stability and error estimates of local discontinuous galerkin methods with implicit explicit time marching for advection diffusion problems the main purpose of this paper is to analyze the stability and error estimates of the local discontinuous galerkin ldg methods coupled with carefully chosen implicit explicit imex runge kutta time discretization up to third order accuracy for solving one dimensional linear advection diffusion equations in the time discretization the advection term is treated explicitly and the diffusion term implicitly there are three highlights of this work the first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the ldg methods the second is that by aid of the aforementioned relationship and the energy method we show that the imex ldg schemes are unconditionally stable for the linear problems in the sense that the time step tau is only required to be upper bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent

rates of clustering in strassen s lil for partial sum processes we provide a detailed description of the rate of clustering in strassen s functional law of the iterated logarithm for partial sum processes necessary and sufficient conditions are given for certain clustering rates in terms of moment type conditions in the process we also derive a new strong approximation of sums of iid random variables by a wiener process

a semi lagrangian discontinuous galerkin dg local dg method for solving convection diffusion equations abstract in this paper we propose an efficient high order semi lagrangian sl discontinuous galerkin dg method for solving linear convection diffusion equations the method generalizes our previous work on developing the sldg method for transport equations 5 making it capable of handling additional diffusion and source terms within the dg framework the solution is evolved along the characteristics while the diffusion term is discretized by the local dg ldg method and integrated along characteristics by implicit runge kutta methods together with source terms the proposed method is named the sldg ldg method and enjoys many attractive features of the dg and sl methods these include the uniformly high order accuracy e g third order in space and in time compact mass conservative and stability under large time stepping size an l 2 stability analysis is provided when the method is coupled with the first order backward euler discretization effectiveness of the method are demonstrated by a group of numerical tests in one and two dimensions

clustering behavior of finite variance partial sum processes clustering rates in strassen s functional law of the iterated logarithm are determined for finite variance partial sum processes in one dimension a general characterization of these rates similar to one recently obtained for onedimensional brownian motion shows that relatively mild moment conditions on a partial sum process lead to high order clustering rates at certain points of the strassen set

domain contraction in nonlinear programming minimizing a quadratic concave objective over a polyhedron using linear underestimating approximations and their dual solutions we develop new results for the nonconvex problem of minimizing a quadratic concave function under linear constraints for solving this problem which has such important management applications as economies of scale fixed charge and quadratic assignment problems some results provide and tighten bounds on the global minimum value while others identify the domains of the variables which may be excluded from further consideration this leads to successive reduction of domains containing a global solution called domain contraction and ideally to a solution with the objective value within some acceptable error level for most results specific problem instances are provided to establish their use in theory in practice experience with solving the problems presented here indicates that a combination of the proposed domain contraction via linear underestimation and its dual information with the branch and bound approach offers a computa

a method for globally minimizing concave functions over convex sets a method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear the algorithm generates linear programs whose solutions minimize the convex envelope of the original function over successively tighter polytopes enclosing the feasible region the algorithm does not involve cuts of the feasible region requires only simplex pivot operations and univariate search computations to be performed allows the objective function to be lower semicontinuous and nonseparable and is guaranteed to converge to the global solution computational aspects of the algorithm are discussed

invariance principles for renewal processes on presente une methodologie generale pour demontrer des principes d invariance pour des processus de renouvellement on obtient des approximations presque sures et d inegalites de probabilites

d c sets d c functions and nonlinear equations ad c set is a set which is the difference of two convex sets we show that any set can be viewed as the image of a d c set under an appropriate linear mapping using this universality we can convert any problem of finding an element of a given compact set in n into one of finding an element of a d c set on the basis of this approach a method is developed for solving a system of nonlinear equations inequations unlike newton type methods our method does not require either convexity differentiability assumptions or an initial approximate solution

concave minimization via conical partitions and polyhedral outer approximation an algorithm is proposed for globally minimizing a concave function over a compact convex set this algorithm combines typical branch and bound elements like partitioning bounding and deletion with suitably introduced cuts in such a way that the computationally most expensive subroutines of previous methods are avoided in each step essentially only few linear programming problems have to be solved some preliminary computational results are reported

convergence and restart in branch and bound algorithms for global optimization application to concave minimization and d c optimization problems a general branch and bound conceptual scheme for global optimization is presented that includes along with previous branch and bound approaches also grid search techniques the corresponding convergence theory as well as the question of restart capability for branch and bound algorithms used in decomposition or outer approximation schemes are discussed as an illustration of this conceptual scheme a finite branch and bound algorithm for concave minimization is described and a convergent branch and bound algorithm based on the previous one is developed for the minimization of a difference of two convex functions

a conservative semi lagrangian multi tracer transport scheme cslam on the cubed sphere grid a conservative multi tracer transport algorithm on the cubed sphere based on the semi lagrangian approach cslam has been developed the scheme relies on backward trajectories and the resulting upstream cells polygons are approximated with great circle arcs biquadratic polynomial functions are used for approximating the density distribution in the cubed sphere grid cells the upstream surface integrals associated with the conservative semi lagrangian scheme are computed as line integrals by employing the gauss green theorem the line integrals are evaluated using a combination of exact integrals and high order gaussian quadrature the upstream cell trajectories information and computation of weights of integrals can be reused for each additional tracer the cslam scheme is extensively tested with various standard benchmark test cases of solid body rotation and deformational flow in both cartesian and spherical geometry and the results are compared with those of other published schemes the cslam scheme is accurate robust and moreover the edges and vertices of the cubed sphere discontinuities do not affect the overall accuracy of the scheme the cslam scheme exhibits excellent convergence properties and has an option for enforcing monotonicity the advantages of introducing cross terms in the fully two dimensional biquadratic density distribution functions are also examined in the context of cartesian as well as the cubed sphere grid which has six local sub domains with discontinuous edges and corners

the darling erdős theorem for sums of i i d random variables we obtain a general darling erd s type theorem for the maximum of appropriately normalized sums of i i d mean zero r v s with finite variances we infer that the darling erd s theorem holds in its classical formulation if and only ife x21 x t o loglogt 1 ast our method is based on an extension of the truncation techniques of feller 1946 to non symmetric r v s as a by product we are able to reprove fundamental results of feller 1946 dealing with lower and upper classes in the hartman wintner lil

normal conical algorithm for concave minimization over polytopes a new conical algorithm is developed for finding the global minimum of a concave function over a polytope to ensure faster convergence and overcome some major drawbacks of previous conical algorithms a normal rather than exhaustive subdivision process is used

a spectral element semi lagrangian sesl method for the spherical shallow water equations a spectral element semi lagrangian sesl method for the shallow water equations on the sphere is presented the sphere is discretized using a hexahedral grid although any grid imaginable can be used as long as it is comprised of quadrilaterals the equations are written in cartesian coordinates to eliminate the pole singularity which plagues the equations in spherical coordinates in a previous paper int j numer methods fluids 35 2001 869 we showed how to construct an explicit eulerian spectral element se model on the sphere we now extend this work to a semi lagrangian formulation the novelty of the lagrangian formulation presented is that the high order se basis functions are used as the interpolation functions for evaluating the values at the lagrangian departure points this makes the method not only high order accurate but quite general and thus applicable to unstructured grids and portable to distributed memory computers the equations are discretized fully implicitly in time in order to avoid having to interpolate derivatives at departure points by incorporating the coriolis terms into the lagrangian derivative the block lu decomposition of the equations results in a symmetric positive definite pseudo helmholtz operator which we solve using the generalized minimum residual method gmres with a fast projection method comput methods appl mech eng 163 1998 193 results for eight test cases are presented to confirm the accuracy and stability of the method these results show that sesl yields the same accuracy as an eulerian spectral element semi implicit sesi while allowing for time steps 10 times as large and being up to 70 more efficient

split invariance principles for stationary processes the results of komlos major and tusnady give optimal wiener approximation of partial sums of i i d random variables and provide an extremely powerful tool in probability and statistical inference recently wu ann probab 35 2007 2294 2320 obtained wiener approximation of a class of dependent stationary processes with finite pth moments 2 0 and liu and lin stochastic process appl 119 2009 249 280 removed the logarithmic factor reaching the komlos major tusnady bound o n 1 p no similar results exist for p 4 and in fact no existing method for dependent approximation yields an a s rate better than o n 1 4 in this paper we show that allowing a second wiener component in the approximation we can get rates near to o n 1 p for arbitrary p 2 this extends the scope of applications of the results essentially as we illustrate it by proving new limit theorems for increments of stochastic processes and statistical tests for short term epidemic changes in stationary processes our method works under a general weak dependence condition covering wide classes of linear and nonlinear time series models and classical dynamical systems

conservative multimoment transport along characteristics for discontinuous galerkin methods a set of algorithms based on characteristic discontinuous galerkin methods is presented for tracer transport on the sphere the algorithms are designed to reduce message passing interface communica

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

the small time chung wichura law for levy processes with non vanishing brownian component we give a small time functional version of chung s other law of the iterated logarithm for levy processes with non vanishing brownian component this is an analogue of the other law of the iterated logarithm at large times for levy processes and random walks with finite variance as extended to a functional version by wichura as one of many possible applications we mention a functional law for a two sided passage time process

quasiconjugates of functions duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint and applications abstract in this paper we introduce a concept of quasiconjugate for functions defined on rn whose values are in r the conjugacy correspondence between functions and their quasiconjugates is one to one and symmetric in a class of quasiconvex functions whose minimizer on rn is located at the origin by using the concept of quasiconjugate we obtain a duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint this duality relationship allows us to establish a primal dual pair in a class of nonconvex optimization problems without the duality gap several applications are given

strong approximations of semimartingales by processes with independent increments strong approximation theorems for continuous time semimartingales are obtained by combining some techniques of the general theory of stochastic processes with some of the direct approximation of dependent random variables by independent ones continuous processes with independent increments whose variance functions increase polynomially or exponentially are considered as approximating processes the basic assumptions of the main results only contain rates of convergence for certain probabilities in particular moment assumptions are not required some almost sure invariance principles for partial sum processes with nonlinear growth of variance and for functionals of markov processes are derived by applying the main results

stability results and strong invariance principles for partial sums of banach space valued random variables a general stability theorem for b valued random variables is obtained which refines a result of kuelbs and zinn our proof is based on two exponential inequalities for sums of independent b valued r v s essentially due to yurinskii and appears particularly simple we then use our theorem to prove strong invariance principles lil results and other related stability results for sums of i i d b valued r v s in the domain of attraction of a gaussian law most of these results seem to be still unknown for real valued r v s

some remarks on a question of strassen strassen s original functional law of the iterated logarithm for partial sums and brownian motion examined convergence and clustering in the sup norm here we address what happens if we use the much larger h norm we provide the answer to a query which appeared at the end of strassen s original paper and also present several contrasting results which are shown to be essentially best possible

on a problem of erdos and taylor let s n n 0 be a centered d dimensional random walk d 3 and consider the so called future infima process j n df inf n s this paper is concerned with obtaining precise integral criteria for a function to be in the levy upper class of j this solves an old problem of erdos and taylor who posed the problem for the simple symmetric random walk on z d d 3 these results are obtained by a careful analysis of the future infima of transient bessel processes and using strong approximations our results belong to a class of ciesielski taylor theorems which relate d and d 2 dimensional bessel processes

feasible partition problem in reverse convex and convex mixed integer programming abstract in this paper we consider the consistent partition problem in reverse convex and convex mixed integer programming in particular we will show that for the considered classes of convex functions both integer and relaxed systems can be partitioned into two disjoint subsystems each of which is consistent and defines an unbounded region the polynomial time algorithm to generate the partition will be proposed and the algorithm for a maximal partition will also be provided

strong approximation for set indexed partial sum processes via kmt constructions ii let x i i in mathbb z d be an array of zero mean independent identically distributed random vectors with values in mathbb r k with finite variance and let mathscr l be a class of borel subsets of lbrack 0 1 rbrack d if for the usual metric mathscr l is totally bounded and has a convergent entropy integral we obtain a strong invariance principle for an appropriately smoothed version of the partial sum process sum i in nu s x i s in mathscr l with an error term depending only on mathscr l and on the tail distribution of x 1 in particular when mathscr l is the class of subsets of lbrack 0 1 rbrack d with alpha differentiable boundaries introduced by dudley we prove that our result is optimal

on the other law of the iterated logarithm a general integral test is established which refines the jain pruitt chung lil for iid random variables as a corollary we obtain that chung s integral test for brownian motion is valid for partial sums of iid random variables satisfyingex21 x t o llt 1 ast

note on solving linear complementarity problems as jointly constrained bilinear programs abstract we investigate refinements of an existing nonconvex programming algorithm that exploits special structure of linear complementarity problems we construct a linear program having an n n working basis which is equivalent to the subproblems generated by the algorithm that generally require 3n 3n basis matrices

strong invariance principles for partial sums of independent random vectors on obtient une estimation dans le theoreme central limite qui est utilisee avec le theoreme de strassen dudley pour demontrer une approximation forte pour des sommes partielles de vecteurs aleatoires i i d a d dimensions

strong approximations for partial sums of i i d b valued r v s in the domain of attraction of a gaussian law we obtain a strong approximation theorem for partial sums of i i d d dimensional r v s with possibly infinite second moments using this result we can extend philipp s strong invariance principle for partial sums of i i d b valued r v s satisfying the central limit theorem tob valued r v s which are only in the domain of attraction of a gaussian law this new strong invariance principle implies a compact as well as a functional law of the iterated logarithm which improve some recent results of kuelbs 1985

a useful estimate in the multidimensional invariance principle an estimate of the convergence speed in the multidimensional invariance principle is obtained using this estimate we can prove strong invariance principles for partial sums of independent not necessarily identically distributed multidimensional random vectors

rates of convergence for the functional lil rates of convergence are obtained for strassen s functional law of the iterated logarithm for polygonal processes under classical conditions

a conical algorithm for globally minimizing a concave function over a closed convex set in this paper we are concerned with the problem of finding the global minimum of a concave function over a closed convex possibly unbounded set in rn the intrinsic difficulty of this problem is due to the fact that a local minimum of the objective function may fail to be a global one which makes the conventional methods of local optimization almost useless

a note on the solution of bilinear programming problems by reduction to concave minimization we present a new algorithm for solving the bilinear programming problem by reduction to concave minimization this algorithm is finite does not assume the boundedness of the constraint set and uses an efficient procedure for checking whether a concave function is bounded below on a given halfline some preliminary computational experience with a computer code for implementing the algorithm on a microcomputer is also reported

a note on the law of the iterated logarithm in hilbert space kuelbs 1975 established a kolmogorov erdos petrowski type integral test for lower and upper classes in the law of the iterated logarithm for sums of i i d hilbert space valued gaussian mean zero random variables we show that this integral test remains valid for sums of i i d pregaussian mean zero random variables satisfying an additional very mild assumption

a branch and bound outer approximation algorithm for concave minimization over a convex set abstract this article presents a new algorithm for solving the problem of globally minimizing a concave function over a compact convex set the algorithm uses both branch and bound and outer approximation one of the major advantages of the algorithm is that the only significant nonlinear computation required can be accomplished using univariate search procedures after establishing the convergence of the algorithm we discuss this and other computational considerations a small example is solved to illustrate the algorithm because of its advantages the algorithm offers significant promise for efficiently minimizing a concave function over a compact convex set