experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

isoperimetry and integrability of the sum of independent banach space valued random variables we develop a new method to study the tails of a sum of independent mean zero banach space valued random variables x i i leq n it relies on a new isoperimetric inequality for subsets of a product of probability spaces in particular we prove that for p geq 1 bigg sum i leq n x i bigg p leq frac kp 1 log p bigg bigg sum i leq n x i bigg 1 max i leq n x i p bigg where k is a universal constant other optimal inequalities for exponential moments are obtained

transference principles for log sobolev and spectral gap with applications to conservative spin systems we obtain new principles for transferring log sobolev and spectral gap inequalities from a source metric measure space to a target one when the curvature of the target space is bounded from below as our main application we obtain explicit estimates for the log sobolev and spectral gap constants of various conservative spin system models consisting of non interacting and weakly interacting particles constrained to conserve the mean spin when the self interaction is a perturbation of a strongly convex potential this partially recovers and partially extends previous results of caputo chafai grunewald landim lu menz otto panizo villani westdickenberg and yau when the self interaction is only assumed to be non strongly convex as in the case of the two sided exponential measure we obtain sharp estimates on the system s spectral gap as a function of the mean spin independently of the size of the system

best constants in martingale version of rosenthal s inequality the following generalization of rosenthal s inequality was proved by burkholder a 1 p s f p d ast p leq f ast p leq b p s f p d ast p for all martingales f n it is known that a p grows like sqrt p as p rightarrow infty in this paper we prove that the growth rate of b p as p rightarrow infty is p ln p

estimates of moments and tails of gaussian chaoses we derive two sided estimates on moments and tails of gaussian chaoses that is random variables of the form ai1 idgi1 gid where gi are i i d mathcal n 0 1 r v s estimates are exact up to constants depending on d only

the asymptotic distributions of generalized u statistics with applications to random graphs we consider the random variable s n v f sum limits i 1 i v leqq n f x i 1 x i v y i 1 i 2 y i v 1 i v

decoupling inequalities for the tail probabilities of multivariate u statistics in this paper we present a decoupling inequality that shows that multivariate u statistics can be studied as sums of conditionally independent random variables this result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration more precisely we get the following result let x j be a sequence of independent random variables on a measurable space mathscr j s and let x j i j 1 ldots k be k independent copies of x i let f i 1i 2 ldots i k be families of functions of k variables taking s times cdots times s into a banach space b cdots then for all n geq k geq 2 t 0 there exist numerical constants c k depending on k only so that p bigg big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 1 i 2 ldots x 1 i k big geq t bigg leq c kp bigg c k big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 2 i 2 ldots x k i k big geq t bigg the reverse bound holds if in addition the following symmetry condition holds almost surely f i 1i 2 cdots i k x i 1 x i 2 ldots x i k f i pi 1 i pi 2 cdots i pi k x i pi 1 x i pi 2 ldots x i pi k for all permutations pi of 1 ldots k

on asymptotic behavior of multilinear eigenvalue statistics of random matrices we prove the law of large numbers and the central limit theorem for analogs of u and v von mises statistics of eigenvalues of random matrices as their size tends to infinity we show first that for a certain class of test functions kernels determining the statistics the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics we then check the conditions of the reduction statements for several most known ensembles of random matrices the reduction phenomenon is well known in statistics dealing with i i d random variables it is of interest that an analogous phenomenon is also the case for random matrices whose eigenvalues are strongly dependent even if the entries of matrices are independent

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

noncommutative martingale deviation and poincare type inequalities with applications we prove a deviation inequality for noncommutative martingales by extending oliveira s argument for random matrices by integration we obtain a burkholder type inequality with satisfactory constant using continuous time we establish noncommutative poincare type inequalities for nice semigroups with a positive curvature condition these results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to bobkov and gotze in the commutative case to demonstrate our setting is general enough we give various examples including certain group von neumann algebras random matrices and classical diffusion processes among others

fast and guaranteed blind multichannel deconvolution under a bilinear system model we consider the multichannel blind deconvolution problem where we observe the output of multiple channels that are all excited with the same unknown input from these observations we wish to estimate the impulse responses of each of the channels we show that this problem is well posed if the channels follow a bilinear model where the ensemble of channel responses is modeled as lying in a low dimensional subspace but with each channel modulated by an independent gain under this model we show how the channel estimates can be found by minimizing a quadratic function over a non convex set we analyze two methods for solving this non convex program and provide performance guarantees for each the first is a method of alternating eigenvectors that breaks the program down into a series of eigenvalue problems the second is a truncated power iteration which can roughly be interpreted as a method for finding the largest eigenvector of a symmetric matrix with the additional constraint that it adheres to our bilinear model as with most non convex optimization algorithms the performance of both of these algorithms is highly dependent on having a good starting point we show how such a starting point can be constructed from the channel measurements our performance guarantees are non asymptotic and provide a sufficient condition on the number of samples observed per channel in order to guarantee channel estimates of certain accuracy our analysis uses a model with a generic subspace that is drawn at random and we show the performance bounds hold with high probability mathematically the key estimates are derived by quantifying how well the eigenvectors of certain random matrices approximate the eigenvectors of their mean we also present a series of numerical results demonstrating that the empirical performance is consistent with the presented theory

solving optimization problems with diseconomies of scale via decoupling we present a new framework for solving optimization problems with a diseconomy of scale in such problems our goal is to minimize the cost of resources used to perform a certain task the cost of resources grows superlinearly as xq q 1 with the amount x of resources used we define a novel linear programming relaxation for such problems and then show that the integrality gap of the relaxation is aq where aq is the q th moment of the poisson random variable with parameter 1 using our framework we obtain approximation algorithms for the minimum energy efficient routing minimum degree balanced spanning tree load balancing on unrelated parallel machines and unrelated parallel machine scheduling with nonlinear functions of completion times problems our analysis relies on the decoupling inequality for nonnegative random variables the inequality states that ie1nxi q cq ie1nyi q where xi are independent nonnegative random variables yi are possibly dependent nonnegative random variables and each yi has the same distribution as xi the inequality was proved by de la pena in 1990 de la pena ibragimov and sharakhmetov showed that cq 2 for qi 1 2 and cq aq1 q for q 2 we show that the optimal constant is cqe aq1 q for any q 1 we then prove a more general inequality for every convex function v elv ie1nxi r elv p ie1n yi r and for every concave function el ie1n xi r el p ie1n yi r where p is a poisson random variable with parameter 1 independent of the random variables yi

bounds on the tail probability of u statistics and quadratic forms the final version of this paper appears in bulletin of american mathematical society 31 1994 223 227 print

decoupling inequalities for polynomial chaos let x x 1 ldots x d be a sequence of independent symmetric identically distributed random vectors with independent components the main subject of this paper is the so called decoupling inequalities i e inequalities of the form begin align e phi cq x x ldots x leq e phi q x 1 x 2 ldots x d leq e phi cq x x ldots x end align where q is a symmetric multilinear form with values in a vector space f with all diagonal terms equal to zero and phi is a convex function on f

a new central limit theorem and decomposition for gaussian polynomials with an application to deterministic approximate counting one of the main results of this paper is a new multidimensional central limit theorem clt for multivariate polynomials under gaussian inputs roughly speaking the new clt shows that any collection of gaussian polynomials with small eigenvalues suitably defined must have a joint distribution which is close to a multidimensional gaussian distribution the new clt is proved using tools from malliavin calculus and stein s method a second main result of the paper which complements the new clt is a new decomposition theorem for low degree multilinear polynomials over gaussian inputs roughly speaking this result shows that up to some small error any such polynomial is very close to a polynomial which can be decomposed into a bounded number of multilinear polynomials all of which have extremely small eigenvalues an important feature of this decomposition theorem is the delicate control obtained between the number of polynomials in the decomposition versus their eigenvalues as the main application of these results we give a deterministic algorithm for approximately counting satisfying assignments of a degree d polynomial threshold function ptf over the domain 1 1 n this is a well studied problem from theoretical computer science more precisely given as input a degree d polynomial p x 1 dots x n over mathbb r n and a parameter epsilon 0 the algorithm approximates begin aligned mathop mathbf pr x sim 1 1 n p x ge 0 end aligned to within an additive pm epsilon in time o d epsilon 1 cdot mathrm poly n d since it is np hard to determine whether the above probability is nonzero any sort of efficient multiplicative approximation is almost certainly impossible even for randomized algorithms note that the running time of the algorithm as a function of n d the number of coefficients of a degree d ptf is a fixed polynomial the fastest previous algorithm for this problem kane corr arxiv 1210 1280 2012 based on constructions of unconditional pseudorandom generators for degree d ptfs runs in time n o d c 1 cdot epsilon c for all c 0

the lower and upper p topological p regular modifications for lattice valued convergence spaces in this paper the notions of lower and upper topological regular modifications with respect to a stratified l generalized convergence structure and those with respect to a stratified l strong generalized convergence structure are defined and studied the basic properties and characterizations are presented in particular it is proved that the lower modifications behave reasonably well relative to final structures whereas the upper modifications exhibit comparable behavior relative to initial structures

monotonicity properties of optimal transportation and the fkg and related inequalities optimal transportation between densities f x g y can be interpreted as a joint probability distribution with marginally f x and g y we prove monotonicity and concavity properties of optimal transportation y x under suitable assumptions on f and g as an application we obtain the fortuin kasteleyn ginibre correlation inequalities as well as some generalizations of the brascamp lieb momentum inequalities

on extremal distributions and sharp l sub p bounds for sums of multilinear forms in this paper we present a study of the problem of approximating the expectations of functions of statistics in independent and dependent random variables in terms of the expectations of functions of the component random variables we present results providing sharp analogues of the burkholder rosenthal inequalities and related estimates for the expectations of functions of sums of dependent nonnegative r v s and conditionally symmetric martingale differences with bounded conditional moments as well as for sums of multilinear forms among others we obtain the following sharp inequalities e nk 1xk t 2max nk 1extk nk 1ak t for all nonnegative r v s x1 xn with e xk x1 xk 1 ak extk k 1 n 1 x003c t x003c 2undefined e nk 1xk t e t 1 max nk 1bk nk 1ask t s for all nonnegative r v s x1 xn with e xsk x1 xk 1 ask e xtk x1 xk 1 bk k 1 n 1 x003c t x003c 2undefined 0 x003c s le t 1undefined or t 2 0 x003c s le 1undefined where 1 is a poisson random variable with parameter 1 as applications new decoupling inequalities for sums of multilinear forms are presented and sharp khintchine marcinkiewicz zygmund inequalities for generalized moving averages are obtained the results can also be used in the study of a wide class of nonlinear statistics connected to problems of long range dependence and in an econometric setup in particular in stabilization policy problems and in the study of properties of moving average and autocorrelation processes the results are based on the iteration of a series of key lemmas that capture the essential extremal properties of the moments of the statistics involved

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

concentration inequalities for non lipschitz functions with bounded derivatives of higher order building on the inequalities for homogeneous tetrahedral polynomials in independent gaussian variables due to r latala we provide a concentration inequality for not necessarily lipschitz functions f mathbb r n rightarrow mathbb r with bounded derivatives of higher orders which holds when the underlying measure satisfies a family of sobolev type inequalities begin aligned vert g mathbb eg vert p le c p vert nabla g vert p end aligned such sobolev type inequalities hold e g if the underlying measure satisfies the log sobolev inequality in which case c p le c sqrt p or the poincare inequality then c p le cp our concentration estimates are expressed in terms of tensor product norms of the derivatives of f when the underlying measure is gaussian and f is a polynomial not necessarily tetrahedral or homogeneous our estimates can be reversed up to a constant depending only on the degree of the polynomial we also show that for polynomial functions analogous estimates hold for arbitrary random vectors with independent sub gaussian coordinates we apply our inequalities to general additive functionals of random vectors in particular linear eigenvalue statistics of random matrices and the problem of counting cycles of fixed length in erd s renyi random graphs obtaining new estimates optimal in a certain range of parameters

exact inequalities for sums of asymmetric random variables with applications let rm bs 1 dots rm bs n be independent identically distributed random variables each having the standardized bernoulli distribution with parameter p in 0 1 let m p 1 p 2 p 2 2 sqrt p p 2 4 p 2 if 0 p le frac12 and m p 1 if frac12 le p 1 let m ge m p let f be such a function that f and f are nondecreasing and convex then it is proved that for all nonnegative numbers c 1 dots c n one has the inequality mathsf e f c 1 mathrm b s 1 dots c n mathrm b s n le mathsf e f big s m mathrm b s 1 dots mathrm bs n big where s m big frac1n sum i 1 n c i 2m big frac1 2m the lower bound m p on m is exact for each p in 0 1 moreover operatorname mathsf e f c 1 mathrm b s 1 dots c n mathrm b s n is schur concave in c 1 2m ldots c n 2m

the category of stratified l filter spaces abstract in this paper the concepts of stratified l filter space complete stratified l filter space and symmetric stratified l kent convergence space are introduced it is shown that 1 the category of stratified l filter spaces with cauchy continuous maps is a strong topological universe 2 the category of complete stratified l filter spaces as a bicoreflective subcategory of the category of stratified l filter spaces is isomorphic to the category of symmetric stratified l kent convergence spaces 3 the category of complete stratified l filter spaces as an isomorphism closed full subcategory of the category of stratified l filter spaces is strongly cartesian closed

msp a class of parallel multistep successive sparse approximate inverse preconditioning strategies we develop a class of parallel multistep successive preconditioning strategies to enhance efficiency and robustness of standard sparse approximate inverse preconditioning techniques the key idea is to compute a series of simple sparse matrices to approximate the inverse of the original matrix studies are conducted to show the advantages of such an approach in terms of both improving preconditioning accuracy and reducing computational cost compared to the standard sparse approximate inverse preconditioners numerical experiments using one prototype implementation to solve a few sparse matrices on a distributed memory parallel computer are reported

completion of stratified l m filter tower spaces the notion of stratified l m filter tower spaces is introduced and the resulting category is shown to be a strong topological universe completions of stratified l m filter tower spaces are considered and a sufficient and necessary condition for a stratified l m filter tower space to have a completion is also given it is proved that the reflective modification of completion for a stratified l m cauchy tower space considered as a stratified l m filter tower space is still a completion for this stratified l m cauchy tower space

random multilinear forms on etudie la convergence des formes multilineaires f n n k x n 1 x n k dans des variables aleatoires independantes symetriques

moment inequalities for functions of independent random variables a general method for obtaining moment inequalities for functions of independent random variables is presented it is a generalization of the entropy method which has been used to derive concentration inequalities for such functions boucheron lugosi and massart ann probab 31 2003 1583 1614 and is based on a generalized tensorization inequality due to latala and oleszkiewicz lecture notes in math 1745 2000 147 168 the new inequalities prove to be a versatile tool in a wide range of applications we illustrate the power of the method by showing how it can be used to effortlessly re derive classical inequalities including rosenthal and kahane khinchine type inequalities for sums of independent random variables moment inequalities for suprema of empirical processes and moment inequalities for rademacher chaos and u statistics some of these corollaries are apparently new in particular we generalize talagrand s exponential inequality for rademacher chaos of order 2 to any order we also discuss applications for other complex functions of independent random variables such as suprema of boolean polynomials which include as special cases subgraph counting problems in random graphs

comparison of moments of sums of independent random variables and differential inequalities fors xi i where i is a sequence of independent symmetric random variables and xi is a sequence of vectors in a normed space we give two methods of proving inequalities e s p 1 p cp q e s q 1 qwith the constantscp qindependent of the sequence xi the methods depend on using differential inequalities of poincare or logarithmic sobolev type the obtained constants are usually better than the ones obtained by other methods

limit theorems for u processes a u process is a collection of u statistics concretely a u process over a family h of kernels h of m variables based on a probability measure p on s s is the collection u n m h p h in mathcal h of u statistics this chapter is devoted to the asymptotic theory of u processes we are interested in finding conditions on h and p ensuring that the law of large numbers or the central limit theorem or the law of the iterated logarithm hold for u n m h p uniformly in h h the theory of empirical processes deals with the same questions for the case m 1 and u process theory is patterned after it this is a relatively new subject at least in the generality presented here h being an arbitrary collection of kernels which are defined on a general measurable space there is therefore a need to indicate its usefulness to this effect a section on applications is added at the end of the chapter section 5 5 there we illustrate the use of each of the main theorems in this chapter by deriving properties of certain multidimensional generalizations of the median and in general of m estimators and by studying estimators of the cumulative hazard and distribution functions of a random variable based on truncated data

lattice valued convergence associated with cns spaces abstract in this paper in the context of meet continuous quantale we present a notion of conical topological lattice valued convergence space which provides the theory of lattice valued convergence associated with such conical neighborhood spaces

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

order of magnitude bounds for expectations of δ2 functions of nonnegative random bilinear forms and generalized u statistics let x 1 y 1 y 2 x n y n be independent nonnegative rv s and let b ij 1 i j n be an array of nonnegative constants we present a method of obtaining the order of magnitude of e 1 i j n b ij x i x j e 1 i j n b ij x i x j for any such x i y j and b ij and any nondecreasing function on 0 with 0 0 and satisfying a 2 growth condition furthermore this technique is extended to provide the order of magnitude of e 1 i j n f j x i y j where f ij x y 1 i j n is any array of nonnegative functions for arbitrary functions g ij x y 1 i j n the aforementioned approximation enables us to identify the order of magnitude of e 1 i j n g ij x i x j whenever decoupling results and khintchine type inequalities apply such as is convex l g ij x i x j l g ji x j x i and eg ij x i x 0 for all x in the range of x j

sparse approximate inverse preconditioners using norm minimization techniques we investigate the use of sparse approximate inverse preconditioners for the iterative solution of unsymmetric linear systems of equations we consider the approximations proposed by cosgrove diaz and griewank internat j comput math 44 1992 pp 91 110 and huckle and grote a new approach to parallel preconditioning with sparse approximate inverses tech report sccm 94 03 stanford university 1994 which are based on norm minimization techniques such methods are of particular interest because of the considerable scope for parallelization we propose a number of enhancements which may improve their performance when run in a sequential environment these methods can perform unfavorably when compared with other techniques however they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment

on diagonal completion of lattice valued diagonal cauchy spaces we define and study diagonal axioms for lattice valued cauchy spaces a completion of a weakly diagonal lattice valued cauchy space is constructed which is weakly diagonal and the coarsest among such completions it is at the same time at least as fine as any weakly regular completion

estimation of moments of sums of independent real random variables for the sum s sum x i of a sequence x i of independent symmetric or nonnegative random variables we give lower and upper estimates of moments of s the estimates are exact up to some universal constants and extend the previous results for particular types of variables x i

u processes rates of convergence on introduit un nouveau processus stochastique une collection de statistiques u indicees par une famille de noyaux symetriques on obtient des conditions pour la convergence presque sure uniforme d une suite de tels processus

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

decoupling inequalities for multilinear forms in independent symmetric random variables on considere x 1 x 2 des copies independantes d une suite x x 1 x 2 de variables aleatoires symetriques independantes et m une forme multilineaire symetrique de rang s sur r n dont les composantes a i1 a is relatives a la base standard de r n satisfont a i1 i s 0

concentration inequalities using the entropy method we investigate a new methodology worked out by ledoux and massart to prove concentration of measure inequalities the method is based on certain modified logarithmic sobolev inequalities we provide some very simple and general ready to use inequalities one of these inequalities may be considered as an exponential version of the efron stein inequality the main purpose of this paper is to point out the simplicity and the generality of the approach we show how the new method can recover many of talagrand s revolutionary inequalities and provide new applications in a variety of problems including rademacher averages rademacher chaos the number of certain small subgraphs in a random graph and the minimum of the empirical risk in some statistical estimation problems

a sparse approximate inverse preconditioner for parallel preconditioning of general sparse matrices a factored sparse approximate inverse is computed and used as a preconditioner for solving general sparse matrices the algorithm is derived from a matrix decomposition algorithm for inverting dense nonsymmetric matrices several strategies and special data structures are proposed to implement the algorithm efficiently sparsity patterns of the factored inverse are exploited to reduce computational cost the preconditioner possesses greater inherent parallelism than traditional preconditioners based on incomplete lu factorizations numerical experiments are used to show the effectiveness and efficiency of the proposed sparse approximate inverse preconditioner

remarks on strong exponential integrability of vector valued random series and triangular arrays some optimal refinements of the results on strong exponential integrability of sums of independent zero mean uniformly bounded random vectors are given

poincare s inequalities and talagrand s concentration phenomenon for the exponential distribution we present a simple proof based on modified logarithmic sobolev inequalities of talagrand s concentration inequality for the exponential distribution we actually observe that every measure satisfying a poincare inequality shares the same concentration phenomenon we also discuss exponential integrability under poincare inequalities and its consequence to sharp diameter upper bounds on spectral gaps

properties of isoperimetric functional and transport entropy inequalities via concentration various properties of isoperimetric functional transport entropy and concentration inequalities are studied on a riemannian manifold equipped with a measure whose generalized ricci curvature is bounded from below first stability of these inequalities with respect to perturbation of the measure is obtained the extent of the perturbation is measured using several different distances between perturbed and original measure such as a one sided l bound on the ratio between their densities wasserstein distances and kullback leibler divergence in particular an extension of the holley stroock perturbation lemma for the log sobolev inequality is obtained and the dependence on the perturbation parameter is improved from linear to logarithmic second the equivalence of transport entropy inequalities with different cost functions is verified by obtaining a reverse jensen type inequality the main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting of independent interest is a new dimension independent characterization of transport entropy inequalities with respect to the 1 wasserstein distance which does not assume any curvature lower bound

a two phase preconditioning strategy of sparse approximate inverse for indefinite matrices a two phase preconditioning strategy based on a factored sparse approximate inverse is proposed for solving sparse indefinite matrices in each phase the strategy first makes the original matrix diagonally dominant to enhance the stability by a shifting method and constructs an inverse approximation of the shifted matrix by utilizing a factored sparse approximate inverse preconditioner the two inverse approximation matrices produced from each phase are then combined to be used as a preconditioner experimental results show that the presented strategy improves the accuracy and the stability of the preconditioner on solving indefinite sparse matrices furthermore the strategy ensures that convergence rate of the preconditioned iterations of the two phase preconditioning strategy is much better than that of the standard sparse approximate inverse ones for solving indefinite matrices

a new stabilization strategy for incomplete lu preconditioning of indefinite matrices a two step combined preconditioning strategy is proposed to construct stable and accurate incomplete lu factorization of indefinite matrices arising from cfd applications this preconditioning procedure is divided into two steps each step is a factorization of a shifted matrix numeral experiments show that a preconditioner can be computed with high accuracy and low fill in and that the new strategy is robust on some difficult indefinite matrix test problems

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

on the role of convexity in isoperimetry spectral gap and concentration we show that for convex domains in euclidean space cheeger s isoperimetric inequality spectral gap of the neumann laplacian exponential concentration of lipschitz functions and the a priori weakest requirement that lipschitz functions have arbitrarily slow uniform tail decay are all quantitatively equivalent to within universal constants independent of the dimension this substantially extends previous results of maz ya cheeger gromov milman buser and ledoux as an application we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume up to constants and under maps which are on average lipschitz we also provide a new characterization up to constants of the spectral gap of a convex domain as one over the square of the average distance from the worst subset having half the measure of the domain in addition we easily recover and extend many previously known lower bounds on the spectral gap of convex domains due to payne weinberger li yau kannan lovasz simonovits bobkov and sodin the proof involves estimates on the diffusion semi group following bakry ledoux and a result from riemannian geometry on the concavity of the isoperimetric profile our results extend to the more general setting of riemannian manifolds with density which satisfy the cd 0 curvature dimension condition of bakry emery

optimum bounds for the distributions of martingales in banach spaces a general device is proposed which provides for extension of exponential inequalities for sums of independent real valued random variables to those for martingales in the 2 smooth banach spaces this is used to obtain optimum bounds of the rosenthal burkholder and chung types on moments of the martingales in 2 smooth banach spaces in turn it leads to best order bounds on moments of sums of independent random vectors in any separable banach spaces although the emphasis is put on infinite dimensional martingales most of the results seem to be new even for one dimensional martingales moreover the bounds on moments of the rosenthal burkholder type seem to be to a certain extent new even for sums of independent real valued random variables analogous inequalities for one dimensional supermartingales are given

limit theorems for functionals of moving averages let x n i 1 a i e n 1 where the e i are i i d with mean 0 and finite second moment and the a i are either summable or regularly varying with index 1 1 2 the sequence x n has short memory in the former case and long memory in the latter for a large class of functions k a new approach is proposed to develop both central n rate and noncentral non n rate limit theorems for s n n 1 n k x n ek x n specifically we show that in the short memory case the central limit theorem holds for s n and in the long memory case s n can be decomposed into two asymptotically uncorrelated parts that follow a central limit and a non central limit theorem respectively further we write the noncentral part as an expansion of uncorrelated components that follow noncentral limit theorems connections with the usual hermite expansion in the gaussian setting are also explored

modified logarithmic sobolev inequalities and transportation inequalities we present a class of modified logarithmic sobolev inequality interpolating between poincare and logarithmic sobolev inequalities suitable for measures of the type exp x or exp x log 2 x 1 2 and which lead to new concentration inequalities these modified inequalities share common properties with usual logarithmic sobolev inequalities as tensorisation or perturbation and imply as well poincare inequality we also study the link between these new modified logarithmic sobolev inequalities and transportation inequalities

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

lattice valued semiuniform convergence spaces in this paper two kinds of lattice valued semiuniform convergence spaces are proposed namely stratified l semiuniform convergence spaces and stratified l ordered semiuniform convergence spaces respectively it is shown that i the category of stratified l semiuniform convergence spaces is topological ii the category of stratified l ordered semiuniform convergence spaces is a bireflective full subcategory of the category of stratified l semiuniform convergence spaces and hence it is topological iii both the category of stratified l semiuniform convergence spaces and that of stratified l ordered semiuniform convergence spaces are cartesian closed iv the category of stratified l semiuniform convergence spaces is extensional v both the category of stratified l semiuniform convergence spaces and that of stratified l ordered semiuniform convergence spaces are closed under the formation of products of quotient mappings in case that l is the two point chain both coincide with the category of semiuniform convergence spaces in the classical case

regularity lattice valued cauchy spaces a category of lattice valued cauchy spaces is defined and its properties are investigated the relationship between this category and the recent work by jager is presented the notion of regularity in the cauchy space context is the primary emphasis here in particular the category consisting of the regular cauchy spaces is shown to be bireflective in the category of all lattice valued cauchy spaces having cauchy continuous maps as its morphisms moreover completions are also investigated

sharp square function inequalities for conditionally symmetric martingales let t be a conditionally symmetric martingale taking values in a hilbert space 1ei and let s i be its square function if vp is the smallest positive zero of the confluent hypergeometric function and ap is the largest positive zero of the parabolic cylinder function of parameter p then the following inequalities are sharp 3 vp iis llp 2 moreover the constants vp and ap for the cases mentioned above are also best possible for the marcinkiewicz paley inequalities for haar functions