bounds on coarsening rates for the lifschitz slyozov wagner equation this paper is concerned with the large time behavior of solutions to the lifschitz slyozov wagner lsw system of equations point wise in time upper and lower bounds on the rate of coarsening are obtained for solutions with fairly general initial data these bounds complement the time averaged upper bounds obtained by dai and pego and the point wise in time upper and lower bounds obtained by niethammer and velasquez for solutions with initial data close to a self similar solution

synchronization and directed percolation in coupled map lattices we study a synchronization mechanism based on one way coupling of all or nothing type applied to coupled map lattices with several different local rules by analyzing the metric and the topological distance between the two systems we found two different regimes a strong chaos phase in which the transition has a directed percolation character and a weak chaos phase in which the synchronization transition occurs abruptly we are able to derive some analytical approximations for the location of the transition point and the critical properties of the system we propose to use the characteristics of this transition as indicators of the spatial propagation of chaoticity

combinatorial entropy power inequalities a preliminary study of the stam region we initiate the study of the stam region defined as the subset of the positive orthant in r 2 n 1 that arises from considering the entropy powers of subset sums of n independent random vectors in a euclidean space of finite dimension we show that the class of fractionally superadditive set functions provides an outer bound to the stam region resolving a conjecture of barron and madiman on the other hand the entropy power of a sum of independent random vectors is not supermodular in any dimension we also develop some qualitative properties of the stam region showing for instance that its closure is a logarithmically convex cone

first order system least squares for the helmholtz equation this paper develops a multilevel least squares approach for the numerical solution of the complex scalar exterior helmholtz equation this second order equation is first recast into an equivalent first order system by introducing several field variables a combination of scaled l2 and h 1 norms is then applied to the residual of this system to create a least squares functional it is shown that in an appropriate hilbert space the homogeneous part of this functional is equivalent to a squared graph norm that is a product norm on the space of individual variables this equivalence to a norm that decouples the variables means that standard finite element discretization techniques and standard multigrid solvers can be applied to obtain optimal performance however this equivalence is not uniform in the wavenumber k which can signal degrading performance of the numerical solution process as k increases to counter this difficulty we obtain a result that characterizes the error components causing performance degradation we do this by defining a finite dimensional subspace of these components on whose orthogonal complement k uniform equivalence is proved for this functional and an analogous functional that is based only on l2 norms this subspace equivalence motivates a nonstandard multigrid method that attempts to achieve optimal convergence uniformly in k we report on numerical experiments that empirically confirm k uniform optimal performance of this multigrid solver we also report on tests of the error in our discretization that seem to confirm optimal accuracy that is free of the so called pollution effect

efficient nonlinear solvers for laplace beltrami smoothing of three dimensional unstructured grids the laplace beltrami system of nonlinear elliptic partial differential equations has utility in the generation of computational grids on complex and highly curved geometry discretization of this system using the finite element method accommodates unstructured grids but generates a large sparse ill conditioned system of nonlinear discrete equations the use of the laplace beltrami approach particularly in large scale applications has been limited by the scalability and efficiency of solvers this paper addresses this limitation by developing two nonlinear solvers based on the jacobian free newton krylov jfnk methodology a key feature of these methods is that the jacobian is not formed explicitly for use by the underlying linear solver iterative linear solvers such as the generalized minimal residual gmres method do not technically require the stand alone jacobian instead its action on a vector is approximated through two nonlinear function evaluations the preconditioning required by gmres is also discussed two different preconditioners are developed both of which employ existing algebraic multigrid amg methods further the most efficient preconditioner overall for the problems considered is based on a picard linearization numerical examples demonstrate that these solvers are significantly faster than a standard newton krylov approach a speedup factor of approximately 26 was obtained for the picard preconditioner on the largest grids studied here in addition these jfnk solvers exhibit good algorithmic scaling with increasing grid size

an alternative least squares formulation of the navier stokes equations with improved mass conservation the focus of this paper is on incompressible flows in three dimensions modeled by least squares finite element methods lsfem and using a novel reformulation of the navier stokes equations lsfem are attractive because the resulting discrete equations yield symmetric positive definite systems of algebraic equations and the functional provides both a local and global error measure on the other hand it has been documented for existing reformulations that certain types of boundary conditions and high aspect ratio domains can yield very poor mass conservation it has also been documented that improved mass conservation with lsfem can be achieved by strengthening the coupling between the pressure and velocity the new reformulation presented here is demonstrated to provide both improved multigrid convergence rates because it is differentially diagonally dominant and improved mass conservation over existing methods because it increases the pressure velocity coupling along the inflow and outflow boundaries

a new entropy power inequality for integer valued random variables the entropy power inequality epi yields lower bounds on the differential entropy of the sum of two independent real valued random variables in terms of the individual entropies versions of the epi for discrete random variables have been obtained for special families of distributions with the differential entropy replaced by the discrete entropy but no universal inequality is known beyond trivial ones more recently the sumset theory for the entropy function yields a sharp inequality h x x prime h x geq 1 over 2 o 1 when x x prime are independent identically distributed i i d with high entropy this paper provides the inequality h x x prime h x geq g h x where x x prime are arbitrary i i d integer valued random variables and where g is a universal strictly positive function on bbr satisfying g 0 0 extensions to nonidentically distributed random variables and to conditional entropies are also obtained

rate of convergence and edgeworth type expansion in the entropic central limit theorem an edgeworth type expansion is established for the entropy distance to the class of normal distributions of sums of i i d random variables or vectors satisfying minimal moment conditions

shannon s monotonicity problem for free and classical entropy we give a short unified proof of the following theorem valid in the context of both classical probability theory and voiculescu s free probability theory let x j 1 x j n be independent resp freely independent n tuples of random variables let z n p n 1 2 x 1 p x n p be their central limit sums then the entropy resp free entropy of the n tuple z n 1 z n n is a monotone function of n the classical case for n 1 is a celebrated result of artstein ball barthe and naor and our proof is an adaptation and simplification of their argument

monotonic convergence in an information theoretic law of small numbers an entropy increasing to the maximum result analogous to the entropic central limit theorem barron 1986 artstein 2004 is obtained in the discrete setting this involves the thinning operation and a poisson limit monotonic convergence in relative entropy is established for general discrete distributions while monotonic increase of shannon entropy is proved for the special class of ultra log concave distributions overall we extend the parallel between the information theoretic central limit theorem and law of small numbers explored by kontoyiannis 2005 and harremoeumls 2007 2008 2009 ingredients in the proofs include convexity majorization and stochastic orders

gaussian approximations in high dimensional estimation abstract several estimation techniques assume validity of gaussian approximations for estimation purposes interestingly these ensemble methods have proven to work very well for high dimensional data even when the distributions involved are not necessarily gaussian we attempt to bridge the gap between this oft used computational assumption and the theoretical understanding of why this works by employing some recent results on random projections on low dimensional subspaces and concentration inequalities

on the rate of convergence in the entropic central limit theorem we study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition

entropy jumps in the presence of a spectral gap it is shown that if x is a random variable whose density satisfies a poincare inequality and y is an independent copy of x then the entropy of x y p 2 is greater than that of x by a fixed fraction of the entropy gap between x and the gaussian of the same variance the argument uses a new formula for the fisher information of a marginal which can be viewed as a local reverse form of the brunn minkowski inequality in its functional form due to prekopa and leindler

a remark on the alexandrov fenchel inequality abstract in this article we give a complex geometric proof of the alexandrov fenchel inequality without using toric compactifications the idea is to use the legendre transform and develop the brascamp lieb proof of the prekopa theorem new ingredients in our proof include an integration of timorin s mixed hodge riemann bilinear relation and a mixed norm version of hormander s l 2 estimate which also implies a non compact version of the khovanski teissier inequality

multilevel first order system least squares for nonlinear elliptic partial differential equations a fully variational approach is developed for solving nonlinear elliptic equations that enables accurate discretization and fast solution methods the equations are converted to a first order system that is then linearized via newton s method first order system least squares fosls is used to formulate and discretize the newton step and the resulting matrix equation is solved using algebraic multigrid amg the approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse level computation a general theory is developed that confirms the usual full multigrid efficiency accuracy comparable to the finest level discretization is achieved at a cost proportional to the number of finest level degrees of freedom in a companion paper the theory is applied to elliptic grid generation egg and supported by numerical results

entropy inequalities for stable densities and strengthened central limit theorems we consider the central limit theorem for stable laws in the case of the standardized sum of independent and identically distributed random variables with regular probability density function by showing decay of different entropy functionals along the sequence we prove convergence with explicit rate in various norms to a levy centered density of parameter lambda 1 this introduces a new information theoretic approach to the central limit theorem for stable laws in which the main argument is shown to be the relative fractional fisher information recently introduced in toscani ricerche mat 65 1 71 91 2016 in particular it is proven that with respect to the relative fractional fisher information the levy density satisfies an analogous of the logarithmic sobolev inequality which allows to pass from the monotonicity and decay to zero of the relative fractional fisher information in the standardized sum to the decay to zero in relative entropy with an explicit decay rate

entropy bounds on abelian groups and the ruzsa divergence over the past few years a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by ruzsa tao and others motivated by analogous results in additive combinatorics this paper extends these earlier results to the case of random variables taking values in mathbb r n or more generally in arbitrary locally compact and polish abelian groups we isolate and study a key quantity the ruzsa divergence between two probability distributions and we show that its properties can be used to extend the earlier inequalities to the present general setting the new results established include several variations on the theme that the entropies of the sum and the difference of two independent random variables severely constrain each other although the setting is quite general the results are already of interest and new for random vectors in mathbb r n in that special case we discuss quantitative bounds for the stability of the equality conditions in the entropy power inequality a reverse entropy power inequality for log concave random vectors an information theoretic analog of the rogers shephard inequality for convex bodies and consequences of some of our results to determinant inequalities for sums of positive definite matrices moreover by considering various multiplicative subgroups of the complex plane one obtains new inequalities for the differential entropies of products and ratios of nonzero complex valued random variables

a coordinate system for gaussian networks this paper investigates network information theory problems where the external noise is gaussian distributed in particular the gaussian broadcast channel with coherent fading and the gaussian interference channel are considered it is shown that in these problems non gaussian code ensembles can achieve higher rates than the gaussian ones it is also shown that the strong shamai laroia conjecture on the gaussian isi channel does not hold in order to analyze non gaussian code ensembles over gaussian networks a geometrical tool using the hermite polynomials is proposed this tool provides a coordinate system to analyze a class of non gaussian input distributions that are invariant over gaussian networks

estimation in gaussian noise properties of the minimum mean square error consider the minimum mean square error mmse of estimating an arbitrary random variable from its observation contaminated by gaussian noise the mmse can be regarded as a function of the signal to noise ratio snr as well as a functional of the input distribution of the random variable to be estimated it is shown that the mmse is concave in the input distribution at any given snr for a given input distribution the mmse is found to be infinitely differentiable at all positive snr and in fact a real analytic function in snr under mild conditions the key to these regularity results is that the posterior distribution conditioned on the observation through gaussian channels always decays at least as quickly as some gaussian density furthermore simple expressions for the first three derivatives of the mmse with respect to the snr are obtained it is also shown that as functions of the snr the curves for the mmse of a gaussian input and that of a non gaussian input cross at most once over all snrs these properties lead to simple proofs of the facts that gaussian inputs achieve both the secrecy capacity of scalar gaussian wiretap channels and the capacity of scalar gaussian broadcast channels as well as a simple proof of the entropy power inequality in the special case where one of the variables is gaussian

a local proof of the dimensional prekopa s theorem abstract the aim of this paper is to find an expression for second derivative of the function t defined by t v t x d x 1 n n where u r and v r n are open bounded subsets and u v r is a c 2 smooth function as a consequence this result gives us a direct proof of the dimensional prekopa s theorem based on a local approach

high order unstructured curved mesh generation using the winslow equations we propose a method to generate high order unstructured curved meshes using the classical winslow equations we start with an initial straight sided mesh in a reference domain and fix the position of the nodes on the boundary on the true curved geometry in the interior of the domain we solve the winslow equations using a new continuous galerkin finite element discretization this formulation appears to produce high quality curved elements which are highly resistant to inversion in addition the corresponding nonlinear equations can be solved efficiently using picard iterations even for highly stretched boundary layer meshes compared to several previously proposed techniques such as optimization and approaches based on elasticity analogies this can significantly reduce the computational cost while producing curved elements of similar quality we show a number of examples in both two and three space dimensions including complex geometries and stretched boundary layers and demonstrate the high quality of the generated meshes and the performance of the nonlinear solver

reverse brunn minkowski and reverse entropy power inequalities for convex measures we develop a reverse entropy power inequality for convex measures which may be seen as an anegeometric inverse of the entropy power inequality of shannon and stam the specialization of this inequality to log concave measures may be seen as a version of milman s reverse brunn minkowski inequality the proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies and on a study of eective supports of convex measures both of which are of independent interest as well as on milman s deep technology of m ellipsoids and on certain information theoretic inequalities as a by product we also give a

heat flow derivatives and minimum mean square error in gaussian noise we connect recent developments on gaussian noise estimation and the minimum mean square error to earlier results on entropy and fisher information heat flow expansion in particular the derivatives of the minimum mean square error with respect to the noise parameter are related to the heat flow derivatives of the fisher information and a special lie algebra structure on iterated gradients the results lead in particular to a partial answer to the minimum mean square error conjecture

first order system least squares fosls for coupled fluid elastic problems mathematical models for the mechanical coupling between a moving fluid and an elastic solid are inherently nonlinear because the shape of the eulerian fluid domain is not known a priori it is at least partially determined by the displacement of the elastic solid in this paper a first order system least squares finite element formulation is used to solve the nonlinear system of model equations using different iteration techniques including an approach where the equations are fully coupled and two other approaches in which the equations are decoupled the discrete linear system of equations is solved using an algebraic multigrid solver as a preconditioner for a conjugate gradient iteration the numerical results show that the approach is optimal in the sense that computational cost is proportional to the degrees of freedom the results also show that the choice of iteration method from fully coupled to fully decoupled does not significantly effect computational cost but it does influence the error in the solution

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

on mass conserving least squares methods least squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations including symmetric positive definite discrete operators and a sharp error measure one of the potential drawbacks especially in three dimensions is that mass conservation is achieved only in a least squares sense and underresolved solutions are especially vulnerable to poor conservation for the stationary navier stokes equations which are typically rewritten as a larger system of first order equations the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used a new first order system reformulation of the navier stokes equations is presented that admits a wider range of mass conserving boundary conditions this new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems a square channel backward facing step and branching tubes with two generations of bifurcations

gaussian optimizers for entropic inequalities in quantum information we survey the state of the art for the proof of the quantum gaussian optimizer conjectures of quantum information theory these fundamental conjectures state that quantum gaussian input states are the solution to several optimization problems involving quantum gaussian channels these problems are the quantum counterpart of three fundamental results of functional analysis and probability the entropy power inequality the sharp young s inequality for convolutions and the theorem gaussian kernels have only gaussian maximizers quantum gaussian channels play a key role in quantum communication theory they are the quantum counterpart of gaussian integral kernels and provide the mathematical model for the propagation of electromagnetic waves in the quantum regime the quantum gaussian optimizer conjectures are needed to determine the maximum communication rates over optical fibers and free space the restriction of the quantum limited gaussian attenuator to input states diagonal in the fock basis coincide

information inequalities for joint distributions with interpretations and applications upper and lower bounds are obtained for the joint entropy of a collection of random variables in terms of an arbitrary collection of subset joint entropies these inequalities generalize shannon s chain rule for entropy as well as inequalities of han fujishige and shearer a duality between the upper and lower bounds for joint entropy is developed all of these results are shown to be special cases of general new results for submodular functions thus the inequalities presented constitute a richly structured class of shannon type inequalities the new inequalities are applied to obtain new results in combinatorics such as bounds on the number of independent sets in an arbitrary graph and the number of zero error source channel codes as well as determinantal inequalities in matrix theory a general inequality for relative entropies is also developed finally revealing connections of the results to literature in economics computer science and physics are explored

island coalescence using parallel first order system least squares on incompressible resistive magnetohydrodynamics this paper investigates the performance of a parallel newton first order system least squares fosls finite element method with local adaptive refinement and algebraic multigrid amg applied to incompressible resistive magnetohydrodynamics in particular an island coalescence test problem is studied that models magnetic reconnection using a reduced two dimensional 2d model of a tokamak fusion reactor the results show that using an appropriate temporal and spatial resolution these methods are capable of resolving the physical instabilities accurately at small computational cost the time dependent nonlinear system of pdes is solved using work equivalent to about 50 60 simple relaxation sweeps gauss seidel iterations per time step experiments show that unless the time step is sufficiently small nonphysical numerical instabilities may occur further decreasing the time step size does not proportionally increase the cost of the computation because amg convergence is improved in addition

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

synchronization and maximum lyapunov exponents of cellular automata we study the synchronization of totalistic one dimensional cellular automata ca the ca with a nonzero synchronization threshold exhibit complex nonperiodic space time patterns and vice versa this synchronization transition is related to directed percolation we also study the maximum lyapunov exponent for ca defined analogous to continuous dynamical systems as the exponential rate of expansion of the linear map induced by the evolution rule of ca constructed with the aid of the boolean derivatives the synchronization threshold is strongly correlated to the maximum lyapunov exponent and we propose approximate relations between these quantities the value of this threshold can be used to parametrize the space time complexity of ca

on berndtsson s generalization of prekopa s theorem we present a new approach to berndtsson s complex extension of prekopa s theorem this approach is inspired by the recent local proof of prekopa s theorem obtained by ball barthe and naor in the complex setting this approach leads to a formula expressing in terms of derivatives of where e z as a consequence we obtain new conditions ensuring that is subharmonic when is plurisubharmonic

fractal rock slope dynamics anticipating a collapse time series of dilatometric measurements of relative displacements on rock cracks on stable and unstable sandstone slopes were analyzed the inherent dynamics of rock slopes lack any significant nonlinearity however the residuals obtained by removing meteorological influences are fat tailed non gaussian fluctuations with short range correlations in the case of stable slopes the fluctuations of unstable slopes exhibit self affine dynamics of fractional brownian motions with power law long range correlations and are characterized by asymptotic power law probability distributions with decay coefficients outside the range of stable l evy distributions

the b conjecture for the gaussian measure of dilates of symmetric convex sets and related problems we show that given a symmetric convex set k rd the function t etk is log concave on r where denotes the standard d dimensional gaussian measure we also comment on the extension of this property to unconditional log concave measures and sets and on the complex case

dimensional variance inequalities of brascamp lieb type and a local approach to dimensional prekopaʼs theorem abstract we give a new approach inspired by hormander s l 2 method to weighted variance inequalities which extend results obtained by bobkov and ledoux it provides in particular a local proof of the dimensional functional forms of the brunn minkowski inequalities we also present several applications of these variance inequalities including reverse holder inequalities for convex functions weighted brascamp lieb inequalities and sharp weighted poincare inequalities for generalized cauchy measures

maximal correlation and the rate of fisher information convergence in the central limit theorem we consider the behaviour of the fisher information of scaled sums of independent and identically distributed random variables in the central limit theorem regime we show how this behaviour can be related to the second largest non trivial eigenvalue of the operator associated with the hirschfeld gebelein renyi maximal correlation we prove that assuming this eigenvalue satisfies a strict inequality an o 1 n rate of convergence and a strengthened form of monotonicity hold

on the entropy of compound distributions on nonnegative integers some entropy comparison results are presented concerning compound distributions on nonnegative integers the main result shows that under a log concavity assumption two compound distributions are ordered in terms of shannon entropy if both the ldquonumbers of claimsrdquo and the ldquoclaim sizesrdquo are ordered accordingly in the convex order several maximum minimum entropy theorems follow as a consequence most importantly two recent results of johnson 2008 on maximum entropy characterizations of compound poisson and compound binomial distributions are proved under fewer assumptions and with simpler arguments

monotonicity thinning and discrete versions of the entropy power inequality we consider the entropy of sums of independent discrete random variables in analogy with shannon s entropy power inequality where equality holds for normals in our case infinite divisibility suggests that equality should hold for poisson variables we show that some natural analogues of the epi do not in fact hold but propose an alternative formulation which does always hold the key to many proofs of shannon s epi is the behavior of entropy on scaling of continuous random variables we believe that renyi s operation of thinning discrete random variables plays a similar role to scaling and give a sharp bound on how the entropy of ultra log concave random variables behaves on thinning in the spirit of the monotonicity results established by artstein ball barthe and naor we prove a stronger version of concavity of entropy which implies a strengthened form of our discrete epi

invariances in variance estimates we provide variants and improvements of the brascamp lieb variance inequality which takes into account the invariance properties of the underlying measure this is applied to spectral gap estimates for log concave measures with many symmetries and to non interacting conservative spin systems

on matrix valued log concavity and related prekopa and brascamp lieb inequalities abstract we propose a new self contained approach to h raufi s extension of prekopa s theorem for matrix valued log concave functions along the way new related inequalities are established in particular a brascamp lieb variance inequality for matrix weights

solution of shannon s problem on the monotonicity of entropy it is shown that if x1 x2 are independent and identically distributed square integrable random variables then the entropy of the normalized sum ent x1 xn over n is an increasing function of n this resolves an old problem which goes back to 6 7 5 the result also has a version for non identically distributed random variables or random vectors

on the entropy power inequality for the renyi entropy of order 0 1 using a sharp version of the reverse young inequality and a renyi entropy comparison result due to fradelizi madiman and wang 2016 the authors derive renyi entropy power inequalities for log concave random vectors when renyi parameters belong to 0 1 furthermore the estimates are shown to be sharp up to absolute constants

monotonic decrease of the non gaussianness of the sum of independent random variables a simple proof artstein ball barthe and naor have recently shown that the non gaussianness divergence with respect to a gaussian random variable with identical first and second moments of the sum of independent and identically distributed i i d random variables is monotonically nonincreasing we give a simplified proof using the relationship between non gaussianness and minimum mean square error mmse in gaussian channels as artstein we also deal with the more general setting of nonidentically distributed random variables

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

the entropy power inequality and mrs gerber s lemma for groups of order 2 n shannon s entropy power inequality can be viewed as characterizing the minimum differential entropy achievable by the sum of two independent random variables with fixed differential entropies the entropy power inequality has played a key role in resolving a number of problems in information theory it is therefore interesting to examine the existence of a similar inequality for discrete random variables in this paper we obtain an entropy power inequality for random variables taking values in a group of order 2 n i e for such a group g we explicitly characterize the function f g x y giving the minimum entropy of the sum of two independent g valued random variables with respective entropies x and y random variables achieving the extremum in this inequality are thus the analogs of gaussians in this case and these are also determined it turns out that f g x y is convex in x for fixed y and by symmetry convex in y for fixed x this is a generalization to groups of order 2 n of the result known as mrs gerber s lemma

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

the relationship between causal and noncausal mismatched estimation in continuous time awgn channels a continuous time finite power process with distribution p is observed through an awgn channel at a given signal to noise ratio snr and is estimated by an estimator that would have minimized the mean square error if the process had distribution q we show that the causal filtering mean square error mse achieved at snr level snr is equal to the average value of the noncausal smoothing mse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr emerging as the bridge for equating these two quantities are mutual information and relative entropy our result generalizes that of guo shamai and verdu 2005 from the nonmismatched case where p q to general p and q among our intermediate results is an extension of duncan s theorem that relates mutual information and causal mmse to the case of mismatched estimation some further extensions and implications are discussed key to our findings is the recent result of verdu on mismatched estimation and relative entropy

relative entropy of cone measures and lp centroid bodies let k be a convex body in r n we introduce a new affine invariant which we call k t hat can be found in three different ways a as a limit of normalized lp affine surface areas b as the relative entropy of the cone measure of k and the cone measure of k c as the limit of the volume difference of k and lp centroid bodies we investigate properties of k and of related new invariant quantities in particular we show new affine isoperimetric inequalities and we show an information inequality for convex bodies

fisher information and the central limit theorem an edgeworth type expansion is established for the relative fisher information distance to the class of normal distributions of sums of i i d random variables satisfying moment conditions the validity of the central limit theorem is studied via properties of the fisher information along convolutions

a free analogue of shannon s problem on monotonicity of entropy abstract we prove a free probability analog of a result of s artstein k ball f barthe a naor solution of shannon s problem on monotonicity of entropy j amer math soc 17 2004 975 982 in particular we prove that if x 1 x 2 are freely independent identically distributed random variables then the function n x 1 x n n is monotone increasing for all n

entropy and the fourth moment phenomenon we develop a new method for bounding the relative entropy of a random vector in terms of its stein factors our approach is based on a novel representation for the score function of smoothly perturbed random variables as well as on the de bruijn s identity of information theory when applied to sequences of functionals of a general gaussian field our results can be combined with the carbery wright inequality in order to yield multidimensional entropic rates of convergence that coincide up to a logarithmic factor with those achievable in smooth distances such as the 1 wasserstein distance in particular our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components with explicit rates of convergence in total variation that are independent of the order of the associated wiener chaoses the results proved in the present paper are outside the scope of other existing techniques such as for instance the multidimensional stein s method for normal approximations

multilevel first order system least squares for elliptic grid generation a new fully variational approach is studied for elliptic grid generation egg it is based on a general algorithm developed in a companion paper a l codd t a manteuffel and s f mccormick siam j numer anal 41 2003 pp 2197 2209 that involves using newton s method to linearize an appropriate equivalent first order system first order system least squares fosls to formulate and discretize the newton step and algebraic multigrid amg to solve the resulting matrix equation the approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse level computation the present paper verifies the assumptions of the companion work and confirms the overall efficiency of the scheme with numerical experiments

information theoretic proofs of entropy power inequalities while most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information up to now shannon s entropy power inequality epi is an exception existing information theoretic proofs of the epi hinge on representations of differential entropy using either fisher information or minimum mean square error mmse which are derived from de bruijn s identity in this paper we first present an unified view of these proofs showing that they share two essential ingredients 1 a data processing argument applied to a covariance preserving linear transformation 2 an integration over a path of a continuous gaussian perturbation using these ingredients we develop a new and brief proof of the epi through a mutual information inequality which replaces stam and blachman s fisher information inequality fii and an inequality for mmse by guo shamai and verdu used in earlier proofs the result has the advantage of being very simple in that it relies only on the basic properties of mutual information these ideas are then generalized to various extended versions of the epi zamir and feder s generalized epi for linear transformations of the random variables takano and johnson s epi for dependent variables liu and viswanath s covariance constrained epi and costa s concavity inequality for the entropy power