maximum entropy and the nearly black object summary maximum entropy me inversion is a non linear inversion technique for inverse problems where the object to be recovered is known to be positive it has been applied in areas ranging from radio astronomy to various forms of spectroscopy sometimes with dramatic success in some cases me has attained an order of magnitude finer resolution and or an order of magnitude smaller noise level than that obtainable by standard linear methods the dramatic successes all seem to occur in cases where the object to be recovered is nearly black essentially zero in the vast majority of samples we show that near blackness is required both for signal to noise enhancements and for superresolution however other methods in particular minimum 1 norm reconstruction may exploit near blackness to an even greater extent

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

binary spatial partitioning of the central difference time integration scheme for explicit fast transient dynamics this paper proposes a generalization of the explicit central difference time integration scheme using a time step variable not only in time but also in space the solution at each element node is advanced in time following local rather than global stability limitations this allows substantial saving of computer time in realistic applications with non uniform meshes especially in multi field problems like fluid structure interactions a binary scheme in space is used time steps are not completely arbitrary but stay in a constant ratio of two when passing from one partition level to the next one this choice greatly facilitates implementation via an integer based logic ensures inherent synchronization and avoids any interpolations necessary in other partitioning schemes in the literature but which may reduce numerical stability the mesh partition is automatically built up and continuously updated by simple spatial adjacency considerations the resulting algorithm deals automatically with large variations in time of stability limits the paper introduces the core spatial partitioning technique in the lagrangian formulation some academic numerical examples allow a detailed comparison with the standard spatially uniform algorithm a final more realistic example shows the application of partitioning in simulations with arbitrary lagrangian eulerian formulation and fully coupled boundary conditions fluid structure interaction copyright 2008 john wiley sons ltd

adapting to unknown sparsity by controlling the false discovery rate we attempt to recover an n dimensional vector observed in white noise where n is large and the vector is known to be sparse but the degree of sparsity is unknown we consider three different ways of defining sparsity of a vector using the fraction of nonzero terms imposing power law decay bounds on the ordered entries and controlling the lp norm for p small we obtain a procedure which is asymptotically minimax for l r loss simultaneously throughout a range of such sparsity classes the optimal procedure is a data adaptive thresholding scheme driven by control of the false discovery rate fdr fdr control is a relatively recent innovation in simultaneous testing ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections in our treatment the fdr control parameter qn also plays a determining role in asymptotic minimaxity if q lim qn 0 1 2 and also qn log n we get sharp asymptotic minimaxity simultaneously over a wide range of sparse parameter spaces and loss functions on the other hand q lim qn 1 2 1 forces the risk to exceed the minimax risk by a factor growing with q to our knowledge this relation between ideas in simultaneous inference and asymptotic decision theory is new our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors these new rules impose complexity penalization of the form 2 log potential model size actual model size we exhibit a close connection with fdr controlling procedures under stringent control of the false discovery rate

a posteriori error estimation for unilateral contact with matching and non matching meshes in this paper we consider the unilateral contact problem between elastic bodies we propose an error estimator based on the concept of error in the constitutive relation in order to evaluate the finite element approximation involving matching and non matching meshes on the contact zone the determination of the a posteriori error estimate is linked to the building of kinematically admissible stress fields and statically admissible stress fields we then propose a finite element method for approximating the unilateral contact problem taking into account matching and non matching meshes on the contact zone then we describe the construction of admissible fields lastly we present optimized computations by using both the error estimates and a convenient mesh adaptivity procedure 2000 elsevier science s a all rights reserved

a variational principle for the formulation of partitioned structural systems a continuum based variational principle is presented for the formulation of the discrete governing equations of partitioned structural systems this application includes coupled substructures as well as subdomains obtained by mesh decomposition the present variational principle is derived by a series of modifications of a hybrid functional originally proposed by atluri for finite element development the interface is treated by a displacement frame and a localized version of the method of lagrange multipliers interior displacements are decomposed into rigid body and deformational components to handle floating subdomains both static and dynamic versions are considered an important application of the present principle is the treatment of nonmatching meshes that arise from various sources such as separate discretization of substructures independent mesh refinement and global local analysis the present principle is compared with that of a globalized version of the multiplier method copyright 2000 john wiley sons ltd

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

special values of multiple polylogarithms historically the polylogarithm has attracted specialists and non specialists alike with its lovely evaluations much the same can be said for euler sums or multiple harmonic sums which within the past decade have arisen in combinatorics knot theory and high energy physics more recently we have been forced to consider multidimensional extensions encompassing the classical polylogarithm euler sums and the riemann zeta function here we provide a general framework within which previously isolated results can now be properly understood applying the theory developed herein we prove several previously conjectured evaluations including an intriguing conjecture of don zagier

analytic and algorithmic aspects of generalized harmonic sums and polylogarithms in recent three loop calculations of massive feynman integrals within quantum chromodynamics qcd and e g in recent combinatorial problems the so called generalized harmonic sums in short s sums arise they are characterized by rational or real numerator weights also different from 1 in this article we explore the algorithmic and analytic properties of these sums systematically we work out the mellin and inverse mellin transform which connects the sums under consideration with the associated poincare iterated integrals also called generalized harmonic polylogarithms in this regard we obtain explicit analytic continuations by means of asymptotic expansions of the s sums which started to occur frequently in current qcd calculations in addition we derive algebraic and structural relations like differentiation with respect to the external summation index and different multi argument relations for the compactification of s sum expressions finally we calculate algebraic relations for infinite s sums or equivalently for generalized harmonic polylogarithms evaluated at special values the corresponding algorithms and relations are encoded in the computer algebra package harmonicsums

rates of convergence for minimum contrast estimators we shall present here a general study of minimum contrast estimators in a nonparametric setting although our results are also valid in the classical parametric case for independent observations these estimators include many of the most popular estimators in various situations such as maximum likelihood estimators least squares and other estimators of the regression function estimators for mixture models or deconvolution the main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters optimal rates depending on entropy conditions are already known at least for some of the models involved and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large but under some circumstances large entropies or changes in the entropy structure due to local perturbations the resulting the rates are only suboptimal counterexamples are constructed which show that the phenomenon is real for non parametric maximum likelihood or regression this proves that under purely metric assumptions our theorem is optimal and that minimum contrast estimators happen to be suboptimal

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

comparison of different feti preconditioners for elastoplasticity this paper illustrates parallel solution of elastoplastic problems with hardening based on the tfeti domain decomposition method with several preconditioning strategies we consider von mises plasticity with isotropic hardening using the return mapping concept for one time step to treat nonlinearity and nonsmoothness we use the semismooth newton method in each newton iteration we solve a linear system of equations using the tfeti domain decomposition method with lumped dirichlet or no preconditioner our permon software is used for the numerical experiments the observed times and numbers of iterations are backed up by the regular condition number estimates both preconditioners accelerate the solution process in terms of the convergence rate for the worst conditioned benchmark the most expensive dirichlet preconditioner gives the lowest computational times this lets us suggest the dirichlet preconditioner as the default choice for engineering problems

iterated elliptic and hypergeometric integrals for feynman diagrams we calculate 3 loop master integrals for heavy quark correlators and the 3 loop quantum chromodynamics corrections to the parameter they obey non factorizing differential equations of second order with more than three singularities which cannot be factorized in mellin n space either the solution of the homogeneous equations is possible in terms of 2f1 gaus hypergeometric functions at rational argument in some cases integrals of this type can be mapped to complete elliptic integrals at rational argument this class of functions appears to be the next one arising in the calculation of more complicated feynman integrals following the harmonic polylogarithms generalized polylogarithms cyclotomic harmonic polylogarithms square root valued iterated integrals and combinations thereof which appear in simpler cases the inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals where the new innermost letter itself is not an iterative integral a new class of iterative integrals is introduced containing letters in which multiple definite integrals appear as factors for the elliptic case we also derive the solution in terms of integrals over modular functions and also modular forms using q product and series representations implied by jacobi s i functions and dedekind s function the corresponding representations can be traced back to polynomials out of lambert eisenstein series having representations also as elliptic polylogarithms a q factorial 1 k logarithms and polylogarithms of q and their q integrals due to the specific form of the physical variable x q for different processes different representations do usually appear numerical results are also presented

lambert w solves the noncommutative varphi 4 φ 4 model the closed dyson schwinger equation for the 2 point function of the noncommutative lambda phi 4 2 model is rearranged into the boundary value problem for a sectionally holomorphic function in two variables we prove an exact formula for a solution in terms of lambert s w function this solution is holomorphic in lambda inside a domain which contains 1 log 4 infty our methods include the hilbert transform perturbation series and lagrange burmann resummation

on density estimation in the view of kolmogorov s ideas in approximation theory the paper deals with upper and lower bounds for the quality of probability density estimation connections are established between these problems and the theory of approximation of functions particularly it is demonstrated how some of kolmogorov s concepts work

constrained minimax estimation of the mean of the normal distribution with known variance in this paper we shall discuss the estimation of the mean of a normal distribution with variance 1 the main question in this work is the existence and computation of a least favorable distribution among all the prior distributions satisfying a given set of constraints in the following we show that if this distribution is bounded from above on some even moment then the least favorable distribution exists and it is either normal or discrete the support of the discrete distribution function does not have any accumulation point the least favorable distribution is normal if and only if the second moment is bounded from above without any other relevant constraint these theorems shed light on the james stein estimator as the minimax estimator for a prior with unknown bounded variance

incorporation of linear multipoint constraints in domain decomposition based iterative solvers part ii blending feti dp and mortar methods and assembling floating substructures two approaches are presented for extending the feti dp domain decomposition based iterative method to the solution of problems with linear multipoint constraints lmpcs both cases of arbitrary lmpcs and lmpcs generated by the mortar method for enforcing a weak continuity of the solution across non matching finite element interfaces are considered in the first approach the lmpcs are addressed during the preconditioning step but enforced only at convergence in the second approach the lmpcs are enforced at each iteration through the solution of an auxiliary coarse problem both extended feti dp solvers are benchmarked on a parallel processor for a wide range of constrained structural mechanics problems the obtained results reveal that both solvers are numerically scalable and complement each other from the memory and cpu performance viewpoints

estimation of a sparse group of sparse vectors summary we consider estimating a sparse group of sparse normal mean vectors based on penalized likelihood estimation with complexity penalties on the number of nonzero mean vectors and the numbers of their significant components which can be performed by a fast algorithm the resulting estimators are developed within a bayesian framework and can be viewed as maximum a posteriori estimators we establish their adaptive minimaxity over a wide range of sparse and dense settings a simulation study demonstrates the efficiency of the proposed approach which successfully competes with the sparse group lasso estimator

lectures on differential equations for feynman integrals over the last year significant progress was made in the understanding of the computation of feynman integrals using differential equations de these lectures give a review of these developments while not assuming any prior knowledge of the subject after an introduction to de for feynman integrals we point out how they can be simplified using algorithms available in the mathematical literature we discuss how this is related to a recent conjecture for a canonical form of the equations we also discuss a complementary approach that is based on properties of the space time loop integrands and explain how the ideas of leading singularities and d log representations can be used to find an optimal basis for the de finally as an application of these ideas we show how single scale integrals can be bootstrapped using the drinfeld associator of a de

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

on optimality of bayesian testimation in the normal means problem we consider a problem of recovering a high dimensional vector observed in white noise where the unknown vector is assumed to be sparse the objective of the paper is to develop a bayesian formalism which gives rise to a family of l 0 type penalties the penalties are associated with various choices of the prior distributions n on the number of nonzero entries of and hence are easy to interpret the resulting bayesian estimators lead to a general thresholding rule which accommodates many of the known thresholding and model selection procedures as particular cases corresponding to specific choices of n furthermore they achieve optimality in a rather general setting under very mild conditions on the prior we also specify the class of priors n for which the resulting estimator is adaptively optimal in the minimax sense for a wide range of sparse sequences and consider several examples of such priors

on bayesian testimation and its application to wavelet thresholding summary we consider the problem of estimating the unknown response function in the gaussian white noise model we first utilize the recently developed bayesian maximum a posteriori testimation procedure of abramovich et al 2007 for recovering an unknown high dimensional gaussian mean vector the existing results for its upper error bounds over various sparse lp balls are extended to more general cases we show that for a properly chosen prior on the number of nonzero entries of the mean vector the corresponding adaptive estimator is asymptotically minimax in a wide range of sparse and dense lp balls the proposed procedure is then applied in a wavelet context to derive adaptive global and level wise wavelet estimators of the unknown response function in the gaussian white noise model these estimators are then proven to be respectively asymptotically near minimax and minimax in a wide range of besov balls these results are also extended to the estimation of derivatives of the response function simulated examples are conducted to illustrate the performance of the proposed level wise wavelet estimator in finite sample situations and to compare it with several existing counterparts

differential galois theory of linear difference equations we present a galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations with this we are able to reprove holder s theorem that the gamma function satisfies no polynomial differential equation and are able to give general results that imply for example that no differential relationship holds among solutions of certain classes of q hypergeometric equations

adaptive minimax regression estimation over sparse l q hulls given a dictionary of mn predictors in a random design regression setting with n observations we construct estimators that target the best performance among all the linear combinations of the predictors under a sparse lq norm 0 q 1 constraint on the linear coefficients besides identifying the optimal rates of convergence our universal aggregation strategies by model mixing achieve the optimal rates simultaneously over the full range of 0 q 1 for any mn and without knowledge of the lq norm of the best linear coefficients to represent the regression function to allow model misspecification our upper bound results are obtained in a framework of aggregation of estimates a striking feature is that no specific relationship among the predictors is needed to achieve the upper rates of convergence hence permitting basically arbitrary correlations between the predictors therefore whatever the true regression function assumed to be uniformly bounded our estimators automatically exploit any sparse representation of the regression function if any to the best extent possible within the lq constrained linear combinations for any 0 q 1 a sparse approximation result in the lq hulls turns out to be crucial to adaptively achieve minimax rate optimal aggregation it precisely characterizes the number of terms needed to achieve a prescribed accuracy of approximation to the best linear combination in an lq hull for 0 q 1 it offers the insight that the minimax rate of lq aggregation is basically determined by an effective model size which is a sparsity index that depends on q mn n and the lq norm bound in an easily interpretable way based on a classical model selection theory that deals with a large number of models

a stochastic deterministic coupling method for continuum mechanics in this paper we present a novel approach that allows to couple a deterministic continuum model with a stochastic continuum one the coupling strategy is performed in the arlequin framework which is based on a volume coupling and a partition of the energy a suitable functional space is chosen for the weak enforcement of the continuity between the two models the choice of this space ensures that the mean of the stochastic solution equals the deterministic solution point wise and enforces appropriate boundary conditions on the stochastic dimension the proof of the existence of the solution of the mixed problem is provided the numerical strategy is also reviewed in particular with a view at the monte carlo method finally examples show the interest of the method and possible strategies for use in adaptive modeling

estimating a bounded normal mean the problem of estimating a normal mean has received much attention in recent years if one assumes however that the true mean lies in a bounded interval the problem changes drastically in this paper we show that if the interval is small approximately two standard deviations wide then the bayes rule against a two point prior is the unique minimax estimator under squared error loss for somewhat wider intervals we also derive sufficient conditions for minimaxity of the bayes rule against a three point prior

minimax estimation of discrete distributions under ell _ 1 loss we consider the problem of discrete distribution estimation under ell 1 loss we provide tight upper and lower bounds on the maximum risk of the empirical distribution the maximum likelihood estimator and the minimax risk in regimes where the support size s may grow with the number of observations n we show that among distributions with bounded entropy h the asymptotic maximum risk for the empirical distribution is 2h ln n while the asymptotic minimax risk is h ln n moreover we show that a hard thresholding estimator oblivious to the unknown upper bound h is essentially minimax however if we constrain the estimates to lie in the simplex of probability distributions then the asymptotic minimax risk is again 2h ln n we draw connections between our work and the literature on density estimation entropy estimation total variation distance ell 1 divergence estimation joint distribution estimation in stochastic processes normal mean estimation and adaptive estimation

a time staggered partitioned coupling algorithm for transient heat conduction we present a time staggered partitioned coupling algorithm for transient heat conduction finite element simulations this algorithm divides a large structural mesh into a number of smaller subdomains solves the individual subdomains separately and couples the solutions to obtain the response to the original problem the proposed algorithm is a mixed multi timestep algorithm and enables arbitrary time integration schemes and meshes to be coupled with different timesteps in the various subdomains in this procedure the solution of each partition is separately evaluated over a system timestep after which the interfacial conditions are enforced making this a staggered algorithm that facilitates parallel computation we present examples showing the feasibility of the coupling algorithm and discuss the merits in terms of convergence and stability

non conforming mesh gluing in the finite elements method a methodology to connect two or more non conforming meshes generated from a geometric model is presented the problem leads to a variational problem subject to constraints the emphasis of the work is on constraints construction two methods namely discrete connection and continuous connection are developed the relation with hybrid formulations is considered and two application examples are presented

nested sums expansion of transcendental functions and multiscale multiloop integrals expansion of higher transcendental functions in a small parameter are needed in many areas of science for certain classes of functions this can be achieved by algebraic means these algebraic tools are based on nested sums and can be formulated as algorithms suitable for an implementation on a computer examples such as expansions of generalized hypergeometric functions or appell functions are discussed as a further application we give the general solution of a two loop integral the so called c topology in terms of multiple nested sums in addition we discuss some important properties of nested sums in particular we show that they satisfy a hopf algebra

deflated preconditioned conjugate gradient solvers for linear elasticity extensions of deflation techniques previously developed for the poisson equation to static elasticity are presented compared to the scalar poisson equation j comput phys 2008 227 24 10196 10208 int j numer meth engng 2010 doi 10 1002 nme 2932 int j numer meth biomed engng 2010 26 1 73 85 the elasticity equations represent a system of equations giving rise to more complex low frequency modes multigrid elsevier amsterdam 2000 in particular the straightforward extension from the scalar case does not provide generally satisfactory convergence however a simple modification allows to recover the remarkable acceleration in convergence and cpu time reached in the scalar case numerous examples and timings are provided in a serial and a parallel context and show the dramatic improvements of up to two orders of magnitude in cpu time for grids with moderate graph depths compared to the non deflated version furthermore a monotonic decrease of iterations with increasing subdomains as well as a remarkable acceleration for very few subdomains are also observed if all the rigid body modes are included copyright 2011 john wiley sons ltd

optimal classification in sparse gaussian graphic model consider a two class classification problem where the number of features is much larger than the sample size the features are masked by gaussian noise with mean zero and covariance matrix sigma where the precision matrix omega sigma 1 is unknown but is presumably sparse the useful features also unknown are sparse and each contributes weakly i e rare and weak to the classification decision by obtaining a reasonably good estimate of omega we formulate the setting as a linear regression model we propose a two stage classification method where we first select features by the method of innovated thresholding it and then use the retained features and fisher s lda for classification in this approach a crucial problem is how to set the threshold of it we approach this problem by adapting the recent innovation of higher criticism thresholding hct we find that when useful features are rare and weak the limiting behavior of hct is essentially just as good as the limiting behavior of ideal threshold the threshold one would choose if the underlying distribution of the signals is known if only somewhat surprisingly when omega is sufficiently sparse its off diagonal coordinates usually do not have a major influence over the classification decision compared to recent work in the case where omega is the identity matrix proc natl acad sci usa 105 2008 14790 14795 philos trans r soc lond ser a math phys eng sci 367 2009 4449 4470 the current setting is much more general which needs a new approach and much more sophisticated analysis one key component of the analysis is the intimate relationship between hct and fisher s separation another key component is the tight large deviation bounds for empirical processes for data with unconventional correlation structures where graph theory on vertex coloring plays an important role

incorporation of linear multipoint constraints in substructure based iterative solvers part 1 a numerically scalable algorithm we consider the iterative solution by a class of substructuring methods of the large scale systems of equations arising from the nite element discretization of structural models with an arbitrary set of linear multipoint constraints we present a methodology for generalizing to such problems numerically scalable substructure based iterative solvers without interfering with their formulations and their well established local and global preconditioners we apply this methodology to the feti method and show that the resulting algorithm is numerically scalable with respect to both the substructure and problem sizes 1998 john wiley sons ltd

a comparison of deflation and the balancing preconditioner in this paper we compare various preconditioners for the numerical solution of partial differential equations we compare the well known balancing preconditioner used in domain decomposition methods with a so called deflation preconditioner we prove that the effective condition number of the deflated preconditioned system is always i e for all deflation vectors and all restrictions and prolongations below the condition number of the system preconditioned by the balancing preconditioner even more we establish that both preconditioners lead to almost the same spectra the zero eigenvalues of the deflation preconditioned system are replaced by eigenvalues which are one if the balancing preconditioner is used moreover we prove that the a norm of the errors of the iterates built by the deflation preconditioner is always below the a norm of the errors of the iterates built by the balancing preconditioner depending on the implementation of the balancing preconditioner the amount of work of one iteration of the deflation preconditioned system is less than or equal to the amount of work of one iteration of the balancing preconditioned system if the amount of work is equal both preconditioners are sensitive with respect to inexact computations finally we establish that the deflation preconditioner and the balancing preconditioner produce the same iterates if one uses certain starting vectors numerical results for porous media flows emphasize the theoretical results

sparse pca via covariance thresholding in sparse principal component analysis we are given noisy observations of a low rank matrix of dimension n p and seek to reconstruct it under additional sparsity assumptions in particular we assume here each of the principal components v1 vr has at most s0 non zero entries we are particularly interested in the high dimensional regime wherein p is comparable to or even much larger than n in an in uential paper johnstone and lu 2004 introduced a simple algorithm that estimates the support of the principal vectors v1 vr by the largest entries in the diagonal of the empirical covariance this method can be shown to identify the correct support with high probability if s0 k1 n log p and to fail with high probability if s0 k2n log p for two constants 0 k1 k2 despite a considerable amount of work over the last ten years no practical algorithm exists with provably better support recovery guarantees here we analyze a covariance thresholding algorithm that was recently proposed by krauthgamer nadler vilenchik et al 2015 on the basis of numerical simulations for the rank one case these authors conjectured that covariance thresholding correctly recover the support with high probability for s0 k n assuming n of the same order as p we prove this conjecture and in fact establish a more general guarantee including higher rank as well as n much smaller than p recent lower bounds berthet and rigollet 2013 ma and wigderson 2015 suggest that no polynomial time algorithm can do significantly better the key technical component of our analysis develops new bounds on the norm of kernel random matrices in regimes that were not considered before using these we also derive sharp bounds for estimating the population covariance and the principal component with l2 loss

a monolithic energy conserving method to couple heterogeneous time integrators with incompatible time steps in structural dynamics a new hybrid multi time method for multi time scales structural dynamics simulations is described a monolithic method in a schur dual domain decomposition framework is proposed and allows to consider heterogeneous time integrators with their own time discretization and possible large ratio between the time steps for each subdomain in the proposed method zero numerical dissipation is ensured at the interface this implies that the global stability of the coupling method is governed by the stability of each time integrator without influence of the interface for that purpose velocity continuity is ensured in a weak sense at the interfaces and time integrators newmark hht simo krenk verlet are introduced in a unified framework incremental velocity formulation furthermore dynamics governing equations are introduced from a weak formulation in time in other words equilibrium equation is no more ensured in a strong sense at a given time step but rather on average on a time interval some numerical examples illustrate the efficiency and the robustness of the proposed method for ratio of time scales close to 1000 without any numerical dissipation at the interfaces

an approach to the connection between subdomains with non matching meshes for transient mechanical analysis this paper describes a general method for coupling non matching linear finite element meshes in transient dynamic analysis we propose a method based on schur s dual formulation whose main advantage is to provide equilibrium as well as kinematic continuity throughout the interface the essence of our work lies in the particular discretization of the space of lagrange multipliers and in the validation of the method through two and three dimensional static calculations as well as two dimensional dynamic calculations an example is also presented and the results are compared to those of the mortar method copyright 2002 john wiley sons ltd

multi scale domain decomposition method for large scale structural analysis with a zooming technique application to plate assembly this article is concerned with a multi scale domain decomposition method ddm based on the feti dp solver for large scale structural elastic analysis and suited to problems that exhibit structural heterogeneities such as plate assemblies in the presence of structural details in this approach once a partition of the global fine mesh into subdomains has been performed all subdomains possess a fine mesh and to optimize the computational time the fine mesh is preserved only in the zones of interest with local phenomena due to discontinuity hole etc while the remaining subdomains are replaced by numerical homogenized coarse elements indeed the multi scale aspect is introduced by the description of subdomains with either a fine or a coarse scale mesh as a result an extension of the feti dp ddm is proposed in this article called herein feti dp micro macro that allows the simultaneous usage of different discretizations fine microscopic mesh for subdomains in zones of interest and coarse macroscopic or homogenized mesh for the complementary part of the structure using this strategy raises the problem of the determination of the stiffness of coarse subdomains and of the incompatible finite element connection between fine and coarse subdomains two approaches collocation and mortar are presented and compared the article ends with patch tests and some numerical examples in 2d and 3d the obtained numerical results exemplify the efficiency and capability of the feti dp micro macro approach and reveal that the mortar approach is more accurate at constant cost than the collocation approach copyright 2009 john wiley sons ltd

superconvergent discontinuous galerkin methods for second order elliptic problems we identify discontinuous galerkin methods for second order elliptic problems in several space dimensions having superconvergence properties similar to those of the raviart thomas and the brezzi douglas marini mixed methods these methods use polynomials of degree k 0 for both the potential as well as the flux we show that the approximate flux converges in l2 with the optimal order of k 1 and that the approximate potential and its numerical trace superconverge in l2 like norms to suitably chosen projections of the potential with order k 2 we also apply element by element postprocessing of the approximate solution to obtain new approximations of the flux and the potential the new approximate flux is proven to have normal components continuous across inter element boundaries to converge in l2 with order k 1 and to have a divergence converging in l2 also with order k 1 the new approximate potential is proven to converge with order k 2 in l2 numerical experiments validating these theoretical results are presented

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

minimax risk over hyperrectangles and implications consider estimating the mean of a standard gaussian shift when that mean is known to lie in an orthosymmetric quadratically convex set in l 2 such sets include ellipsoids hyperrectangles and l p bodies with p 2 the minimax risk among linear estimates is within 25 of the minimax risk among all estimates the minimax risk among truncated series estimates is within a factor 4 44 of the minimax risk this implies that the difficulty of estimation a statistical quantity is measured fairly precisely by the n widths a geometric quantity if the set is not quadratically convex as in the case of l p bodies with p 2 things change appreciably minimax linear estimators may be out performed arbitrarily by nonlinear estimates the ordinary kolmogorov n widths still determine the difficulty of linear estimation but the difficulty of nonlinear estimation is tied to the inner bernstein n widths which can be far smaller essential use is made of a new heuristic that the difficulty of the hardest rectangular subproblem is equal to the difficulty of the full problem

unbiased estimation of a sparse vector in white gaussian noise the problem studied in this paper is unbiased estimation of a sparse nonrandom vector corrupted by additive white gaussian noise it is shown that while there are infinitely many unbiased estimators for this problem none of them has uniformly minimum variance therefore the focus is placed on locally minimum variance unbiased lmvu estimators simple closed form lower and upper bounds on the variance of lmvu estimators or equivalently on the barankin bound bb are derived these bounds allow an estimation of the threshold region separating the low signal to noise ratio snr and high snr regimes and they indicate the asymptotic behavior of the bb at high snr in addition numerical lower and upper bounds are derived these are tighter than the closed form bounds and thus characterize the bb more accurately numerical studies compare the proposed characterizations of the bb with established biased estimation schemes and demonstrate that while unbiased estimators perform poorly at low snr they may perform better than biased estimators at high snr an interesting conclusion of this analysis is that the high snr behavior of the bb depends solely on the value of the smallest nonzero entry of the sparse vector and that this type of dependence is also exhibited by the performance of certain practical estimators

numerical implementation of two nonconforming finite element methods for unilateral contact we consider the finite element approximation of the unilateral contact problem between elastic bodies we are interested in a practical problem which often occurs in finite element computations concerning two independently discretized bodies in unilateral contact it follows that the nodes of both bodies located on the contact surface do not fit together we present two different approaches in order to define unilateral contact on nonmatching meshes the first is an extension of the mortar finite element method to variational inequalities that defines the contact in a global way on the contrary the second one expresses local node on segment contact conditions in both cases the theoretical approximation properties are given then we implement and compare the two methods

model selection and sharp asymptotic minimaxity we obtain sharp minimax results for estimation of an n dimensional normal mean under quadratic loss the estimators are chosen by penalized least squares with a penalty that grows like ck log n k for k equal to the number of nonzero elements in the estimating vector for a wide range of sparse parameter spaces we show that the penalized estimator achieves the exact minimax rate with the correct multiplication constant if and only if c equals 2 our results unify the theory obtained by many other authors for penalized estimation of normal means in particular we establish that a conjecture by abramovich et al ann stat 34 584 653 2006 is true

stabilized lagrange multiplier methods for bilateral elastic contact with friction in most finite element fe codes contact is checked only at the nodes corresponding to the use of pointwise constraints however this approach might not be stable in case the bodies coming into contact have non matching grids at the contact interface to alleviate this problem we propose a stabilized lagrange multiplier method based on a global polynomial multiplier for the finite element solution of non linear elastic contact problems with non matching grids in particular our approach allows us to avoid integrating products of different finite element basis functions on the surface meshes at the contact zone

expansion around half integer values binomial sums and inverse binomial sums i consider the expansion of transcendental functions in a small parameter around rational numbers this includes in particular the expansion around half integer values i present algorithms which are suitable for an implementation within a symbolic computer algebra system the method is an extension of the technique of nested sums the algorithms allow in addition the evaluation of binomial sums inverse binomial sums and generalizations thereof

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

resurrecting the asymptotics of linear recurrences abstract once on the forefront of mathematical research in america the asymptotics of the solutions of linear recurrence equations is now almost forgotten especially by the people who need it most namely combinatorists and computer scientists here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her or his asymptotics tool kit

minimax risk over l p balls for l p error consider estimating the mean vector from datan n 2 i withl q norm loss q 1 when is known to lie in ann dimensionall p ball p 0 for largen the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp q obvious exceptions aside the limiting ratio equals 1 only ifp q 2 our arguments are mostly indirect involving a reduction to a univariate bayes minimax problem whenp q simple non linear co ordinatewise threshold rules are asymptotically minimax at small signal to noise ratios and within a bounded factor of asymptotic minimaxity in general we also give asymptotic evaluations of the minimax linear risk our results are basic to a theory of estimation in besov spaces using wavelet bases to appear elsewhere

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

on the choice of smoothing parameter threshold and truncation in nonparametric regression by non linear wavelet methods summary concise asymptotic theory is developed for non linear wavelet estimators of regression means in the context of general error distributions general designs general normalizations in the case of stochastic design and non structural assumptions about the mean the influence of the tail weight of the error distribution is addressed in the setting of choosing threshold and truncation parameters mainly the tail weight is described in an extremely simple way by a moment condition previous work on this topic has generally imposed the much more stringent assumption that the error distribution be normal different approaches to correction for stochastic design are suggested these include conventional kernel estimation of the design density in which case the interaction between the smoothing parameters of the non linear wavelet estimator and the linear kernel method is described

general maximum likelihood empirical bayes estimation of normal means we propose a general maximum likelihood empirical bayes gmleb method for the estimation of a mean vector based on observations with i i d normal errors we prove that under mild moment conditions on the unknown means the average mean squared error mse of the gmleb is within an infinitesimal fraction of the minimum average mse among all separable estimators which use a single deterministic estimating function on individual observations provided that the risk is of greater order than log n 5 n we also prove that the gmleb is uniformly approximately minimax in regular and weak l p balls when the order of the length normalized norm of the unknown means is between log n k1 n 1 p 2 and n log n k2 simulation experiments demonstrate that the gmleb outperforms the james stein and several state of the art threshold estimators in a wide range of settings without much down side

multi time step explicit implicit method for non linear structural dynamics we present a method with domain decomposition to solve time dependent non linear problems this method enables arbitrary numeric schemes of the newmark family to be coupled with different time steps in each subdomain this coupling is achieved by prescribing continuity of velocities at the interface we are more specifically interested in the coupling of implicit explicit numeric schemes taking into account material and geometric non linearities the interfaces are modelled using a dual schur formulation where the lagrange multipliers represent the interfacial forces unlike the continuous formulation the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements velocities and accelerations at the interfaces we show that within the framework of the newmark family of numeric schemes continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged to prove this stability we use an energy method i e a global method over the whole time interval in order to verify the algorithms properties then we propose to extend this to non linear situations in the following cases implicit linear explicit non linear explicit non linear explicit non linear and implicit non linear explicit non linear finally we present some examples showing the feasibility of the method copyright 2001 john wiley sons ltd

recovering edges in ill posed inverse problems optimality of curvelet frames we consider a model problem of recovering a function f x 1 x 2 from noisy radon data the function f to be recovered is assumed smooth apart from a discontinuity along a c 2 curve that is an edge we use the continuum white noise model with noise level varepsilon traditional linear methods for solving such inverse problems behave poorly in the presence of edges qualitatively the reconstructions are blurred near the edges quantitatively they give in our model mean squared errors mses that tend to zero with noise level varepsilon only as o varepsilon 1 2 as varepsilon to 0 a recent innovation nonlinear shrinkage in the wavelet domain visually improves edge sharpness and improves mse convergence to o varepsilon 2 3 however as we show here this rate is not optimal in fact essentially optimal performance is obtained by deploying the recently introduced tight frames of curvelets in this setting curvelets are smooth highly anisotropic elements ideally suited for detecting and synthesizing curved edges to deploy them in the radon setting we construct a curvelet based biorthogonal decomposition of the radon operator and build curvelet shrinkage estimators based on thresholding of the noisy curvelet coefficients in effect the estimator detects edges at certain locations and orientations in the radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain we prove that the curvelet shrinkage can be tuned so that the estimator will attain within logarithmic factors the mse o varepsilon 4 5 as noise level varepsilon to 0 this rate of convergence holds uniformly over a class of functions which are c 2 except for discontinuities along c 2 curves and except for log terms is the minimax rate for that class our approach is an instance of a general strategy which should apply in other inverse problems we sketch a deconvolution example

variation diminishing transformations a direct approach to total positivity and its statistical applications abstract karlin has shown that the variation diminishing property possessed by totally positive distributions allows a unified and straightforward presentation of many basic properties of hypothesis tests his work in fact reveals that total positivity is equivalent to a suitably formulated variation diminishing property this largely expository article begins with the latter concept and consequently gives a more direct account of the theory this approach avoids the extensive mathematical preliminaries previously required and isolates the more important statistical property

optimal shrinkage of eigenvalues in the spiked covariance model we show that in a common high dimensional covariance model the choice of loss function has a profound effect on optimal estimation in an asymptotic framework based on the spiked covariance model and use of orthogonally invariant estimators we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker that acts elementwise on the sample eigenvalues indeed to each loss function there corresponds a unique admissible eigenvalue shrinker dominating all other shrinkers the shape of the optimal shrinker is determined by the choice of loss function and crucially by inconsistency of both eigenvalues and eigenvectors of the sample covariance matrix details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature including the stein entropy divergence frechet bhattacharya matusita frobenius norm operator norm nuclear norm and condition number losses

nonparametric reconstruction of a multifractal function from noisy data we estimate a real valued function f of d variables subject to additive gaussian perturbation at noise level varepsilon 0 under l loss for 1 the main novelty is that f can have an extremely varying local smoothness exhibiting a so called multifractal behaviour the results of jaffard on the frisch parisi conjecture suggest to link the singularity spectrum of f to besov properties of the signal that can be handled by wavelet thresholding for denoising purposes we prove that the optimal minimax rate of estimation of multifractal functions with singularity spectrum d h has explicit representation varepsilon 2v d bullet pi with v d bullet pi min h frac h left d d h right pi 2h d the minimum is taken over a specific domain and the rate is corrected by logarithmic factors in some cases in particular the usual rate varepsilon 2s 2s d is retrieved for monofractal functions with spectrum reduced to a single value s irrespectively of more interestingly the sparse case of estimation over single besov balls has a new interpretation in terms of multifractal analysis

a linear wavelet filter for parametric imaging with dynamic pet describes a new filter for parametric images obtained from dynamic positron emission tomography pet studies the filter is based on the wavelet transform following the heuristics of a previously published method that are here developed into a rigorous theoretical framework it is shown that the space time problem of modeling a dynamic pet sequence reduces to the classical one of estimation of a normal multivariate vector of independent wavelet coefficients that under least squares risk can be solved by straightforward application of well established theory from the study of the distribution of wavelet coefficients of pet images it is inferred that a james stein linear estimator is more suitable for the problem than traditional nonlinear procedures that are incorporated in standard wavelet filters this is confirmed by the superior performance of the james stein filter in simulation studies compared to a state of the art nonlinear wavelet filter and a nonstationary filter selected from literature finally the formal framework is interpreted for the practitioner s point of view and advantages and limitations of the method are discussed

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

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

wavelet shrinkage of a noisy dynamical system with non linear noise impact by filtering wavelet coefficients it is possible to construct a good estimate of a pure signal from noisy data especially for a simple linear noise influence donoho and johnstone 1994 have already defined an optimal filter design in the sense of a minimization of the error made when estimating the pure signal we set here a different framework where the influence of the noise is non linear in particular we propose a method to filter the wavelet coefficients of a discrete dynamical system disrupted by a weak noise in order to construct good estimates of the pure signal including bayes estimate minimax estimate oracular estimate or thresholding estimate we present the example of a logistic and a lorenz chaotic dynamical system as well as an adaptation of our technique in order to show empirically the robustness of the thresholding method in presence of leptokurtic noise moreover we test both the hard and the soft thresholding and also another kind of smoother thresholding which seems to have almost the same reconstruction power as the hard thresholding finally besides the tests on an estimated dataset the method is tested on financial data oil prices and nok usd exchange rate

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

nonparametric estimation of transfer functions rates of convergence and adaptation the paper deals with estimating the transfer functions of stable linear time invariant systems under stochastic assumptions we adopt a nonparametric minimax approach for measuring the estimation accuracy the quality of an estimator is measured by its worst case error over a family of transfer functions the families with polynomially and exponentially decaying impulse response sequences are considered we establish nonasymptotic upper bounds on the accuracy of the least squares estimator for finite impulse response approximation it is shown that the attainable estimation accuracy is determined essentially by the rate at which the true impulse response tends to zero lower bounds on the estimation accuracy are presented an adaptive estimator which does not exploit any a priori information about the true system is developed

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

higher criticism thresholding optimal feature selection when useful features are rare and weak in important application fields today genomics and proteomics are examples selecting a small subset of useful features is crucial for success of linear classification analysis we study feature selection by thresholding of feature z scores and introduce a principle of threshold selection based on the notion of higher criticism hc for i 1 2 p let i denote the two sided p value associated with the ith feature z score and i denote the ith order statistic of the collection of p values the hc threshold is the absolute z score corresponding to the p value maximizing the hc objective i p i i p 1 i p we consider a rare weak rw feature model where the fraction of useful features is small and the useful features are each too weak to be of much use on their own hc thresholding hct has interesting behavior in this setting with an intimate link between maximizing the hc objective and minimizing the error rate of the designed classifier and very different behavior from popular threshold selection procedures such as false discovery rate thresholding fdrt in the most challenging rw settings hct uses an unconventionally low threshold this keeps the missed feature detection rate under better control than fdrt and yields a classifier with improved misclassification performance replacing cross validated threshold selection in the popular shrunken centroid classifier with the computationally less expensive and simpler hct reduces the variance of the selected threshold and the error rate of the constructed classifier results on standard real datasets and in asymptotic theory confirm the advantages of hct

minimax risk of matrix denoising by singular value thresholding an unknown m by n matrix x 0 is to be estimated from noisy measurements y x 0 z where the noise matrix z has i i d gaussian entries a popular matrix denoising scheme solves the nuclear norm penalization problem operatorname min x y x f 2 2 lambda x where x denotes the nuclear norm sum of singular values this is the analog for matrices of ell 1 penalization in the vector case it has been empirically observed that if x 0 has low rank it may be recovered quite accurately from the noisy measurement y in a proportional growth framework where the rank r n number of rows m n and number of columns n all tend to infty proportionally to each other r n m n rightarrow rho m n n rightarrow beta we evaluate the asymptotic minimax mse mathcal m rho beta lim m n n rightarrow infty inf lambda sup operatorname rank x leq r n operatorname mse x 0 hat x lambda our formulas involve incomplete moments of the quarter and semi circle laws beta 1 square case and the marcenko pastur law beta 1 nonsquare case for finite m and n we show that mse increases as the nonzero singular values of x 0 grow larger as a result the finite n worst case mse a quantity which can be evaluated numerically is achieved when the signal x 0 is infinitely strong the nuclear norm penalization problem is solved by applying soft thresholding to the singular values of y we also derive the minimax threshold namely the value lambda rho which is the optimal place to threshold the singular values all these results are obtained for general nonsquare nonsymmetric real matrices comparable results are obtained for square symmetric nonnegative definite matrices

minimax estimation of functionals of discrete distributions we propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters and elaborate on the case of discrete distributions where the support size s is unknown and may be comparable with or even much larger than the number of observations n we treat the respective regions where the functional is nonsmooth and smooth separately in the nonsmooth regime we apply an unbiased estimator for the best polynomial approximation of the functional whereas in the smooth regime we apply a bias corrected version of the maximum likelihood estimator mle we illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures 1 the entropy h p sum i 1 s p i ln p i and 2 f alpha p sum i 1 s p i alpha alpha 0 we obtain the minimax l 2 rates for estimating these functionals in particular we demonstrate that our estimator achieves the optimal sample complexity n asymp s ln s for entropy estimation we also demonstrate that the sample complexity for estimating f alpha p 0 is n asymp s 1 alpha ln s which can be achieved by our estimator but not the mle for 1 we show the minimax l 2 rate for estimating f alpha p is n ln n 2 alpha 1 for infinite support size while the maximum l 2 rate for the mle is n 2 alpha 1 for all the above cases the behavior of the minimax rate optimal estimators with n samples is essentially that of the mle plug in rule with n ln n samples which we term effective sample size enlargement we highlight the practical advantages of our schemes for the estimation of entropy and mutual information we compare our performance with various existing approaches and demonstrate that our approach reduces running time and boosts the accuracy moreover we show that the minimax rate optimal mutual information estimator yielded by our framework leads to significant performance boosts over the chow liu algorithm in learning graphical models the wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable

minimax estimation of the mean of a normal distribution when the parameter space is restricted if x is a n theta 1 random variable let rho m be the minimax risk for estimation with quadratic loss subject to theta leq m then rho m 1 pi 2 m 2 o m 2 we exhibit estimates which are asymptotically minimax to this order as well as approximations to the least favorable prior distributions the approximate least favorable distributions correct to order m 2 have density m 1 cos 2 big frac pi 2m s big s leq m rather than the naively expected uniform density on lbrack m m rbrack we also show how our results extend to estimation of a vector mean and give some explicit solutions

minimax sparse principal subspace estimation in high dimensions we study sparse principal components analysis in high dimensions where p the number of variables can be much larger than n the number of observations and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix we prove optimal non asymptotic lower and upper bounds on the minimax subspace estimation error under two different but related notions of lq subspace sparsity for 0 q 1 our upper bounds apply to general classes of covariance matrices and they show that lq constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions 1 introduction principal components analysis pca was introduced in the early 20th century pearson 1901 hotelling 1933 and is arguably the most well known and widely used technique for dimension reduction it is part of the mainstream statistical repertoire and is routinely used in numerous and diverse areas of application however contemporary applications often involve much higher dimensional data than envisioned by the early developers of pca in such high dimensional situations where the number of variables p is of the same order or much larger than the number of observations n serious difficulties emerge standard pca can produce inconsistent estimates of the principal directions of variation and lead to unreliable conclusions johnstone and lu 2009 paul 2007 nadler 2008 the principal directions of variation correspond to the eigenvectors of the covariance matrix and in high dimensions consistent estimation of the eigenvectors is generally not possible without additional assumptions about the covariance matrix or its eigenstructure much of the recent development in pca has focused on methodology that applies the concept of sparsity to eigenvector estimation some examples include jolliffe trendafilov and uddin

iterated binomial sums and their associated iterated integrals we consider finite iterated generalized harmonic sums weighted by the binomial 2kk in numerators and denominators a large class of these functions emerges in the calculation of massive feynman diagrams with local operator insertions starting at 3 loop order in the coupling constant and extends the classes of the nested harmonic generalized harmonic and cyclotomic sums the binomially weighted sums are associated by the mellin transform to iterated integrals over square root valued alphabets the values of the sums for n and the iterated integrals at x 1 lead to new constants extending the set of special numbers given by the multiple zeta values the cyclotomic zeta values and special constants which emerge in the limit n of generalized harmonic sums we develop algorithms to obtain the mellin representations of these sums in a systematic way they are of importance for the derivation of the asymptotic expansion of these sums and their analytic continuation to n c the associated convolution

the massless higher loop two point function we introduce a new method for computing massless feynman integrals analytically in parametric form an analysis of the method yields a criterion for a primitive feynman graph g to evaluate to multiple zeta values the criterion depends only on the topology of g and can be checked algorithmically as a corollary we reprove the result due to bierenbaum and weinzierl that the massless 2 loop 2 point function is expressible in terms of multiple zeta values and generalize this to the 3 4 and 5 loop cases we find that the coefficients in the taylor expansion of planar graphs in this range evaluate to multiple zeta values but the non planar graphs with crossing number 1 may evaluate to multiple sums with 6th roots of unity our method fails for the five loop graphs with crossing number 2 obtained by breaking open the bipartite graph k3 4 at one edge

a time and space mortar method for coupling linear modal subdomains and non linear subdomains in explicit structural dynamics this paper deals with an extension of dynamic subdomain methods it relies on time multi scale algorithms in the particular case of explicit dynamics it introduces new opportunities in terms of proper modeling on each subdomain adapted to its effective loading in order to reduce the calculation time of the simulations treatment of non matching meshes and local modal projection methods are thus especially discussed the efficiency of the method is then illustrated with two large scale examples

asymptotic minimax risk for sup norm loss solution via optimal recovery we study the problem of estimating an unknown function on the unit interval or itsk th derivative with supremum norm loss when the function is observed in gaussian white noise and the unknown function is known only to obey lipschitz smoothness k 0 we discuss an optimization problem associated with the theory ofoptimal recovery although optimal recovery is concerned with deterministic noise chosen by a clever opponent the solution of this problem furnishes the kernel of the minimax linear estimate for gaussian white noise moreover this minimax linear estimator is asymptotically minimax among all estimates we sketch also applications to higher dimensions and to indirect measurement e g deconvolution problems

on micro to macro connections in domain decomposition multiscale methods abstract micro to macro connection techniques constitute a key ingredient in the formulation of multiscale strategies in this contribution several connection methods are explored for the concurrent multiscale analysis of brittle heterogeneous materials particularly these techniques are investigated in a domain decomposition strong coupling multiscale framework the structural component under analysis is partitioned into a number of non overlapping domains and a fine scale resolution is assigned therein in an adaptive manner exploiting a zoom in technique mesh refinement is employed in the domains where crack coalescence and growth take place the original contribution presented in this manuscript consists in the study of different strong and weak locality constraints that connect coarse and fine resolution domains standard collocation and average compatibility are considered and serve as a basis for the development of two new interscale links the influence of different locality constraints is studied in terms of the mechanical response and the error distribution is compared to a full fine scale analysis

harmonic sums and polylogarithms generated by cyclotomic polynomials the computation of feynman integrals in massive higher order perturbative calculations in renormalizable quantum field theories requires extensions of multiply nested harmonic sums which can be generated as real representations by mellin transforms of poincare iterated integrals including denominators of higher cyclotomic polynomials we derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations the analytic continuation of cyclotomic harmonic sums to complex values of n is performed using analytic representations we also consider special values of the cyclotomic harmonic polylogarithms at argument x 1 resp for the cyclotomic harmonic sums at n which are related to colored multiple zeta values deriving various of their relations based on the stuffle and shuffle alge bras and three multiple ar gument relations we also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums basis representations are derived for weight w 1 2 sums up to cyclotomy l 20

feti dp bddc and block cholesky methods the feti dp and bddc algorithms are reformulated using block cholesky factorizations an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations instead of introducing lagrange multipliers to enforce the coarse level primal continuity constraints in these algorithms a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom with the new formulation of these algorithms a simplified proof is provided that the spectra of a pair of feti dp and bddc algorithms with the same set of primal constraints are essentially the same numerical experiments for a two dimensional laplace s equation also confirm this result copyright 2005 john wiley sons ltd