unbounded perturbation of the exponential dichotomy for evolution equations abstract in this paper we prove that the exponential dichotomy for evolution equations in banach spaces is not destroyed if we perturb the equation by small unbounded linear operator this is done by employing a skew product semiflow technique and a perturbation principle from linear operator theory finally we apply these results to partial parabolic equations and functional differential equations

infinite horizon optimization we consider the general problem of choosing a discounted cost minimizing infinite sequence of decisions from a closed subset of the product space formed by a sequence of arbitrary compact metric spaces examples include equipment replacement production planning and more generally infinite stage mathematical programs it is shown that the optimal costs for finite horizon approximating problems converge to the optimal infinite horizon cost as the horizons diverge to infinity moreover the existence of a unique algorithmically optimal i e accumulation point solution is shown to be a necessary and sufficient condition for convergence in the product topology i e policy convergence of all finite horizon optima under the weaker condition of hausdorff convergence of the sets of finite horizon optima to the set of infinite horizon optima we show how to force policy convergence through a natural tie breaking rule finally a forward algorithm is presented which in the presence of a unique infinite horizon optimum is guaranteed to converge

finite dimensional approximation in infinite dimensional mathematical programming we consider the problem of approximating an optimal solution to a separable doubly infinite mathematical program p with lower staircase structure by solutions to the programs p n obtained by truncating after the first n variables and n constraints of p viewing the surplus vector variable associated with the nth constraint as a state and assuming that all feasible states are eventually reachable from any feasible state we show that the efficient set of all solutions optimal to all possible feasible surplus states for p n converges to the set of optimal solutions to p a tie breaking algorithm which selects a nearest point efficient solution for p n is shown for convex programs to converge to an optimal solution to p a stopping rule is provided for discovering a value of n sufficiently large to guarantee any prespecified level of accuracy the theory is illustrated by an application to production planning

solution regularity and smooth dependence for abstract equations and applications to hyperbolic pdes first we present a generalized implicit function theorem for abstract equations of the type f u 0 we suppose that u0 is a solution for 0 and that f is smooth for all but mainly we do not suppose that f u is smooth for all u we state conditions such that for all 0 there exists exactly one solution u u0 that u is smooth in a certain abstract sense and that the data to solution map u is smooth then we apply this to time periodic solutions of first order hyperbolic systems tuj aj x xuj bj t x u 0 and second order hyperbolic equations t2u a x 2 x2u b t x u tu xu 0 here we have to prevent small divisors from coming up moreover we need smooth dependence of bj and b on t to get smooth dependence of the solution on this is completely different to the case of parabolic pdes

experiments with a local adaptive grid h refinement for the finite difference solution of bvps in singularly perturbed second order odes abstract despite many years of the development of adaptive grid methods there is still a need for re evaluation tuning and comparisons of the various approaches to grid adaptation in this study we seek to identify the most reliable and computationally inexpensive variant of the iterative grid refinement de refinement algorithm for the finite difference solution of singularly perturbed bvps in second order odes that is also easily extendable to evolutionary pdes conventional three point discretisations of the first and second derivatives are used errors of the discrete solution and its first derivative are simultaneously controlled the grid refinement is accomplished by adding nodes at inter nodal positions and de refinement by removing previously added nodes in reverse order computational experiments are performed for 15 examples of odes assuming several alternative choices of the a posteriori error estimators grid refinement indicators and regridding strategies the most satisfactory results are obtained by combining deferred approach to error estimation based on higher order reference derivative approximations for truncation error evaluation interpolation based indicator for grid refinement and regridding strategy that uses the mean indicator value as the indicator threshold for refinement

moving least squares via orthogonal polynomials a method for moving least squares interpolation and differentiation is presented in the framework of orthogonal polynomials on discrete points this yields a robust and efficient method which can avoid singularities and breakdowns in the moving least squares method caused by particular configurations of nodes in the system the method is tested by applying it to the estimation of first and second derivatives of test functions on random point distributions in two and three dimensions and by examining in detail the evaluation of second derivatives on one selected configuration the accuracy and convergence of the method are examined with respect to length scale point separation and the number of points used the method is found to be robust accurate and convergent

provably near optimal sampling based policies for stochastic inventory control models in this paper we consider two fundamental inventory models the single period newsvendor problem and its multiperiod extension but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions under the assumption that the demand distributions are given explicitly these models are well studied and relatively straightforward to solve however in most real life scenarios the true demand distributions are not available or they are too complex to work with thus a sampling driven algorithmic framework is very attractive both in practice and in theory we shall describe how to compute sampling based policies that is policies that are computed based only on observed samples of the demands without any access to or assumptions on the true demand distributions moreover we establish bounds on the number of samples required to guarantee that with high probability the expected cost of the sampling based policies is arbitrarily close i e with arbitrarily small relative error compared to the expected cost of the optimal policies which have full access to the demand distributions the bounds that we develop are general easy to compute and do not depend at all on the specific demand distributions

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

lyapunov based boundary control for a class of hyperbolic lotka volterra systems this paper considers a boundary feedback control problem for two first order nonlinearly coupled hyperbolic partial differential equations with lotka volterra type coupling boundary control action is used on one equation to drive the state at the end of the spatial domain to a desired constant reference value static and dynamic boundary controllers are designed based on a special lyapunov functional that is related to an entropy function the time derivative of the entropy function is made strictly negative by an appropriate choice of boundary conditions a unique classical solution is shown to exist globally in time and asymptotic exponential convergence to the desired steady state solution is shown in the norm the boundary control design is illustrated with simulations

an approximation scheme for stochastic integer programs arising in capacity expansion planning for capacity expansion forms a crucial part of the strategic level decision making in many applications consequently quantitative models for economic capacity expansion planning have been the subject of intense research however much of the work in this area has been restricted to linear cost models and or limited degree of uncertainty to make the problems analytically tractable this paper addresses a stochastic capacity expansion problem where the economies of scale in expansion costs are handled via fixed charge cost functions and forecast uncertainties in the problem parameters are explicitly considered by specifying a set of scenarios the resulting formulation is a multistage stochastic integer program we develop a fast linear programming based approximation scheme that exploits the decomposable structure and is guaranteed to produce feasible solutions for this problem through a probabilistic analysis we prove that the optimality gap of the heuristic solution almost surely vanishes asymptotically as the problem size increases

an ellam scheme for advection diffusion equations in two dimensions we develop an eulerian lagrangian localized adjoint method ellam to solve two dimensional advection diffusion equations with all combinations of inflow and outflow dirichlet neumann and flux boundary conditions the ellam formalism provides a systematic framework for implementation of general boundary conditions leading to mass conservative numerical schemes the computational advantages of the ellam approximation have been demonstrated for a number of one dimensional transport systems practical implementations of ellam schemes in multiple spatial dimensions that require careful algorithm development are discussed in detail in this paper extensive numerical results are presented to compare the ellam scheme with many widely used numerical methods and to demonstrate the strength of the ellam scheme

approximation algorithms for stochastic inventory control models we consider two classical stochastic inventory control models the periodic review stochastic inventory control problem and the stochastic lot sizing problem the goal is to coordinate a sequence of orders of a single commodity aiming to supply stochastic demands over a discrete finite horizon with minimum expected overall ordering holding and backlogging costs in this paper we address the important problem of finding computationally efficient and provably good inventory control policies for these models in the presence of correlated nonstationary time dependent and evolving stochastic demands this problem arises in many domains and has many practical applications in supply chain management our approach is based on a new marginal cost accounting scheme for stochastic inventory control models combined with novel cost balancing techniques specifically in each period we balance the expected cost of overordering i e costs incurred by excess inventory against the expected cost of underordering i e costs incurred by not satisfying demand on time this leads to what we believe to be the first computationally efficient policies with constant worst case performance guarantees for a general class of important stochastic inventory control models that is there exists a constant c such that for any instance of the problem the expected cost of the policy is at most c times the expected cost of an optimal policy in particular we provide a worst case guarantee of two for the periodic review stochastic inventory control problem and a worst case guarantee of three for the stochastic lot sizing problem our results are valid for all of the currently known approaches in the literature to model correlation and nonstationarity of demands over time

solution and forecast horizons for infinite horizon nonhomogeneous markov decision processes we consider a nonhomogeneous infinite horizon markov decision process mdp problem with multiple optimal first period policies we seek an algorithm that given finite data delivers an optimal first period policy such an algorithm can thus recursively generate within a rolling horizon procedure an infinite horizon optimal solution to the original problem however it can happen that no such algorithm exists i e the mdp is not well posed equivalently it is impossible to solve the problem with a finite amount of data assuming increasing marginal returns in actions with respect to states and stochastically increasing state transitions with respect to actions we provide an algorithm that is guaranteed to solve the given mdp whenever it is well posed this algorithm determines in finite time a forecast horizon for which an optimal solution delivers an optimal first period policy as an application we solve all well posed instances of the time varying version of the classic asset selling problem

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

using stochastic approximation methods to compute optimal base stock levels in inventory control problems in this paper we consider numerous inventory control problems for which the base stock policies are known to be optimal and we propose stochastic approximation methods to compute the optimal base stock levels the existing stochastic approximation methods in the literature guarantee that their iterates converge but not necessarily to the optimal base stock levels in contrast we prove that the iterates of our methods converge to the optimal base stock levels moreover our methods continue to enjoy the well known advantages of the existing stochastic approximation methods in particular they only require the ability to obtain samples of the demand random variables rather than to compute expectations explicitly and they are applicable even when the demand information is censored by the amount of available inventory

classical bounded and almost periodic solutions to quasilinear first order hyperbolic systems in a strip abstract we consider boundary value problems for quasilinear first order one dimensional hyperbolic systems in a strip the boundary conditions are supposed to be of a smoothing type in the sense that the l 2 generalized solutions to the initial boundary value problems become eventually c 2 smooth for any initial l 2 data we investigate small global classical solutions and obtain the existence and uniqueness result under the condition that the evolution family generated by the linearized problem has exponential dichotomy on r we prove that the dichotomy survives under small perturbations in the leading coefficients of the hyperbolic system assuming that the coefficients of the hyperbolic system are almost periodic we prove that the bounded solution is almost periodic also

fdem how we make the fdm more flexible than the fem the finite difference element method fdem is a black box solver for the solution of nonlinear systems of elliptic and parabolic pdes an algorithm has been developed to generate on an unstructured fem grid difference formulas of arbitrary consistency order q from the difference of difference formulas of different consistency order an estimate of the discretization error is obtained an error equation permits the explicit following of all errors and gives the prescriptions for the selfadaptation of the method coupled domains with different pdes and different nonmatching grids that slide relative to each other can be treated and a global error estimate is computed thus we get an fdm that is in all aspects more flexible than the fem the whole code is efficiently parallelized on distributed memory parallel computers

classical solutions in sobolev spaces for a class of hyperbolic lotka volterra systems this paper considers the global classical solvability well posedness of a mixed initial boundary value problem for semilinear hyperbolic systems with nonlinear reaction coupling of lotka volterra type the reaction nonlinearity is not globally lipschitz in l 2 and has lipschitz properties depending on an l infty norm bound the well posedness problem is reformulated in an abstract setting as a modified cauchy problem with homogeneous boundary conditions and solved based on the banach contraction mapping theorem extra regularity of the local solutions in sobolev spaces is shown based on moser type inequalities it is shown that global existence of classical solutions holds if a uniform a priori bound on the l infty norm of the solution and boundary term exists

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

a stochastic programming approach for planning horizons of infinite horizon capacity planning problems planning horizon is a key issue in production planning different from previous approaches based on markov decision processes we study the planning horizon of capacity planning problems within the framework of stochastic programming we first consider an infinite horizon stochastic capacity planning model involving a single resource linear cost structure and discrete distributions for general stochastic cost and demand data non markovian and non stationary we give sufficient conditions for the existence of an optimal solution furthermore we study the monotonicity property of the finite horizon approximation of the original problem we show that the optimal objective value and solution of the finite horizon approximation problem will converge to the optimal objective value and solution of the infinite horizon problem when the time horizon goes to infinity these convergence results together with the integrality of decision variables imply the existence of a planning horizon we also develop a useful formula to calculate an upper bound on the planning horizon then by decomposition we show the existence of a planning horizon for a class of very general stochastic capacity planning problems which have complicated decision structure

exponential dichotomy for hyperbolic systems with periodic boundary conditions abstract we investigate evolution families generated by general linear first order hyperbolic systems in one space dimension with periodic boundary conditions we state explicit conditions on the coefficient functions that are sufficient for the existence of exponential dichotomies on r in the space of continuous periodic functions

perturbations of superstable linear hyperbolic systems abstract the paper deals with initial boundary value problems for linear non autonomous first order hyperbolic systems whose solutions stabilize to zero in a finite time we prove that problems in this class remain exponentially stable in l 2 as well as in c 1 under small bounded perturbations to show this for c 1 we prove a general smoothing result implying that the solutions to the perturbed problems become eventually c 1 smooth for any l 2 initial data

a singularity avoiding moving least squares scheme for two dimensional unstructured meshes moving least squares interpolation schemes are in widespread use as a tool for numerical analysis on scattered data in particular they are often employed when solving partial differential equations on unstructured meshes which are typically needed when the geometry defining the domain is complex it is known that such schemes can be singular if the data points in the stencil happen to be in certain special geometric arrangements however little research has specifically addressed this issue in this paper a moving least squares scheme is presented which is an appropriate tool for use when solving partial differential equations in two dimensions and the precise conditions under which singularities occur are identified the theory is used to develop a stencil building algorithm which automatically detects singular stencils and corrects them in an efficient manner while attempting to maintain stencil symmetry as closely as possible finally the scheme is applied in a convection diffusion equation solver and an incompressible navier stokes solver and the results are shown to compare favourably with known analytical solutions and previously published results

solving nonstationary infinite horizon dynamic optimization problems monotonicity of optimal solutions to finite horizon dynamic optimization problems is used to prove the existence of a forecast horizon i e a long enough planning horizon that ensures that a first period optimal action for the infinite horizon and the finite horizon problem agree regardless of problem parameter changes in the tail the existence of extremal monotone optimal solutions motivates a stopping rule that is ensured to detect the minimal forecast horizon to be used in a rolling horizon procedure to exactly solve the problem

conditions for the discovery of solution horizons we present necessary and sufficient conditions for discrete infinite horizon optimization problems with unique solutions to be solvable these problems can be equivalently viewed as the task of finding a shortest path in an infinite directed network we provide general forward algorithms with stopping rules for their solution the key condition required is that of weak reachability which roughly requires that for any sequence of nodes or states it must be possible from optimal states to reach states close in cost to states along this sequence moreover the costs to reach these states must converge to zero applications are considered in optimal search undiscounted markov decision processes and deterministic infinite horizon optimization

adaptive meshless centres and rbf stencils for poisson equation we consider adaptive meshless discretisation of the dirichlet problem for poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions new meshless stencil selection and adaptive refinement algorithms are proposed in 2d numerical experiments show that the accuracy of the solution is comparable with and often better than that achieved by the mesh based adaptive finite element method

hopf bifurcation for semilinear dissipative hyperbolic systems abstract we consider boundary value problems for semilinear hyperbolic systems of the type t u j a j x x u j b j x u 0 x 0 1 j 1 n with smooth coefficient functions a j and b j such that b j x 0 0 for all x 0 1 r and j 1 n we state conditions for hopf bifurcation i e for existence local uniqueness up to phase shifts smoothness and smooth dependence on of time periodic solutions bifurcating from the zero stationary solution furthermore we derive a formula which determines the bifurcation direction the proof is done by means of a liapunov schmidt reduction procedure for this purpose fredholm properties of the linearized system and implicit function theorem techniques are used

conditions for the existence of planning horizons we consider the general class of discounted problems involving sequential decision making over an unbounded horizon under the assumptions of a finite set of policy alternatives at each decision time and costs that are eventually uniformly bounded by some exponential existence of an optimal infinite horizon strategy is established moreover it is shown that there is a finite horizon version whose first policy decision agrees with an infinite horizon optimal policy under the additional assumption that the infinite horizon optimal strategy is eventually cyclic the stronger result follows that the infinite horizon optimal solution is unique for almost all interest rates and that therefore a planning horizon exists algorithmic implications are explored for a restricted subclass of these problems

a perturbation theorem for exponential dichotomies suppose the linear equation x a t x has an exponential dichotomy and suppose b t is close to a t in the following sense on any interval of length 2t b t is close to some translate a t of a t actually the conditions in the paper are slightly weaker than this then if t is sufficiently large the equation y b t y also has an exponential dichotomy this generalises the usual roughness theorem for exponential dichotomies

how we solve pdes a finite difference method on an unstructured finite element mesh which we call finite difference element method fdem is presented the fdem program package will be a black box solver for nonlinear systems of elliptic and parabolic pdes with mesh refinement and automatic control of the consistency order in each space grid point in this paper we present the solution method with examples for 2 d systems of elliptic pdes

concepts of forecast and decision horizons applications to dynamic stochastic optimization problems in this paper we develop a rigorous new framework for the concepts of forecast and decision horizons these concepts are conditional in nature and in turn enable us to unify the existing concepts of strong and weak horizons moreover we are able to precisely state the questions of existence of forecast decision horizons finally we are able to further develop the theory of existence of forecast decision horizons in the discounted case and procedures to compute these horizons

reformulation of the multiperiod milp model for capacity expansion of chemical processes the problem of selecting processes and capacity expansion policies for a chemical complex consisting of continuous chemical processes can be formulated as a multiperiod mixed integer linear programming milp problem based on a variable disaggregation technique which exploits lot sizing substructures we propose two reformulations of the conventional milp model the first one is an nlp reformulation which very quickly yields good suboptimal solutions the second is an milp reformulation for exact solutions which leads to up to an order of magnitude faster computational results for large problems due to its tighter linear programming relaxation