moderate deviation principles for weakly interacting particle systems moderate deviation principles for empirical measure processes associated with weakly interacting markov processes are established two families of models are considered the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump markov processes with a countable state space for both cases the moderate deviation principle is formulated in terms of a large deviation principle ldp with an appropriate speed function for suitably centered and normalized empirical measure processes for the first family of models the ldp is established in the path space of an appropriate schwartz distribution space whereas for the second family the ldp is proved in the space of l 2 the hilbert space of square summable sequences valued paths proofs rely on certain variational representations for exponential functionals of brownian motions and poisson random measures

the random quadratic assignment problem the quadratic assignment problem qap is one of the most difficult of all com binatorial optimization problems here we use an abbreviated application of the statistical mechanics replica method to study the asymptotic behavior of instances in which the en tries of at least one of the two matrices that specify the problem are chosen from a random distributionp s urprisingly the qap has not been studied before using the replica method despite the fact that the qap was first proposed over 50 years ago and the replica method was developed over 30 years ago we find simple forms forc min andc max t he costs of the minimal and maximum solutions respectively notable features of our results are the sym metry of the results forc min andc max and their dependence onponly through its mean and standard deviation independent of the details ofp

a fast and stable well balanced scheme with hydrostatic reconstruction for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom it is a hyperbolic system of conservation laws that approximately describes various geophysical flows such as rivers coastal areas and oceans when completed with a coriolis term or granular flows when completed with friction numerical approximate solutions to this system may be generated using conservative finite volume methods which are known to properly handle shocks and contact discontinuities however in general these schemes are known to be quite inaccurate for near steady states as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions this difficulty can be overcome by using the so called well balanced schemes we describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well balanced scheme from any given numerical flux for the homogeneous problem whenever the initial solver satisfies some classical stability properties it yields a simple and fast well balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

moderate deviation principles for stochastic differential equations with jumps moderate deviation principles for stochastic differential equations driven by a poisson random measure prm in finite and infinite dimensions are obtained proofs are based on a variational representation for expected values of positive functionals of a prm

moderate deviations for empirical measures of markov chains lower bounds we obtain upper bounds for moderate deviations of empirical measures of markov chains with general state space in the topology under the minimal assumption of geometric ergodicity the regeneration split chain method and a result on projective systems are employed in the proof

stochastic analysis of the quadratic assignment problem let a and b be m m matrices the aim is to maximize q i jaijb i j over all permutations of 1 2 m we define a simple greedy algorithm that constructs a permutation that gives a large q assuming that the entries of a are i i d with a symmetric distribution and that the entries of b are independent of a we show under some weak moment conditions eq k 1m3 2 log m 1 2 where the constant k does not depend on m this contrasts with the fact that max km3 2 log m 1 2 with overwhelming probability

discontinuous galerkin methods with nodal and hybrid modal nodal triangular quadrilateral and polygonal elements for nonlinear shallow water flow abstract we present a comprehensive assessment of nodal and hybrid modal nodal discontinuous galerkin dg finite element solutions on a range of unstructured meshes to nonlinear shallow water flow with smooth solutions the nodal dg methods on triangles and a tensor product nodal basis on quadrilaterals are considered the hybrid modal nodal dg methods utilize two different synergistic polynomial bases on polygons in realizing the dg discretization orthogonal basis functions constructed by the gram schmidt process are used as trial and test functions in a dg weak formulation and a nodal basis is used as an efficient means for area integration these are implemented on triangular quadrilateral and polygonal elements in addition we discuss aspects to be considered in order to achieve the so called well balanced property that preserves steady state at rest with a spatially varying bed the performance in terms of accuracy and computational cost is demonstrated using h and p convergence studies on a nonlinear problem with a manufactured solution and the nonlinear stommel problem with flat and non flat beds to assess the performance of quadrilateral and polygonal elements in comparison to triangular elements we consider a setting in which a quadrilateral mesh a mixed triangular quadrilateral mesh and polygonal mesh are derived from a given triangular mesh and vice versa the tests conducted reveal the merit of using the quadrilateral elements in terms of computational cost per accuracy and computing time more importantly the numerical results clearly show that high order schemes significantly improve the cost performance for a given level of accuracy with cubic or bi cubic interpolants particularly achieving dramatic improvements in accuracy as compared to linear and quadratic interpolants with diminishing benefit as p 3

averaging principle of sde with small diffusion moderate deviations consider the following stochastic differential equation in rd dxetxe0 b xet t e dt e a xet t e dwt x0 where the random environment t is an exponentially ergodic markov process independent of the wiener process wt with invariant probability measure and e is some small parameter in this paper we prove the moderate deviations for the averaging principle of xe that is deviations of xet around its limit averaging system x t given by d dx t b x t dt and x 0 x0 where b x e b x more precisely we obtain the large deviation estimation about et xet x te h e t 0 1 in the space of continuous trajectories as e decreases to 0 where h e is some deviation scale satisfying 1 h e e 1 2 our strategy will be first to show the exponential tightness and then the local moderate deviation principle which relies on some new method involving a conditional schilder s theorem and a moderate deviation principle for inhomogeneous integral functionals of markov processes previously established by the author

semi discrete discontinuous galerkin methods and stage exceeding order strong stability preserving runge kutta time discretizations this paper investigates the use of a special class of strong stability preserving ssp runge kutta time discretization methods in conjunction with discontinuous galerkin dg finite element spatial discretizatons the class of ssp methods investigated here is defined by the property that the number of stages s is greater than the order k of the method from analysis cfl conditions for the linear l 2 stability of the methods defined using the s k ssp schemes are obtained that are less restrictive than those of the standard so called rkdg methods that use s k ssp runge kutta schemes the improvement in the cfl conditions for linear stability of the methods more than offsets the additional work introduced by the increased number of stages given that the cfl conditions for linear stability are what must be respected in practice in order to maintain high order accuracy the use of the s k ssp schemes results in rkdg methods that are more efficient than those previously defined furthermore with the application of a slope limiter the nonlinear stability properties of the forward euler method and the dg spatial discretization which have been previously proven are preserved with these methods under less restrictive cfl conditions than those required for linear stability thus more efficient rkdg methods that possess the same favorable accuracy and stability properties of the standard rkdg methods are obtained numerical results verify the cfl conditions for stability obtained from analysis and demonstrate the efficiency advantages of these new rkdg methods

asymptotic properties of the quadratic assignment problem for the general quadratic assignment problem as well as for a planar version of this problem we extend earlier work by burkard and fincke to prove that the ratio of the maximal to the minimal solution value converges to 1 almost surely in fact any solution value can almost surely be written asymptotically as a simple explicitly given function of the problem size theoretical analysis and computational experiments reveal the convergence to be relatively fast

heuristics for biquadratic assignment problems and their computational comparison abstract the biquadratic assignment problem biqap is a generalization of the quadratic assignment problem qap as for any hard optimization problem also for biqap a reasonable effort to cope with the problem is trying to derive heuristics which solve it suboptimally and which possibly yield a good trade off between the solution quality and the time and memory requirements in this paper we describe several heuristics for biqaps in particular pair exchange algorithms improvement methods and variants of simulated annealing and taboo search we implement these heuristics as c codes and analyze their performances

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

an asymptotical study of combinatorial optimization problems by means of statistical mechanics the analogy between combinatorial optimization and statistical mechanics has proven to be a fruitful object of study simulated annealing a metaheuristic for combinatorial optimization problems is based on this analogy in this paper we show how a statistical mechanics formalism can be utilized to analyze the asymptotic behavior of combinatorial optimization problems with sum objective function and provide an alternative proof for the following result under a certain combinatorial condition and some natural probabilistic assumptions on the coefficients of the problem the ratio between the optimal solution and an arbitrary feasible solution tends to one almost surely as the size of the problem tends to infinity so that the problem of optimization becomes trivial in some sense whereas this result can also be proven by purely probabilistic techniques the above approach allows one to understand why the assumed combinatorial condition is essential for such a type of asymptotic behavior

artificial boundary layers in discontinuous galerkin solutions to shallow water equations in channels in this work we consider the application of discontinuous galerkin dg solutions to open channel flow problems governed by two dimensional shallow water equations swe with solid curved wall boundaries on which the no normal flow boundary conditions are prescribed a commonly used approach consists of straightforwardly imposing the no normal flow condition on the linear approximation of curved walls numerical solutions indicate clearly that this approach could lead to unfavorable results and that a proper treatment of the no normal flow condition on curved walls is crucial for an accurate dg solution to the swe in the test case used errors introduced through the commonly used approach result in artificial boundary layers of one grid size thickness in the velocity field and a corresponding over prediction of the surface elevation in the upstream direction these significant inaccuracies which render the coarse mesh solution unreliable appear in all dg schemes employed including those using linear quadratic and cubic dg polynomials the issue can be alleviated by either using an approach accounting for errors introduced by the geometric approximation or an approach that accurately represents the geometry

the asymptotic behaviour of quadratic sum assignment problems a statistical mechanics approach abstract in this paper we apply the statistical mechanics formalism to study quadratic sum assignment problems qsap this formalism allows us to exhibit in the limit of large problems the asymptotic behaviour of the optimal value of the cost function the conclusions of the study are confirmed by the results of metropolis computer simulations

on random quadratic bottleneck assignment problems the relative error between best and worst solution of quadratic bottleneck assignment problems with cost coefficientsdijpq aipbjq is considered whereaip is either arbitrarily given or corresponds to a distance in the plane it is shown that the relative error is bounded by a function m tending to zero with probability tending to one asm provided the data are uniformly distributed this implies that any algorithm for the above mentioned problems yields asymptotically an arbitrarily small relative error with probability tending to one

a wetting and drying treatment for the runge kutta discontinuous galerkin solution to the shallow water equations this paper proposes a wetting and drying treatment for the piecewise linear runge kutta discontinuous galerkin approximation to the shallow water equations the method takes a fixed mesh approach as opposed to mesh adaptation techniques and applies a post processing operator to ensure the positivity of the mean water depth within each finite element in addition special treatments are applied in the numerical flux computation to prevent an instability due to excessive drying the proposed wetting and drying treatment is verified through comparisons with exact solutions and convergence rates are examined the obtained orders of convergence are close to or approximately equal to 1 for solutions with discontinuities and are improved for smooth solutions the combination of the proposed wetting and drying treatment and a tvb slope limiter is also tested and is found to be applicable on condition that they are applied exclusively to an element at the same runge kutta step

moderate deviations for diffusions with brownian potentials we present precise moderate deviation probabilities in both quenched and annealed settings for a recurrent diffusion process with a brownian potential our method relies on fine tools in stochastic calculus including kotani s lemma and lamperti s representation for exponential functionals in particular our result for quenched moderate deviations is in agreement with a recent theorem of comets and popov probab theory related fields 126 2003 571 609 who studied the corresponding problem for sinai s random walk in random environment

error estimates to smooth solutions of runge kutta discontinuous galerkin methods for scalar conservation laws in this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for runge kutta discontinuous galerkin rkdg methods where the time discretization is the second order explicit total variation diminishing tvd runge kutta method error estimates for the mathbb p 1 piecewise linear elements are obtained under the usual cfl condition tau leq gamma h for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions where h and tau are the maximum element lengths and time steps respectively and the positive constant gamma is independent of h and tau however error estimates for higher order mathbb p k k geq 2 elements need a more restrictive time step tau leq gamma h 4 3 we remark that this stronger condition is indeed necessary as the method is linearly unstable under the usual cfl condition tau leq gamma h for the mathbb p k elements of degree k geq 2 error estimates of o h k 1 2 tau 2 are obtained for general monotone numerical fluxes and optimal error estimates of o h k 1 tau 2 are obtained for upwind numerical fluxes

on moderate deviations for martingales let x n x t n f t n 0 t 1 be square integrable martingales with the quadratic characteristics x n n 1 2 we prove that the large deviations relation p x 1 n r 1 r 1 holds true for r growing to infinity with some rate depending on l 2 n z 0 t 1 x t n 2 28 and n 2 n e x n 1 1 1 where 0 and l 2 n 0 n 2 n 0 as n the exact bound for the remainder is also obtained

martingale inequalities and np complete problems we use martingale inequalities in the probabilistic analysis of stochastic models of np complete problems we give examples of applications to the bin packing and traveling salesman problems in both cases we obtain sharp bounds on pr un eun t where un is the objective function value of the problem with n random data

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

intelligent local search strategies for solving facility layout problems with the quadratic assignment problem formulation abstract in this paper we implement a tabu search heuristic a probabilistic tabu search heuristic a simulated annealing heuristic and a hybrid tabu search heuristic to solve facility layout problems with the quadratic assignment problem formulation classical problems are used as a benchmark for performance comparison extensive computational experiments were conducted and the results were statistically analyzed to observe the behavior of each heuristic and to discriminate the performance of each heuristic the computational results strongly support the competitiveness of intelligent heuristics for the design of facilities the hybrid tabu search heuristic combining the advantages from both tabu search and simulated annealing can generally outperform their parent algorithms

a thermodynamically motivated simulation procedure for combinatorial optimization problems abstract exchange algorithms are an important class of heuristics for hard combinatorial optimization problems as e g salesman problems or quadratic assignment problems in kirkpatrick s and cerny s exchange algorithms for the travelling salesman problem and placement problem they propose to perform an exchange not only if the objective function value decreases by this exchange but also in certain cases if the objective function value increases an exchange increasing the objective function value is performed stochastically depending on the size of the increment computational tests with quadratic assignment problems revealed an excellent behaviour in such an approach suboptimal solutions differing 1 2 from the best known solutions are obtained by a simple program in short time by starting this program several times with different starting values all known minimal objective function values were reached thus this approach is well suited also for smaller computers and leads in short time to acceptable solutions

moderate deviations and associated laplace approximations for sums of independent random vectors let xj be an i i d sequence of banach space valued r v s and let sn z 1 xj for certain positive sequences bn oo we determine the exact asymptotic behavior of e exp b2 n d sn bn where d is a smooth function we also prove a large deviation principle for 2y sn bn

the n city travelling salesman problem statistical mechanics and the metropolis algorithm in any n city travelling salesman problem there are n 1 mathord left vphantom 2 right kern nulldelimiterspace 2 possible tours we use the metropolis algorithm to generate a sequence of such tours this sequence may be viewed as the random evolution of a physical system in contact with a heat bath as the temperature is lowered the tours generated approach the optimal tour it appears that for large n one arrives within a few percent of the optimal solution in better than quadratic time

random assignment problems abstract analysis of random instances of optimization problems provides valuable insights into the behavior and properties of problem s solutions feasible region and optimal values especially in large scale cases a class of problems that have been studied extensively in the literature using the methods of probabilistic analysis is represented by the assignment problems and many important problems in operations research and computer science can be formulated as assignment problems this paper presents an overview of the recent results and developments in the area of probabilistic assignment problems including the linear and multidimensional assignment problems quadratic assignment problem etc

quadratic assignment problems abstract this paper surveys quadratic assignment problems qap at first several applications of this problem class are described and mathematical formulations of qaps are given then some exact solution methods and good heuristics are outlined their computational behaviour is illustrated by numerical results further recent results on the asymptotic probabilistic behaviour of qaps are outlined