling schemes namely latin hypercube sampling and randomized quasi monte carlo qmc the novelty of our work arises from the fact that while there has been some work on the use of variance reduction techniques and qmc methods in stochastic optimization none of the existing work to the best of our knowledge has provided a theoretical study on the effect of these techniques on rates of convergence for the optimization problem we present numerical results for some two stage stochastic programs from the literature to illustrate the discussed ideas

a nonsmooth regularization approach based on shearlets for poisson noise removal in roi tomography abstract due to its potential to lower exposure to x ray radiation and reduce the scanning time region of interest roi computed tomography ct is particularly appealing for a wide range of biomedical applications to overcome the severe ill posedness caused by the truncation of projection measurements ad hoc strategies are required since traditional ct reconstruction algorithms result in instability to noise and may give inaccurate results for small roi to handle this difficulty we propose a nonsmooth convex optimization model based on l1 shearlet regularization whose solution is addressed by means of the variable metric inexact line search algorithm vmila a proximal gradient method that enables the inexact computation of the proximal point defining the descent direction we compare the reconstruction performance of our strategy against a smooth total variation stv approach by using both poisson noisy simulated data and real data from fan beam ct geometry the results show that while for synthetic data both shearets and stv perform well for real data the proposed nonsmooth shearlet based approach outperforms stv since the localization and directional properties of shearlets allow to detect finer structures of a textured image finally our approach appears to be insensitive to the roi size and location

a combinatorial central limit theorem for randomized orthogonal array sampling designs let x be a random vector uniformly distributed on the unit cube and f 0 1 3 r be a measurable function an objective of many computer experiments is to estimate e f x by computing f at a set of points in 0 1 3 there is a design issue in choosing these points recently owen and tang independently suggested using randomized orthogonal arrays in the choice of such a set this paper investigates the convergence rate to normality of the distribution of the average of a set of f values taken from one of these designs

the semidiscrete filtered backprojection algorithm is optimal for tomographic inversion the filtered backprojection algorithm is probably the most often used reconstruction algorithm in two dimensional computerized tomography for a semidiscrete version in the parallel scanning geometry we prove optimal l2 convergence rates for density distributions in sobolev spaces additionally we show l2 convergence without rates when the density distribution is only in l2 the key to success is a new representation of the filtered backprojection which enables us to apply techniques from approximation theory our analysis provides further a modification of the shepp logan reconstruction filter with an improved convergence behavior numerical experiments in the fully discrete setting reproduce the theoretical predictions

low discrepancy constructions in the triangle most quasi monte carlo research focuses on sampling from the unit cube many problems especially in computer graphics are defined via quadrature over the unit triangle quasi monte carlo methods for the triangle have been developed by pillards and cools j comput appl math 174 2005 pp 29 42 and by brandolini et al a koksma hlawka inequality for simplices in trends in harmonic analysis springer 2013 pp 33 46 this paper presents two quasi monte carlo constructions in the triangle with a vanishing discrepancy the first is a version of the van der corput sequence customized to the unit triangle it is an extensible digital construction that attains a discrepancy below 12 sqrt n the second construction rotates an integer lattice through an angle whose tangent is a quadratic irrational number it attains a discrepancy of o log n n which is the best possible rate previous work strongly indicated that such a discrepancy was possible but no constructions were available s

the mean square discrepancy of scrambled t s sequences the discrepancy arises in the worst case error analysis for quasi monte carlo quadrature rules low discrepancy sets yield good quadrature rules this article considers the mean square discrepancies for scrambled lam t m s nets and t s sequences in base b it is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple monte carlo sampling if the reproducing kernel defining the discrepancy satisfies a lipschitz condition with respect to one of its variables separately then the asymptotic order of the root mean square discrepancy is o n 1 log n s 1 2 for scrambled nets if the reproducing kernel satisfies a lipschitz condition with respect to both of its variables then the asymptotic order of the root mean square discrepancy is o n 3 2 log n s 1 2 for scrambled nets for an arbitrary number of points taken from a t s sequence the root mean square discrepancy appears to be no better than o n 1 log n s 1 2 regardless of the smoothness of the reproducing kernel

on the numerical integration of walsh series by number theoretic methods in analogy to the theory of good lattice points for the numerical integration of rapidly converging fourier series a theory for the fast numerical integration of walsh series is developed the basis for this theory is provided by a class of very well distributed point sets in the s dimensional unit cube the so called t m s nets

three dimensional 8 node and 20 node refined hybrid isoparametric elements in this paper a new hybrid variational principle with independent variables of strain stress and displacement and with a weaker constraint condition of interelement continuity is proposed based on this functional a general formulation of a refined hybrid isoparametric element method has been established by the orthogonal approach the present formulation is a rational approach to be adopted for deriving high performance three dimensional hybrid isoparametric elements even up to the higher order 20 node element several numerical examples are presented to show that the present elements rgh8 8 node and rgh20 20 node have high accuracy excellent computational efficiency and less sensitivity to mesh distortion

orthogonal array based latin hypercubes abstract in this article we use orthogonal arrays oa s to construct latin hypercubes besides preserving the univariate stratification properties of latin hypercubes these strength r oa based latin hypercubes also stratify each r dimensional margin therefore such oa based latin hypercubes provide more suitable designs for computer experiments and numerical integration than do general latin hypercubes we prove that when used for integration the sampling scheme with oa based latin hypercubes offers a substantial improvement over latin hypercube sampling

the patch test for elements the main purpose of this paper is to emphasize the necessity of convergence tests in general and the patch test in particular following a brief introduction the patch test is described with examples this it is hoped will help to remove any lingering misunderstanding of this very useful convergence test some positive contributions of the test in the formulation and improvement of elements are also discussed

an engineers defence of the patch test this convergence criterion is proved for any legalizable finite element formulation i e any model for which a conforming displacement version is possible using the same nodal variables engineers may therefore take control of and responsibility for convergence for most of the elements in service today stummel s counter example is explained as a technical misunderstanding due to inadequate documentation of the patch test it is suggested without proof that the test properly applied is universal provided that it is always combined with an adequate test of stability

local tomography in electron microscopy we present a new local tomographic algorithm applicable to electron microscope tomography our algorithm applies to the standard data acquisition method single axis tilting as well as to more arbitrary acquisition methods including double axis and conical tilt using microlocal analysis we put the reconstructions in a mathematical context explaining which singularities are stably visible from the limited data given by the data collection protocol in the electron microscope finally we provide reconstructions of real specimens from electron tomography data

interior tomography with continuous singular value decomposition the long standing interior problem has important mathematical and practical implications the recently developed interior tomography methods have produced encouraging results a particular scenario for theoretically exact interior reconstruction from truncated projections is that there is a known subregion in the region of interest roi in this paper we improve a novel continuous singular value decomposition svd method for interior reconstruction assuming a known subregion first two sets of orthogonal eigen functions are calculated for the hilbert and image spaces respectively then after the interior hilbert data are calculated from projection data through the roi they are projected onto the eigen functions in the hilbert space and an interior image is recovered by a linear combination of the eigen functions with the resulting coefficients finally the interior image is compensated for the ambiguity due to the null space utilizing the prior subregion knowledge experiments with simulated and real data demonstrate the advantages of our approach relative to the projection onto convex set type interior reconstructions