a class of parallel two level nonlinear schwarz preconditioned inexact newton algorithms we propose and test a new class of two level nonlinear additive schwarz preconditioned inexact newton algorithms aspin the two level aspin combines a local nonlinear additive schwarz preconditioner and a global linear coarse preconditioner this approach is more attractive than the two level method introduced in cai keyes and marcinkowski nonlinear additive schwarz preconditioners and applications in computational fluid dynamics international journal numerical methods in fluids 40 2002 1463 1470 which is nonlinear on both levels since the coarse part of the global function evaluation requires only the solution of a linear coarse system rather than a nonlinear coarse system derived from the discretization of original partial dierential equations the overall computational cost is reduced considerably our parallel numerical results based on an incompressible lid driven flow problem show that the new two level aspin is quite scalable with respect to the number of processors and the fine mesh size when the coarse mesh size is fine enough and in addition the convergence is not sensitive to the reynolds numbers

omnibus tests for comparison of competing risks with adjustment for covariate effects summary this article develops omnibus tests for comparing cause specific hazard rates and cumulative incidence functions at specified covariate levels confidence bands for the difference and the ratio of two conditional cumulative incidence functions are also constructed the omnibus test is formulated in terms of a test process given by a weighted difference of estimates of cumulative cause specific hazard rates under cox proportional hazards models a simulation procedure is devised for sampling from the null distribution of the test process leading to graphical and numerical techniques for detecting significant differences in the risks the approach is applied to a cohort study of type specific hiv infection rates

inference for a nonlinear counting process regression model martingale and counting process techniques are applied to the problem of inference for general conditional hazard functions this problem was first studied by beran who introduced a class of estimators for the conditional cumulative hazard and survival functions in the special case of time independent covariates here the covariate can be time dependent the classical i i d assumptions are relaxed by replacing them with certain asymptotic stability assumptions and models involving recurrent failures are included this is done within the framework of a general nonparametric counting process regression model important examples of the model include right censored survival data semi markov processes an illness death process with duration dependence and age dependent birth and death processes

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

a nonlinearly preconditioned inexact newton algorithm for steady state lattice boltzmann equations most existing methods for calculating the steady state solution of the lattice boltzmann equations are based on pseudo time stepping which often requires a large number of time steps especially for high reynolds number problems to calculate the steady state solution directly without the time integration in this paper we propose and study a nonlinearly preconditioned inexact newton algorithm with a domain decomposition based linear solver for parallelization more precisely the proposed algorithmic framework involves an implicit second order discretization a two level inexact newton method and a nonlinear elimination preconditioner to accelerate the convergence of newton iteration a nonstandard pollution removing coarse space is introduced for the two level method numerical experiments are presented to demonstrate the robustness and efficiency of the algorithm especially for problems at a high reynolds number a comparison is also included to show the superiority of the proposed approach over o

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

confidence bands for cumulative incidence curves under the additive risk model in the context of competing risks the cumulative incidence function is often used to summarize the cause specific failure time data as an alternative to the proportional hazards model the additive risk model is used to investigate covariate effects by specifying that the subject specific hazard function is the sum of a baseline hazard function and a regression function of covariates based on such a formulation we present an approach to constructing simultaneous confidence intervals for the cause specific cumulative incidence function of patients with given risk factors a melanoma data set is used for the purpose of illustration

prediction of cumulative incidence function under the proportional hazards model in the presence of dependent competing risks in survival analysis the cox model can be utilized to examine the covariate effects on the cause specific hazard function for the failure type of interest for this situation the cumulative incidence function provides an intuitively appealing summary curve for marginal probabilities of this particular event in this paper we show how to construct confidence intervals and bands for such a function under the cox model for future patients with certain covariates our proposals are illustrated with data from a prostate cancer trial

a multilevel hybrid newton krylov schwarz method for the bidomain model of electrocardiology abstract a multilevel hybrid newton krylov schwarz nks method is constructed and studied numerically for implicit time discretizations of the bidomain reaction diffusion system in three dimensions this model describes the bioelectrical activity of the heart by coupling two degenerate parabolic equations with a stiff system of ordinary differential equations the nks bidomain solver employs an outer inexact newton iteration to solve the nonlinear finite element system originating at each time step of the implicit discretization the jacobian update during the newton iteration is solved by a krylov method employing a multilevel hybrid overlapping schwarz preconditioner additive within the levels and multiplicative among the levels several parallel tests on linux clusters are performed showing that the convergence of the method is independent of the number of subdomains scalability the discretization parameters and the number of levels optimality

endpoint bilinear restriction theorems for the cone and some sharp null form estimates recently wolff 25 obtained a nearly sharp l 2 bilinear restriction theorem for bounded subsets of the cone in general dimension we obtain the endpoint of wolff s estimate and generalize to the case when one of the subsets is large as a consequence we are able to deduce some nearly sharp l p null form estimates

subdivision based nonlinear multiscale cloth simulation cloth simulation is an important topic for many applications in computer graphics animation and augmented virtual reality the mechanical behavior of cloth objects can be modeled by the kirchhoff

some tests for comparing cumulative incidence functions and cause specific hazard rates abstract we consider the competing risks problem with the available data in the form of times and causes of failure in many practical situations e g in reliability testing it is important to know whether two risks are equal or whether one is more serious than the other we propose some distribution free tests for comparing cumulative incidence functions and cause specific hazard rates against ordered alternatives without making any assumptions on the nature of dependence between the risks both the censored and the uncensored cases are studied the performance of the proposed tests is assessed in a simulation study as an illustration we compare the risks of two types of cancer mortality thymic lymphoma and reticulum cell carcinoma in a strain of laboratory mice

checking the cox model with cumulative sums of martingale based residuals summary this paper presents a new class of graphical and numerical methods for checking the adequacy of the cox regression model the procedures are derived from cumulative sums of martingale based residuals over follow up time and or covariate values the distributions of these stochastic processes under the assumed model can be approximated by zero mean gaussian processes each observed process can then be compared both visually and analytically with a number of simulated realizations from the approximate null distribution these comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are special attention is given to checking the functional form of a covariate the form of the link function and the validity of the proportional hazards assumption an omnibus test consistent against any model misspecification is also studied the proposed techniques are illustrated with two real data sets

isogeometric block feti dp preconditioners for the stokes and mixed linear elasticity systems abstract the aim of this work is to construct and analyze a feti dp type domain decomposition preconditioner for isogeometric discretizations of the stokes and mixed linear elasticity systems this method extends to the isogeometric analysis context the preconditioner previously proposed by tu and li 2015 for finite element discretizations of the stokes system the resulting isogeometric feti dp algorithm is proven to be scalable in the number of subdomains and has a quasi optimal convergence rate bound which is polylogarithmic in the ratio of subdomain and element sizes extensive two dimensional numerical experiments validate the theory investigate the behavior of the preconditioner with respect to both the spline polynomial degree and regularity and show its robustness with respect to domain deformation material incompressibility and presence of elastic coefficient discontinuities across subdomain interfaces

multilevel schwarz methods for elliptic partial differential equations we investigate multilevel schwarz domain decomposition preconditioners to efficiently solve linear systems arising from numerical discretizations of elliptic partial differential equations by the finite element method in our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner we start from two level preconditioners with either aggregative or interpolative coarse level components then we focus on a strategy to increase the number of levels for all preconditioners we consider the additive residual update and its multiplicative variants within and between levels moreover we compare the preconditioners behaviour regarding scalability and rate of convergence numerical results are provided for elliptic boundary value problems including a convection diffusion problem when suitable stabilization becomes necessary c 2011 elsevier b v all rights reserved

nonlinear feti dp and bddc methods new nonlinear feti dp dual primal finite element tearing and interconnecting and bddc balancing domain decomposition by constraints domain decomposition methods are introduced in all these methods in each iteration local nonlinear problems are solved on the subdomains the new approaches can significantly reduce communication and show a significantly improved performance especially for problems with localized nonlinearities compared to a standard newton krylov feti dp or bddc approach moreover the coarse space of the nonlinear feti dp methods can be used to accelerate the newton convergence it is also found that the new nonlinear feti dp and nonlinear bddc methods are not as closely related as in the linear context numerical results for the p laplace operator are presented

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

toward extremely scalable nonlinear domain decomposition methods for elliptic partial differential equations the solution of nonlinear problems e g in material science requires fast and highly scalable parallel solvers finite element tearing and interconnecting dual primal feti dp domain decomposition methods are parallel solution methods for implicit problems discretized by finite elements recently nonlinear versions of the well known feti dp methods for linear problems have been introduced in these methods the nonlinear problem is decomposed before linearization this approach can be viewed as a strategy to further localize computational work and to extend the parallel scalability of feti dp methods toward extreme scale supercomputers here a recent nonlinear feti dp method is combined with an approach that allows an inexact solution of the feti dp coarse problem we combine the nonlinear feti dp domain decomposition method with an algebraic multigrid amg method and thus obtain a hybrid nonlinear domain decomposition multigrid method we consider scalar nonlinear problems as well as nonlinear hyp

a hybrid multilevel schwarz method for the bidomain model abstract a hybrid multilevel schwarz method is studied numerically for the anisotropic bidomain model in both two and three dimensions this multiscale system models the electrical activity of the heart and it consists of two degenerate parabolic non linear reaction diffusion equations coupled with a stiff system of ordinary differential equations the numerical discretization of the whole system by finite elements in space and semi implicit methods in time generates ill conditioned linear systems that must be solved at each time step the multilevel algorithm studied employs a hierarchy of nested meshes with overlapping schwarz preconditioners on each level and is additive within the levels and multiplicative among the levels we perform several parallel tests on two linux clusters showing that the convergence of the method is independent of the number of subdomains scalability the discretization parameters and the number of levels optimality moreover the comparison with the traditional block jacobi ilu parallel preconditioner and the v cycle multigrid parallel preconditioner shows that on a whole heart cycle simulation the proposed method attains the best performances in terms of cpu times

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

the endpoint case of the bennett carbery tao multilinear kakeya conjecture we prove the endpoint case of the multilinear kakeya conjecture of bennett carbery and tao the proof uses the polynomial method introduced by dvir

nonlinear feti dp and bddc methods a unified framework and parallel results parallel newton krylov feti dp finite element tearing and interconnecting dual primal domain decomposition methods are fast and robust solvers e g for nonlinear implicit problems in structural mechanics in these methods the nonlinear problem is first linearized and then decomposed into loosely coupled linear problems which can be solved in parallel by changing the order of the operations new parallel communication can be constructed where the loosely coupled local problems are nonlinear we discuss different nonlinear feti dp methods which are equivalent when applied to linear problems but which show a different performance for nonlinear problems moreover a new unified framework is introduced which casts all nonlinear feti dp domain decomposition approaches discussed in the literature into a single algorithm furthermore the equivalence of nonlinear feti dp methods to specific nonlinearly right preconditioned newton krylov methods is shown for the methods using nested newton iteration

proportional hazards tests and diagnostics based on weighted residuals summary nonproportional hazards can often be expressed by extending the cox model to include time varying coefficients e g for a single covariate the hazard function for subject i is modelled as exp fl t zi t a common example is a treatment effect that decreases with time we show that the function 3 t can be directly visualized by smoothing an appropriate residual plot also many tests of proportional hazards including those of cox 1972 gill schumacher 1987 harrell 1986 lin 1991 moreau o quigley mesbah 1985 nagelkerke oosting hart 1984 o quigley pessione 1989 schoenfeld 1980 and wei 1984 are related to time weighted score tests of the proportional hazards hypothesis and can be visualized as a weighted least squares line fitted to the residual plot

estimates of absolute cause specific risk in cohort studies in this paper we study methods for estimating the absolute risk of an event c1 in a time interval t1 t2 given that the individual is at risk at t1 and given the presence of competing risks we discuss some advantages of absolute risk for measuring the prognosis of an individual patient and some difficulties of interpretation for comparing two treatment groups we also discuss the importance of the concept of absolute risk in evaluating public health measures to prevent disease variance calculations permit one to gauge the relative importance of random and systematic errors in estimating absolute risk efficiency calculations were also performed to determine how much precision is lost in estimating absolute risk with a nonparametric approach or with a flexible piecewise exponential model rather than a simple exponential model and other calculations indicate the extent of bias that arises with the simple exponential model when that model is invalid such calculations suggest that the more flexible models will be useful in practice simulations confirm that asymptotic methods yield reliable variance estimates and confidence interval coverages in samples of practical size

covariate analysis of competing risks data with log linear models a general system of log linear modeling is proposed for analysis of competing risks data with discrete covariates the instantaneous cause specific failure rates approximated by step functions are analyzed by techniques for multidimensional contingency tables censored observations are accommodated counts of failures of each type and the amount of follow up are summarized in two arrays in which each cell denotes a distinct combination of failure type time interval and covariate value maximum likelihood estimators for the parameters of the model are derived by iterative proportional fitting the resulting estimates of the number of failures in each cell are used for goodness of fit tests the principal advantages of this approach are its simple display of data its computational ease for the fitting and comparison of models and its provision of explicit goodness of fit tests interpretation of the models is facilitated by reference to several alternative models for survivorship and competing risks the basic model is extended to incorporate stochastic covariates whose values change during follow up and to accommodate quantitative covariates

a class of tests for the equality of k cause specific hazard rates in a competing risks model a class of asymptotically distribution free tests for the equality of the k risks in a competing risks model is considered the tests can be derived by martingale theory or by a marginal likelihood approach when the risks are dependent the quantities to be compared would be the cause specific hazard rates or equivalently the pseudosurvivor functions related to the cause specific hazard rates this class of tests allows one to compare k risks simultaneously for k 2 is able to handle possibly right censored data and is extremely simple to apply choices of the optimal weight functions leading to the locally most powerful rank tests for a general class of semiparametric alternative hypotheses are discussed the performances of the tests are studied by simulation under various alternative hypotheses

overlapping schwarz methods with a standard coarse space for almost incompressible linear elasticity low order finite element discretizations of the linear elasticity system suffer increasingly from locking effects and ill conditioning when the material approaches the incompressible limit if only the displacement variables are used mixed finite elements using both displacement and pressure variables provide a well known remedy but they yield larger and indefinite discrete systems for which the design of scalable and efficient iterative solvers is challenging two level overlapping schwarz preconditioners for the almost incompressible system of linear elasticity discretized by mixed finite elements with discontinuous pressures are constructed and analyzed the preconditioned systems are accelerated either by a gmres generalized minimum residual method applied to the resulting discrete saddle point problem or by a pcg preconditioned conjugate gradient method applied to a positive definite although extremely ill conditioned reformulation of the problem obtained by eliminating all pressure variab

an improved bound on the minkowski dimension of besicovitch sets in mathbb r 3 a besicovitch set is a set which contains a unit line segment in any direction it is known that the minkowski and hausdorfi dimensions of such a set must be greater than or equal to 5 2i n 3 in this paper we show that the minkowski dimension must in fact be greater than 5 2 for some absolute constant 0 one observation arising from the argument is that besicovitch sets of near minimal dimension have to satisfy certain strong properties which we call stickiness planiness and graininess

a semiparametric mixture model for analyzing clustered competing risks data a very general class of multivariate life distributions is considered for analyzing failure time clustered data that are subject to censoring and multiple modes of failure conditional on cluster specific quantities the joint distribution of the failure time and event indicator can be expressed as a mixture of the distribution of time to failure due to a certain type or specific cause and the failure type distribution we assume here the marginal probabilities of various failure types are logistic functions of some covariates the cluster specific quantities are subject to some unknown distribution that causes frailty the unknown frailty distribution is modeled nonparametrically using a dirichlet process in such a semiparametric setup a hybrid method of estimation is proposed based on the i i d weighted chinese restaurant algorithm that helps us generate observations from the predictive distribution of the frailty the monte carlo ecm algorithm plays a vital role for obtaining the estimates of the parameters that assess the extent of the effects of the causal factors for failures of a certain type a simulation study is conducted to study the consistency of our methodology the proposed methodology is used to analyze a real data set on hiv infection of a cohort of female prostitutes in senegal

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100