information theoretic determination of minimax rates of convergence we present some general results determining minimax bounds on statistical risk for density estimation based on certain information theoretic considerations these bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence

imposing dirichlet boundary conditions in the extended finite element method this paper is devoted to the imposition of dirichlet type conditions within the extended finite element method x fem this method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces imposing neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh observed numerically in earlier papers on the contrary much less work has been devoted to dirichlet boundary conditions for the x fem or the limiting case of stiff boundary conditions in this paper we introduce a strategy to impose dirichlet boundary conditions while preserving the optimal rate of convergence the key aspect is the construction of the correct lagrange multiplier space on the boundary as an application we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding rtm process while avoiding remeshing the case of inner conditions is also discussed as well as two important practical cases material interfaces and phase transformation front capturing copyright 2006 john wiley sons ltd

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

adaptive remeshing and h p domain decomposition abstract the h p version of the finite element method combines local mesh refinement with an increase of the shape functions polynomial degree this paper presents a method which uses an h p approximation for a domain decomposition with fully overlapping domains allowing c 0 continuous approximation for geometrically incompatible mesh refinement to be obtained the method is applied to a reaction diffusion problem resulting from semiconductor process simulation in numerical examples it is demonstrated that this approach is well suited for problems with sharp interior layers or shock like behaviour

on the minimization of convex functionals of probability distributions under band constraints the problem of minimizing convex functionals of probability distributions is solved under the assumption that the density of every distribution is bounded from above and below a system of sufficient and necessary first order optimality conditions as well as a bound on the optimality gap of feasible candidate solutions are derived based on these results two numerical algorithms are proposed that iteratively solve the system of optimality conditions on a grid of discrete points both algorithms use a block coordinate descent strategy and terminate once the optimality gap falls below the desired tolerance while the first algorithm is conceptually simpler and more efficient it is not guaranteed to converge for objective functions that are not strictly convex this shortcoming is overcome in the second algorithm which uses an additional outer proximal iteration and which is proven to converge under mild assumptions two examples are given to demonstrate the theoretical usefulness of the optimality conditions as well as the high efficiency and accuracy of the proposed numerical algorithms

stress constrained shape and topology optimization with fixed mesh a b spline finite cell method combined with level set function abstract in this paper we develop an efficient and flexible design method that integrates the b spline finite cell method b spline fcm and the level set function lsf for stress constrained shape and topology optimization any structure of complex geometry is embedded within an extended regular and fixed eulerian mesh no matter how the structure is optimized high order b spline shape functions are further implemented to ensure precisions of stress analysis and sensitivity analysis meanwhile level set functions i e implicit functions are used to enable topological changes of the considered structure through smooth boundary variations involved parameters rather than the conventional discrete form of lsf are directly taken as design variables to facilitate the numerical computing process to be specific the lsf is constructed by means of r functions that incorporate cubic splines as implicit functions to offer flexibilities for shape optimization within the framework of fixed mesh while the compactly supported radial basis functions cs rbfs are employed as implicit functions for stress constrained topology optimization it is shown the proposed fcm lsf method is a convenient approach that makes it possible to calculate stress and stress sensitivities with high precision representative examples of shape and topology optimization with and without stress constraints are solved with success demonstrating the advantages of the fcm lsf method

applying the hp d version of the fem to locally enhance dimensionally reduced models abstract a modification of the hp d method is presented which allows to locally enhance a dimensionally reduced finite element approximation with an ansatz based on a more accurate i e dimensionally higher model the procedure is based on a hierarchical domain decomposition which is applied to couple discretizations of different types of mechanical models like beams plates and solids thereby the error of the underlying mathematical model can be reduced in critical regions by introducing locally a physically more sophisticated model two examples are given to demonstrate the basic properties of the proposed method in the first example the solution of a one dimensional beam model is improved locally by a two dimensional finite element computation in the second example the local enhancement of a reissner mindlin plate by a fully three dimensional continuum approach is considered yielding a significant reduction of the model error

on estimating a density using hellinger distance and some other strange facts on s interesse ici aux possibles vitesses d estimation d une densite a support compact dans m sous des hypotheses de regularite lorsque la perte est mesuree par le carre de la distance de hellinger on regardera aussi le cas connu des normes mathbb l q pour 1 q 2 et le risque est le risque minimax sur la famille on donne une methode generale permettant de traiter les problemes dans le cadre de la theorie de l approximation sous des conditions concernant l entropie metrique et l e capacite des familles a estimer les rapports entre regularite et entropie metrique etant bien connus nous pourrons aussi traiter les cas classiques et d autres qui le sont moins sous des conditions de bornes inferieures les vitesses sont celles observees pour la norme mathbb l q mais elles different dans le cas general on montre aussi que les restrictions sur la compacite du support ou la regularite sont indispensables et que leur absence mene a l impossibilite d obtenir une estimation raisonnable en ce sens que n importe quelle suite d estimateurs sera arbitrairement mauvaise en un point au moins un resultat analogue est vrai sous des conditions de regularite

tweedie s formula and selection bias we suppose that the statistician observes some large number of estimates zi each with its own unobserved expectation parameter i the largest few of the zi s are likely to substantially overestimate their corresponding i s this being an example of selection bias or regression to the mean tweedie s formula first reported by robbins in 1956 offers a simple empirical bayes approach for correcting selection bias this article investigates its merits and limitations in addition to the methodology tweedie s formula raises more general questions concerning empirical bayes theory discussed here as relevance and empirical bayes information there is a close connection between applications of the formula and james stein estimation

a hierarchical nxfem for fictitious domain simulations we suggest a fictitious domain method based on the nitsche xfem method of comput meth appl mech engrg 2002 191 5537 5552 that employs a band of elements adjacent to the boundary in contrast the classical fictitious domain method uses lagrange multipliers on a line surface where the boundary condition is to be enforced the idea can be seen as an extension of the chimera method of esaim math model numer anal 2003 37 495 514 but with a hierarchical representation of the discontinuous solution field the hierarchical formulation is better suited for moving fictitious boundaries since the stiffness matrix of the underlying structured mesh can be retained during the computations our technique allows for optimal convergence properties irrespective of the order of the underlying finite element method

principal inertia components and applications we explore properties and applications of the principal inertia components pics between two discrete random variables x and y the pics lie in the intersection of information and estimation theory and provide a fine grained decomposition of the dependence between x and y moreover the pics describe which functions of x can or cannot be reliably inferred in terms of mmse given an observation of y we demonstrate that the pics play an important role in information theory and they can be used to characterize information theoretic limits of certain estimation problems in privacy settings we prove that the pics are related to the fundamental limits of perfect privacy

axisymmetric pressure boundary loading for finite deformation analysis using p fem follower loads i e loads which depend on the boundary displacements by definition frequently occur in finite deformation boundary value problems restricting to axisymmetrical applications we provide analytical and numerical solutions for a set of problems in compressible neo hookean materials so to serve as benchmark problems for verifying the accuracy and efficiency of various fe methods for follower load applications thereafter the weak formulation for the follower load in 3 d domain is reduced to an axisymmetrical setting and subsequently consistently linearized in the framework of p fems exploiting the blending function mapping techniques the set of axisymmetric benchmark solutions is compared to numerical experiments in which the results obtained by a p fem code are compared to these obtained by a state of the art commercial h fem code and to the exact results these demonstrate the efficiency and accuracy of p fems when applied to problems in finite deformations with follower loads 2006 elsevier b v all rights reserved

distributed lagrange multipliers based on fictitious domain method for second order elliptic problems in this article we further investigate the solution of linear second order elliptic boundary value problems by distributed lagrange multipliers based fictitious domain methods the following issues are addressed i derivation of the fictitious domain formulations ii finite element approximation iii iterative solution of the resulting finite dimensional problems of the saddle point type by preconditioned conjugate gradient and lanczos algorithms

fictitious domain approach with hp finite element approximation for incompressible fluid flow we consider the application of fictitious domain approach combined with least squares spectral elements for the numerical solution of fluid dynamic incompressible equations fictitious domain methods allow problems formulated on a complicated shaped domain to be solved on a simpler domain containing least squares spectral element method has been used to develop the discrete model as this scheme combines the generality of finite element methods with the accuracy of spectral methods moreover the least squares methods have theoretical and computational advantages in the algorithmic design and implementation this paper presents the formulation and validation of the fictitious domain least squares spectral element approach for the steady incompressible navier stokes equations the convergence of the approximated solution is verified solving two dimensional benchmark problems demonstrating the predictive capability of the proposed formulation

an embedded dirichlet formulation for 3d continua this paper presents a new approach for imposing dirichlet conditions weakly on non fitting finite element meshes such conditions also called embedded dirichlet conditions are typically but not exclusively encountered when prescribing dirichlet conditions in the context of the extended finite element method xfem the methods key idea is the use of an additional stress field as the constraining lagrange multiplier function the resulting mixed hybrid formulation is applicable to 1d 2d and 3d problems the method does not require stabilization for the lagrange multiplier unknowns and allows the complete condensation of these unknowns on the element level furthermore only non zero diagonal terms are present in the tangent stiffness which allows the straight forward application of state of the art iterative solvers like algebraic multigrid amg techniques within this paper the method is applied to the linear momentum equation of an elastic continuum and to the transient incompressible navier stokes equations steady and unsteady benchmark computations show excellent agreement with reference values the general formulation presented in this paper can also be applied to other continuous field problems

arimoto renyi conditional entropy and bayesian m ary hypothesis testing this paper gives upper and lower bounds on the minimum error probability of bayesian m ary hypothesis testing in terms of the arimoto renyi conditional entropy of an arbitrary order alpha the improved tightness of these bounds over their specialized versions with the shannon conditional entropy alpha 1 is demonstrated in particular in the case where m is finite we show how to generalize fano s inequality under both the conventional and list decision settings as a counterpart to the generalized fano s inequality allowing m to be infinite a lower bound on the arimoto renyi conditional entropy is derived as a function of the minimum error probability explicit upper and lower bounds on the minimum error probability are obtained as a function of the arimoto renyi conditional entropy for both positive and negative alpha furthermore we give upper bounds on the minimum error probability as functions of the renyi divergence in the setup of discrete memoryless channels we analyze the exponentially vanishing decay of the arimoto renyi conditional entropy of the transmitted codeword given the channel output when averaged over a random coding ensemble

local shrinkage rules levy processes and regularized regression we use l evy processes to generate joint prior distributions and therefore penalty functions for a location parameter b b1 bp as p grows large this generalizes the class of local global shrinkage rules based on scale mixtures of normals illuminates new connections among disparate methods and leads to new results for computing posterior means and modes under a wide class of priors we extend this framework to large scale regularized regression problems where p n and provide comparisons with other methodologies

on the multi level splitting of finite element spaces in this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases we show that the condition number of such a stiffness matrix behaves like o log 2 where is the condition number of the stiffness matrix with respect to a nodal basis in the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like o left left log frac 1 h right 2 right instead of o left left frac 1 h right 2 right for a nodal basis the proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization especially we do not need the quasiuniformity of the employed triangulations as the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis our results mean that the method of conjugate gradients needs onlyo log n steps ando n log n computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures heren denotes the dimension of the finite element space and of the discrete linear problem to be solved

a new lower bound for multiple hypothesis testing the purpose of this correspondence is to give a new easily tractable and sharp lower bound for the maximal error in multiple hypothesis testing with an application to nonasymptotic lower bounds for the minimax risk of estimators

some inequalities for information divergence and related measures of discrimination inequalities which connect information divergence with other measures of discrimination or distance between probability distributions are used in information theory and its applications to mathematical statistics ergodic theory and other scientific fields we suggest new inequalities of this type often based on underlying identities as a consequence we obtain certain improvements of the well known pinsker inequality our study depends on two measures of discrimination called capacitory discrimination and triangular discrimination the discussion contains references to related research and comparison with other measures of discrimination e g ali silvey csiszar 1996 1966 divergences and in particular the hellinger distance

cbs based topology optimization including design dependent body loads abstract in this paper the inherent problem of the so called parasitic effect of low density region caused by design dependent loads is investigated for topology optimization a cbs closed b splines based method is developed to solve efficiently the problem in the way of shape optimization compared to the standard density method design variables are unattached to the finite element model and defined by control parameters dominating the parametric equation of the cbs as a result design dependent loads associated with the material layout of a structure are made change indirectly by the boundary variation of the structure to favour structural reanalyses and sensitivity analysis the implicit form of the cbs i e level set function lsf is used in conjunction with the fixed computing grid of finite cell method fcm a variety of design dependent body loads is considered for the mathematical formulation of the cbs based method typical examples are given to illustrate that the cbs based method is free of low density regions in the optimized topology owing to the proper definitions of design variables

a comparison of fictitious domain methods appropriate for spectral hp element discretisations abstract the fictitious domain finite cell and fat boundary method which are commonly adopted for fluid structure interaction fsi problems have been considered and implemented in a spectral hp element code for a 1d test problem the finite cell and fat boundary method are extended for the general case of a deformable solid in a fluid and analysed furthermore a new implicit variation of the fat boundary method is proposed the 1d framework allows for a clear and detailed analysis of the different approaches and to highlight the similarities and differences between the methods exponential p convergence which is typical for smooth solutions discretised with spectral hp elements is demonstrated for all methods except the classic fictitious domain formulation this overview can be considered as a starting point to study the capability advantages and disadvantages of these methods for fluid structure interaction problems in a more realistic setting and in higher dimensions

weighted extended b spline approximation of dirichlet problems we describe a new finite element method which uses weighted extended b splines on a regular grid as basis functions for solving dirichlet problems on bounded domains in arbitrary dimensions this web method does not require any grid generation and can be implemented very efficiently it yields smooth high order accurate approximations with relatively low dimensional subspaces

elimination of nuisance parameters with reference priors summary the problem of eliminating nuisance parameters is tackled from different points of view standard likelihood techniques such as profile likelihood and its modifications are compared with a bayesian analysis based on reference priors berger bernardo 1992a examples are considered to illustrate what happens in multiparameter problems in particular the frequentist coverage probabilities of the credible sets derived from the reference priors are shown to be better than those computed from a likelihood approach advantages and disadvantages of bayesian and likelihood analyses are discussed

weakly enforced essential boundary conditions for nurbs embedded and trimmed nurbs geometries on the basis of the finite cell method summary enforcing essential boundary conditions plays a central role in immersed boundary methods nitsche s idea has proven to be a reliable concept to satisfy weakly boundary and interface constraints we formulate an extension of nitsche s method for elasticity problems in the framework of higher order and higher continuity approximation schemes such as the b spline and non uniform rational basis spline version of the finite cell method or isogeometric analysis on trimmed geometries furthermore we illustrate a significant improvement of the flexibility and applicability of this extension in the modeling process of complex 3d geometries with several benchmark problems we demonstrate the overall good convergence behavior of the proposed method and its good accuracy we provide extensive studies on the stability of the method its influence parameters and numerical properties and a rearrangement of the numerical integration concept that in many cases reduces the numerical effort by a factor two a newly composed boundary integration concept further enhances the modeling process and allows a flexible discretization independent introduction of boundary conditions finally we present our strategy in the framework of the modeling and isogeometric analysis process of trimmed non uniform rational basis spline geometries copyright 2013 john wiley sons ltd

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

an isogeometric design through analysis methodology based on adaptive hierarchical refinement of nurbs immersed boundary methods and t spline cad surfaces we explore hierarchical refinement of nurbs as a basis for adaptive isogeometric and immersed boundary analysis we use the principle of b spline subdivision to derive a local refinement procedure which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions we test hierarchical refinement of nurbs for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases using the b spline version of the finite cell method we illustrate the potential of immersed boundary methods as a seamless isogeometric design through analysis procedure for complex engineering parts defined by t spline cad surfaces specifically a ship propeller and an automobile wheel we show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features

implicit boundary method for finite element analysis using non conforming mesh or grid in the finite element method fem a mesh is used for representing the geometry of the analysis and for representing the test and trial functions by piece wise interpolation recently analysis techniques that use structured grids have been developed to avoid the need for a conforming mesh the boundaries of the analysis domain are represented using implicit equations while a structured grid is used to interpolate functions such a method for analysis using structured grids is presented here in which the analysis domain is constructed by boolean combination of step functions implicit equations of the boundary are used in the construction of trial and test functions such that essential boundary conditions are guaranteed to be satisfied furthermore these functions are constructed such that internal elements through which no boundary passes have the same stiffness matrix this approach has been applied to solve linear elastostatic problems and the results are compared with analytical and finite element analysis solutions to show that the method gives solutions that are similar to the fem in quality but is less computationally expensive for dense mesh grid and avoids the need for a conforming mesh copyright 2007 john wiley sons ltd

exact imposition of inhomogeneous dirichlet boundary conditions based on weighted finite cell method and level set function abstract the imposition of inhomogeneous dirichlet boundary conditions idbcs is essential in numerical analysis of a structure it is especially difficult and no longer straightforward whenever non conformal mesh is used to discretize a structure the original contribution of this paper is to develop a weighted finite cell method fcm with high computing accuracy the commonly used transfinite interpolation in the computer aided design cad community is now extended to define boundary value function so that the idbcs are imposed exactly unlike existing weak imposition forms and weighted interpolation methods it is shown that the weighting function and boundary value function involved in the weighted fcm can directly be applied to the physical field independently of the mesh discretization besides level set function lsf is used to represent the geometry of the considered physical domain to verify the effectiveness convergence performance and the generality of the proposed method numerical examples ranging from purely elastic to thermoelastic problems are illustrated computing results are compared with analytical and fea solutions

imposition of essential boundary conditions in the material point method summary there is increasing interest in the material point method mpm as a means of modelling solid mechanics problems in which very large deformations occur e g in the study of landslides and metal forming however some aspects vital to wider use of the method have to date been ignored in particular methods for imposing essential boundary conditions in the case where the problem domain boundary does not coincide with the background grid element edges in this paper we develop a simple procedure originally devised for standard finite elements for the imposition of essential boundary conditions to the mpm expanding its capabilities to boundaries of any inclination to the authors knowledge this is the first time that a method has been proposed that allows arbitrary dirichlet boundary conditions zero and non zero values at any inclination to be imposed in the mpm the method presented in this paper is different from other mpm boundary approximation approaches in that i the boundaries are independent of the background mesh ii artificially stiff regions of material points are avoided and iii the method does not rely on spurious mirroring of the problem domain to imposed symmetry the main contribution of this work is equally applicable to standard finite elements and the mpm this article is protected by copyright all rights reserved

the finite cell method for three dimensional problems of solid mechanics this article presents a generalization of the recently proposed finite cell method to three dimensional problems of linear elasticity the finite cell method combines ideas from embedding or fictitious domain methods with the p version of the finite element method besides supporting a fast simple generation of meshes it also provides high convergence rates mesh generation for a boundary representation of solids and for voxel based data obtained from ct scans is addressed in detail in addition the implementation of non homogeneous neumann boundary conditions and the computation of cell matrices based on a composed integration is presented the performance of the proposed method is demonstrated by three numerical examples including the elastostatic computation of a human bone biopsy

approximation dans les espaces metriques et theorie de l estimation we investigate the relations between the speed of estimation and the metric structure of the parameter space especially in the case when its metric dimension is infinite given some distance d on generally hellinger distance in the case of n i i d variables we consider the minimax risk for n observations r n q mathop inf sup limits t n theta in theta text ie theta d q theta t n t n being any estimate of we shall look for functions r such that for positive constants c 1 q and c 2 q c 1 r q n r n q c2 r q n r n is the speed of estimation and we shall show under fairly general conditions including i i d variables and regular cases of markov chains and stationnary gaussian processes that r n is determined up to multiplicative constants by the metric structure of we shall also give a construction for some sort of universal estimates the risk of which is bounded by c 2 r q n in all cases where the preceding theory applies

outlier models and prior distributions in bayesian linear regression summary bayesian inference in regression models is considered using heavy tailed error distri butions to accommodate outliers the particular class of distributions that can be con structed as scale mixtures of normal distributions are examined and use is made of them as both error models and prior distributions in bayesian linear modelling includ ing simple regression and more complex hierarchical models with structured priors depending on unknown hyperprior parameters the modelling of outliers in nominally normal linear regression models using alternative error distributions which are heavy tailed relative to the normal provides an automatic means of both detecting and accommodating possibly aberrant observations such realistic models do however often lead to analytically intractable analyses with complex posterior distributions in several dimensions that are difficult to summarize and understand in this paper we consider a special yet rather wide class of heavy tailed unimodal and symmetric error distributions for which the analyses though apparently intractable can be examined in some depth by exploiting certain properties of the assumed error form the distributions concerned are those that can be con structed as scale mixtures of normal distributions in his paper concerning location parameters de finetti 1961 discusses such distributions and suggests the hypothetical interpretation that each observation is taken using an instrument with normal error but each time chosen at random from a collection of instruments of different precisions the distribution of the

a new fictitious domain approach inspired by the extended finite element method the purpose of this paper is to present a new fictitious domain approach inspired by the extended finite element method introduced by moes dolbow and belytschko in internat j numer methods engrg 46 1999 pp 131 150 an optimal method is obtained thanks to an additional stabilization technique some a priori estimates are established and numerical experiments illustrate different aspects of the method the presentation is made on a simple poisson problem with mixed neumann and dirichlet boundary conditions the extension to other problems or boundary conditions is quite straightforward

shape optimization of dirichlet boundaries based on weighted b spline finite cell method and level set function abstract this paper addresses extended shape optimization problems where structural supports i e dirichlet boundaries and free boundaries are simultaneously optimized unlike traditional fem the weighted b spline finite cell method fcm is applied as structural analysis tool and combined with the level set function lsf to take into account dirichlet boundary condition dbc automatically through penalization of the displacement field the proposed shape optimization method achieves a comprehensive integration of fixed grid b spline shape function and level set function as both the structure and dirichlet boundaries are described in the form of lsf any modification of dirichlet boundaries can be made in a straightforward way as easily as free boundaries by changing continuous design variables meanwhile the computing accuracy is ensured within the framework of fixed grid owing to the quadtree refinements of boundary cells stress related shape optimization problems are finally solved to demonstrate the merit and validity of the proposed optimization method in dealing with shape optimization of dirichlet boundaries

lower bounds for the minimax risk using f divergences and applications lower bounds involving f divergences between the underlying probability measures are proved for the minimax risk in estimation problems our proofs just use simple convexity facts special cases and straightforward corollaries of our bounds include well known inequalities for establishing minimax lower bounds such as fano s inequality pinsker s inequality and inequalities based on global entropy conditions two applications are provided a new minimax lower bound for the reconstruction of convex bodies from noisy support function measurements and a different proof of a recent minimax lower bound for the estimation of a covariance matrix

a fictitious domain decomposition method for the solution of partially axisymmetric acoustic scattering problems part 2 neumann boundary conditions we present a fictitious domain decomposition method for the fast solution of acoustic scattering problems characterized by a partially axisymmetric sound hard scatterer we apply this method to the solution of a mock up submarine problem and highlight its computational advantages and intrinsic parallelism a key component of our method is an original idea for addressing a neumann boundary condition in the general framework of a fictitious domain method this idea is applicable to many other linear partial differential equations besides the helmholtz equation copyright 2003 john wiley sons ltd

information based complexity feedback and dynamics in convex programming we study the intrinsic limitations of sequential convex optimization through the lens of feedback information theory in the oracle model of optimization an algorithm queries an oracle for noisy information about the unknown objective function and the goal is to approximately minimize every function in a given class using as few queries as possible we show that in order for a function to be optimized the algorithm must be able to accumulate enough information about the objective this in turn puts limits on the speed of optimization under specific assumptions on the oracle and the type of feedback our techniques are akin to the ones used in statistical literature to obtain minimax lower bounds on the risks of estimation procedures the notable difference is that unlike in the case of i i d data a sequential optimization algorithm can gather observations in a controlled manner so that the amount of information at each step is allowed to change in time in particular we show that optimization algorithms often obey the law of diminishing returns the signal to noise ratio drops as the optimization algorithm approaches the optimum to underscore the generality of the tools we use our approach to derive fundamental lower bounds for a certain active learning problem overall the present work connects the intuitive notions of information in optimization experimental design estimation and active learning to the quantitative notion of shannon information

essential boundary conditions and multi point constraints in finite element analysis the application of boundary conditions and other constraints to the stiffness matrix and load vector is an integral part of any finite element code this process is usually trivial but can present difficulties to the point where some codes disallow certain combinations of commonly occurring boundary conditions a general approach to this problem is presented and applied to representative examples necessary and sufficient conditions are obtained under which constraints can be applied sequentially bounds are obtained for the condition number of the reduced system showing it is essentially the same as for the original problem

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

spectral domain embedding for elliptic pdes in complex domains spectral methods are a class of methods for solving partial differential equations pdes when the solution of the pde is analytic it is known that the spectral solutions converge exponentially as a function of the number of modes used the basic spectral method works only for regular domains such as rectangles or disks domain decomposition methods spectral element methods extend the applicability of spectral methods to more complex geometries an alternative is to embed the irregular domain into a regular one this paper uses the spectral method with domain embedding to solve pdes on complex geometry the running time of the new algorithm has the same order as that for the usual spectral collocation method for pdes on regular geometry the algorithm is extremely simple and can handle dirichlet neumann boundary conditions as well as nonlinear equations

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

imposing essential boundary conditions in mesh free methods imposing essential boundary conditions is a key issue in mesh free methods the mesh free interpolation does not verify the kronecker delta property and therefore the imposition of prescribed values is not as straightforward as for the finite element method the aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in galerkin based mesh free methods special attention is paid to the mesh free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the galerkin weak form particular examples are used to analyze and compare their performance in different situations

an implicit boundary finite element method with extension to frictional sliding boundary conditions and elasto plastic analyses abstract implicit boundary methods which enrich the interpolation structure with implicit weight functions are straightforward methods for the enforcement of dirichlet boundary conditions in this article we follow the implicit boundary method that uses approximate step functions the step boundary method developed by kumar et al and provide modifications that have several advantages roller boundary conditions have wide practical applications in engineering however the step boundary method for roller boundary conditions with inclinations has yet to be fully formulated through to the final linear system of equations thus we provide a complete derivation that leads to simplified stiffness matrices compared to the original approach which can be implemented directly in fictitious domain finite element analysis the approach is then extended we believe for the first time to the nonlinear cases of frictional boundary conditions and elasto plastic material behaviour the proposed formulation and procedures are validated on a number of example problems that test different aspects of the method

default bayesian analysis with global local shrinkage priors we provide a framework for assessing the default nature of a prior distribution using the property of regular variation which we study for global local shrinkage priors in particular we show that the horseshoe priors originally designed to handle sparsity are regularly varying and thus are appropriate for default bayesian analysis to illustrate our methodology we discuss four problems of noninformative priors that have been shown to be highly informative for nonlinear functions in each case we show that global local horseshoe priors perform as required global local shrinkage priors can separate a low dimensional signal from high dimensional noise even for nonlinear functions

the hp d adaptive finite cell method for geometrically nonlinear problems of solid mechanics summary the finite cell method fcm combines the fictitious domain approach with the p version of the finite element method and adaptive integration for problems of linear elasticity it offers high convergence rates and simple mesh generation irrespective of the geometric complexity involved this article presents the integration of the fcm into the framework of nonlinear finite element technology however the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map as a consequence of the weak penalization nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries leading to a low algebraic rate of convergence therefore the fcm approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the hp d method which synergetically uses the h adaptivity of the integration scheme numerical experiments show that the hp d overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map the hp d adaptive fcm is thus able to restore high convergence rates for the geometrically nonlinear case while preserving the easy meshing property of the original fcm accuracy and performance of the present scheme are demonstrated by several benchmark problems in one two and three dimensions and the nonlinear simulation of a complex foam sample copyright 2011 john wiley sons ltd

analytical and computational assessment of locking in the hp finite element method locking is the phenomenon by which the numerical approximation of parameter dependent problems deteriorates for values of the parameter close to a limiting value in this paper we give a definition of locking and develop precise computable and analytic ways to quantify it using the example of nearly incompressible elasticity we show by means of computational and theoretical results the difference between the h version and php version in combatting locking our results establish the superiority of high order elements both h p and hp when the standard variational form is used we also discuss other issues such as curved elements mixed methods and locking phenomena for problems over anisotropic materials and over thin domains

fictitious domain finite element methods using cut elements i a stabilized lagrange multiplier method we propose a fictitious domain method where the mesh is cut by the boundary the primal solution is computed only up to the boundary the solution itself is defined also by nodes outside the domain

multiscale computations with a combination of the h and p versions of the finite element method the numerical treatment of structural problems is often difficult if the global system behavior is affected by phenomena on different length scales when simulating multiscale problems it is essential to find a discretization that equally reflects all aspects of the problem with sufficient accuracy the present paper is concerned with the formulation of a finite element procedure which combines the p method to resolve global features of a problem with an h method to resolve local features thus the advantageous properties of both standard procedures can be optimally exploited the implementation provides the further advantage that the numerical treatment of the problems on the different length scales can be done with independent finite element discretizations to solve the overall problem an iteration scheme with an adaptive solution strategy is developed benchmark problems and an example with relevance to soil mechanics are presented

shell finite cell method a high order fictitious domain approach for thin walled structures abstract this article presents a generalization of the recently proposed finite cell method to thin walled structures this approach uses a combination of well known fictitious domain methods with high order hierarchical ansatz spaces known from the p version of the finite element method whereas the original concept embeds a three dimensional structure in a simple domain being meshed into a grid of cube shaped cells the extension presented in this paper applies the fictitious domain approach to a two dimensional master domain defined in the parameter plane of the geometric model implementation details are discussed and numerical benchmark problems show the high accuracy and computational efficiency of the new approach it is also remarked that the present approach can easily be carried over to isogeometric analysis opening an attractive possibility to simulate trimmed nurbs surfaces

an unfitted finite element method using discontinuous galerkin in this paper we present a new approach to simulations on complex shaped domains the method is based on a discontinuous galerkin dg method using trial and test functions defined on a structured grid essential boundary conditions are imposed weakly via the dg formulation this method offers a discretization where the number of unknowns is independent of the complexity of the domain we will show numerical computations for an elliptic scalar model problem in 2 and 3 convergence rates for different polynomial degrees are studied

reference prior bayesian analysis for normal mean products abstract two reference priors for the product of means of n normal distributions with common known variance are developed one of them induces an improper posterior distribution and therefore is not of much interest the other is a generalized form of the n 2 case derived by berger and bernardo the latter is compared with the uniform prior the jeffreys prior in posterior inference and the optimal frequentist coverage criterion the reference prior is shown to be better than the uniform prior in the sense of correct frequentist coverage probability of the posterior quantile by numerical computation the computation was performed by gibbs sampling for n 3 and n 10 furthermore it is shown that the reference prior is among the asymptotic optimal frequentist coverage probability priors under a transformation of the parameter space such that the parameter of interest and the nuisance parameters are orthogonal in an example the bayesian credible interval for the product of normal means using the re

optimal rates of convergence for convex set estimation from support functions nitely many noisy measurements of its support function the solution is based on appropriate regularizations of the least squares estimator both xed and random designs are considered 1 introduction we study the nonparametric estimation problem of estimating a compact convex set in euclidean space from noisy support function measurements the support function hk of a compact convex subset k of r d d 2 is dened for u in the unit sphere s d 1 fx2 r d x 2 x 2 1g by

sharp inequalities for f divergences f divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability mathematical statistics and information theory such as kullback leibler divergence chi squared divergence squared hellinger distance total variation distance and so on in this paper we study the problem of maximizing or minimizing an f divergence between two probability measures subject to a finite number of constraints on other f divergences we show that these infinite dimensional optimization problems can all be reduced to optimization problems over small finite dimensional spaces which are tractable our results lead to a comprehensive and unified treatment of the problem of obtaining sharp inequalities between f divergences we demonstrate that many of the existing results on inequalities between f divergences can be obtained as special cases of our results we also improve on some existing non sharp inequalities

interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems abstract the focus of this paper is on interpolation schemes for fictitious domain and topology optimization approaches with structures undergoing large displacements numerical instability in the finite element simulations can often be observed due to excessive distortion in low stiffness regions a new energy interpolation scheme is proposed in order to stabilize the numerical simulations the elastic energy density in the solid and void regions is interpolated using the elastic energy densities for large and small deformation theory respectively the performance of the proposed method is demonstrated for a challenging test geometry as well as for topology optimization of minimum compliance and compliant mechanisms the effect of combining the proposed interpolation scheme with different hyperelastic material models is investigated as well numerical results show that the proposed approach alleviates the problems in the low stiffness regions and for the simulated cases results in stable topology optimization of structures undergoing large displacements

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

exact and approximate posterior moments for a normal location parameter the forms of first and second posterior moments for a normal location parameter are identified for a rather general class of prior distributions exact and approximate illustrations are given where the prior distribution is double exponential or student t respectively

an extended finite element method lagrange multiplier based approach for fluid structure interaction abstract this paper presents a new fixed grid fluid structure interaction scheme that can be applied to the interaction of most general structures with incompressible flow it is based on an extended finite element method xfem based strategy the extended eulerian fluid field and the lagrangian structural field are partitioned and iteratively coupled using lagrange multiplier techniques for non matching grids the approach allows the simulation of the interaction of thin and bulky structures exhibiting large deformations finally qualitative examples and a benchmark computation demonstrate key features and accuracy of the method

volume ratio sparsity and minimaxity under unitarily invariant norms this paper studies non asymptotic minimax estimation of high dimensional matrices and provides tight minimax rates for a large collection of loss functions in a variety of problems via information theoretic methods based on the convex geometry of finite dimensional banach spaces we first develop a volume ratio approach for determining minimax estimation rates of unconstrained mean matrices under all unitarily invariant norm losses which turn out to only depend on the norm of identity matrix in addition we establish the minimax rates for estimating normal mean matrices with submatrix sparsity where the sparsity constraint introduces an additional term in the rate which in contrast to the unconstrained case is determined by the smoothness lipschitz constant of the norm this method is also applicable to the low rank matrix completion problem and extends well beyond the additive noise model in particular it yields tight rates in covariance matrix estimation and poisson rate matrix estimation problems for all unitarily invariant norms

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

the finite cell method for linear thermoelasticity the recently introduced finite cell method fcm combines the fictitious domain idea with the benefits of high order finite elements while previous publications concentrated on single field applications this paper demonstrates that the advantages of the method carry over to the multi physical context of linear thermoelasticity the ability of the method to converge with exponential rates is illustrated in detail with a benchmark problem a second example shows that the finite cell method correctly captures the thermoelastic state of a complex problem from engineering practice both examples additionally verify that also for two field problems dirichlet boundary conditions can be weakly imposed on non conforming meshes by the proposed extension of nitsche s method

outliers and credence for location parameter inference abstract heavy tailed distributions are important for modeling problems in which there may be outlying observations or parameters examples of their use are given in o hagan 1988 this article develops some general theory based on the notion of credence for inference about unknown location parameters in the case of known variances a density on the real line is defined to have credence c if it is bounded above and below by positive multiples of 1 x 2 c 2 for instance a t distribution with d degrees of freedom has credence 1 d i prove that the credence of a sum of independent random variables is the minimum of their individual credences and that the credence of a posterior density of a location parameter is the sum of the credences of the prior and the observations more generally when independent information sources are combined their credences add when groups of information sources conflict outlier rejection occurs with the group having the greatest total credence dominating all others

convergence of algorithms for reconstructing convex bodies and directional measures we investigate algorithms for reconstructing a convex body k in r n from noisy measurements of its support function or its brightness function in k directions u 1 u k the key idea of these algorithms is to construct a convex polytope p k whose support function or brightness function best approximates the given measurements in the directions u 1 u k in the least squares sense the measurement errors are assumed to be stochastically independent and gaussian it is shown that this procedure is strongly consistent meaning that almost surely p k tends to k in the hausdorff metric as k here some mild assumptions on the sequence u i of directions are needed using results from the theory of empirical processes estimates of rates of convergence are derived which are first obtained in the l 2 metric and then transferred to the hausdorff metric along the way a new estimate is obtained for the metric entropy of the class of origin symmetric zonoids contained in the unit ball similar results are obtained for the convergence of an algorithm that reconstructs an approximating measure to the directional measure of a stationary fiber process from noisy measurements of its rose of intersections in k directions u 1 u k here the dudley and prohorov metrics are used the methods are linked to those employed for the support and brightness function algorithms via the fact that the rose of intersections is the support function of a projection body

f divergence inequalities this paper develops systematic approaches to obtain f divergence inequalities dealing with pairs of probability measures defined on arbitrary alphabets functional domination is one such approach where special emphasis is placed on finding the best possible constant upper bounding a ratio of f divergences another approach used for the derivation of bounds among f divergences relies on moment inequalities and the logarithmic convexity property which results in tight bounds on the relative entropy and bhattacharyya distance in terms of chi 2 divergences a rich variety of bounds are shown to hold under boundedness assumptions on the relative information special attention is devoted to the total variation distance and its relation to the relative information and relative entropy including reverse pinsker inequalities as well as on the e gamma divergence which generalizes the total variation distance pinsker s inequality is extended for this type of f divergence a result which leads to an inequality linking the relative entropy and relative information spectrum integral expressions of the renyi divergence in terms of the relative information spectrum are derived leading to bounds on the renyi divergence in terms of either the variational distance or relative entropy

a fictitious domain approach with spread interface for elliptic problems with general boundary conditions the aim of this article is to solve second order elliptic problems in an original physical domain using a fictitious domain method with a spread interface approach the main idea of the fictitious domain approach consists in immersing the original domain of study into a geometrically bigger and simpler one called fictitious domain as the spatial discretization is being performed in the fictitious domain this method allows the use of structured meshes the discretization is not boundary fitted to the original physical domain this paper describes several ways to impose dirichlet robin or neumann boundary conditions on a spread immersed interface without locally modifying the numerical scheme and without using lagrange multipliers the numerical applications focus on difiusion and convection problems in the unit disk with dirichlet or robin boundary conditions for such problems analytical solutions can be determined for a correctly chosen source term the numerical resolution is performed using a q1 finite element scheme the spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vinicity of the immersed boundary the l2 norm of the errors is computed in order to evaluate the capability of the method immersed boundaries are found in many industrial applications like two phase flow simulations fluid structure interaction etc this article represents a first step towards the simulation of these kinds of applications

generalizing the fano inequality the fano inequality gives a lower bound on the mutual information between two random variables that take values on an m element set provided at least one of the random variables is equiprobable the authors show several simple lower bounds on mutual information which do not assume such a restriction in particular this ran be accomplished by replacing log m with the infinite order renyi entropy in the fano inequality applications to hypothesis testing are exhibited along with bounds on mutual information in terms of the a priori and a posteriori error probabilities

weak impositions of dirichlet boundary conditions in solid mechanics a critique of current approaches and extension to partially prescribed boundaries abstract in this article we first review various approaches developed to date for the weak imposition of dirichlet boundary conditions in fictitious domain analysis for elasticity problems the hellinger reissner h r principle the linked lagrange multiplier llm method the implicit boundary method and the fat boundary method are discussed along with the well known lagrange multiplier penalty and nitsche s methods we state these approaches in a common form starting with energy functionals and weak forms and discretise using the fictitious domain finite element method previous formulations of these methods were in general developed for full prescription along the dirichlet boundary which generally implies no local effect of boundary inclination however partially prescribed conditions such as the structural roller boundary condition with inclination have wide practical applications in engineering here we provide techniques of imposing such boundary conditions in these methods in detail for those methods that contain algorithmic parameters such as the penalty and nitsche s methods extra computation or empirical estimation is necessary to decide values of the parameters and hence we discuss parametric and convergence behaviours through numerical examples to provide guidance on the choice of parameters

information recovery from pairwise measurements this paper is concerned with jointly recovering n node variables left x i right 1 leq i leq n from a collection of pairwise difference measurements imagine we acquire a few observations taking the form of x i x j the observation pattern is represented by a measurement graph mathcal g with an edge set mathcal e such that x i x j is observed if and only if i j in mathcal e to account for noisy measurements in a general manner we model the data acquisition process by a set of channels with given input output transition measures employing information theoretic tools applied to channel decoding problems we develop a unified framework to characterize the fundamental recovery criterion which accommodates general graph structures alphabet sizes and channel transition measures in particular our results isolate a family of minimum channel divergence measures to characterize the degree of measurement corruption which together with the size of the minimum cut of mathcal g dictates the feasibility of exact information recovery for various homogeneous graphs the recovery condition depends almost only on the edge sparsity of the measurement graph irrespective of other graphical metrics alternatively the minimum sample complexity required for these graphs scales like n log n mathsf hel 1 2 min for certain information metric mathsf hel 1 2 min defined in the main text as long as the alphabet size is not super polynomial in n we apply our general theory to three concrete applications including the stochastic block model the random corruption model and the haplotype assembly problem our theory leads to orderwise tight recovery conditions for all these scenarios

covering numbers for convex functions in this paper we study the covering numbers of the space of convex and uniformly bounded functions in multidimension we find optimal upper and lower bounds for the epsilon covering number of cal c a b d b in the l p metric 1 leq p in terms of the relevant constants where d geq 1 a b 0 and cal c a b d b denotes the set of all convex functions on a b d that are uniformly bounded by b we summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

implicit boundary method for analysis using uniform b spline basis and structured grid several analysis techniques such as extended finite element method x fem have been developed recently which use structured grid for the analysis implicit boundary method uses implicit equations of the boundary to apply boundary conditions in x fem framework using structured grids solution structures for test and trial functions are constructed using implicit equations such that the boundary conditions are satisfied even if there are no nodes on the boundary in this paper this method is applied for analysis using uniform b spline basis defined over a structured grid solution structures that are c1 or c2 continuous throughout the analysis domain can be constructed using b spline basis functions as a structured grid does not conform to the geometry of the analysis domain the boundaries of the analysis domain are defined independently using equations of the boundary curves surfaces compared with conforming mesh it is easier to generate structured grids that overlap the geometry and the elements in the grid are regular shaped and undistorted numerical examples are presented to demonstrate the performance of these b spline elements the results are compared with analytical solutions as well as with traditional finite element solutions convergence studies for several examples show that b spline elements provide accurate solutions with fewer elements and nodes compared with traditional fem they also provide continuous stress and strain in the analysis domain thus eliminating the need for smoothing stress strain results copyright 2008 john wiley sons ltd

phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms we propose strongly consistent algorithms for reconstructing the characteristic function 1k of an unknown convex body k in r n from possibly noisy measurements of the modulus of its fourier transform c 1k this represents a complete theoretical solution to the phase retrieval problem for characteristic functions of convex bodies the approach is via the closely related problem of reconstructing k from noisy measurements of its covariogram the function giving the volume of the intersection of k with its translates in the many known situations in which the covariogram determines a convex body up to reflection in the origin and when the position of the body is fixed our algorithms use o k n noisy covariogram measurements to construct a convex polytope pk that approximates k or its reflection k in the origin by recent uniqueness results this applies to all planar convex bodies all three dimensional convex polytopes and all symmetric and most in the sense of baire category arbitrary convex bodies in all dimensions two methods are provided and both are shown to be strongly consistent in the sense that almost surely the minimum of the hausdorff distance between pk and k tends to zero as k tends to infinity

the horseshoe estimator for sparse signals this paper proposes a new approach to sparsity called the horseshoe estimator the horseshoe is a close cousin of other widely used bayes rules arising from for example double exponential and cauchy priors in that it is a member of the same family of multivariate scale mixtures of normals but the horseshoe enjoys a number of advantages over existing approaches including its robustness its adaptivity to dierent sparsity patterns and its analytical tractability we prove two theorems that formally characterize both the horseshoe s adeptness at large outlying signals and its super ecient rate of convergence to the correct estimate of the sampling density in sparse situations finally using a combination of real and simulated data we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full bayesian model averaging approach using a discrete mixture prior to model signals and noise

estimating a product of means bayesian analysis with reference priors abstract suppose that we observe x n 1 and independently y n 1 and are concerned with inference mainly estimation and confidence statements about the product of means this problem arises most obviously in situations of determining area based on measurements of length and width it also arises in other practical contexts however for instance in gypsy moth studies the hatching rate of larvae per unit area can be estimated as the product of the mean of egg masses per unit area times the mean number of larvae hatching per egg mass approximately independent samples can be obtained for each mean see southwood 1978 noninformative prior bayesian approaches to the problem are considered in particular the reference prior approach of bernardo 1979 an appropriate reference prior for the problem is developed and relatively easily implementable formulas for posterior moments e g the posterior mean and variance and credible sets are derived comparisons with alternative noninf

imposing dirichlet boundary conditions with nitsche s method and spline based finite elements a key challenge while employing non interpolatory basis functions in finite element methods is the robust imposition of dirichlet boundary conditions the current work studies the weak enforcement of such conditions for b spline basis functions with application to both second and fourth order problems this is achieved using concepts borrowed from nitsche s method which is a stabilized method for imposing constraints on surfaces conditions for the stability of the system of equations are derived for each class of problem stability parameters in the nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the dirichlet boundary the approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not through several numerical examples the approach is shown to yield optimal rates of convergence copyright 2010 john wiley sons ltd