reliable and robust a posteriori error estimation for singularly perturbed reaction diffusion problems problems with singular perturbations exhibit solutions with strong boundary layers and other types of local behavior such features lend themselves to adaptive solution methods the quality of any adaptive algorithm ultimately rests on the reliability and robustness of the a posteriori error control an estimator that has proved to be one of the most reliable is the equilibrated residual method the main property of the estimator is that it bounds the true error from above however the method is not robust in the singularly perturbed limit the current work generalizes the error estimator based on the equilibrated residuals and coincides with the standard method in the unperturbed limit it is shown that the new method is robust in the singularly perturbed limit while maintaining reliability yielding a guaranteed upper bound on the true error finally the application of the estimator to the problem of controlling the spatial error in rothe s method for the time discretization of a simple parabolic problem is included

note multiplicative interval variation of objective function coefficients in linear programming traditional sensitivity analysis of linear programming objective function coefficients concerns itself with variations in single parameters in a recent book gal developed a theoretical framework to determine the effect of multiple variations in the parameters in this paper we develop a methodology to implement and extend the theoretical results in the earlier work for interval variations of objective function coefficients this methodology uses necessary and sufficient conditions to determine all optimal solutions to the linear programming problem for objective function coefficients within given intervals the resulting procedure has been coded for computer use and tested on various types of test problems computational results and an example are presented

flux recovery and a posteriori error estimators conforming elements for scalar elliptic equations in this paper we first study two flux recovery procedures for the conforming finite element approximation to general second order elliptic partial differential equations one is accurate in a weighted l 2 norm studied in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 for linear elements and the other is accurate in a weighted h mathrm div norm up to the accuracy of the current finite element approximation for the l 2 recovered flux we introduce and analyze an a posteriori error estimator that is more accurate than the explicit residual based estimator based on the h mathrm div recovered flux we introduce two a posteriori error estimators one estimator may be regarded as an extension of the recovery based estimator studied in z cai and s zhang siam j numer anal 47 2009 pp 2132 2156 to higher order conforming elements the global reliability and the local efficiency bounds for this estimator are established provided that the underlying problem is neither convection nor reaction dominant the other is proved to be exact locally and globally on any given mesh with no regularity assumptions with respect to a norm depending on the underlying problem numerical results on test problems for these estimators are also presented

guaranteed and robust discontinuous galerkin a posteriori error estimates for convection diffusion reaction problems we propose and study a posteriori error estimates for convection diffusion reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior penalty discontinuous galerkin methods our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction we first derive locally computable estimates for the error measured in the energy semi norm these estimates are evaluated using h div w conforming diffusive and convective flux reconstructions thereby extending the previous work on pure diffusion problems the resulting estimates are semi robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local peclet and damkohler numbers fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy semi norm a dual norm of the skew symmetric part of the differential operator and a suitable contribution of the interelement jumps of the discrete solution numerical experiments are presented to illustrate the theoretical results

right inverses of nonsymmetric levy processes we analyze the existence and properties of right inverses k for nonsymmetric levy processes x extending recent work of evans 7 in the symmetric setting first both x and x have right inverses if and only if x is recurrent and has a nontrivial gaussian component our main result is then a description of the excursion measure n z of the strong markov process z x l reflected process where lt inf x 0 kx t specifically n z is essentially the restriction of n x to the excursions starting negative second when only asking for right inverses of x a certain strength of asymmetry is needed millar s 9 notion of creeping turns out necessary but not sufficient for the existence of right inverses we analyze this both in the bounded and unbounded variation case with a particular emphasis on results in terms of the levy khintchine characteristics 1 introduction consider a real valued levy process that is a continuous time process with stationary independent increments and cadlag paths right inverses of levy processes have first been studied in the symmetric setting by evans 7 he defines an increasing process k to be a right inverse of a levy process x if it satisfies x kx x for all x 0 the minimal such k turns out to be a subordinator i e an increasing levy process specifically evans characterized the symmetric levy processes that have right inverses he introduced lt inf x 0 kx t and the reflected process z x l and showed that z is strong markov with local time l at zero he gave some fluctuation type identities for z and l and proved formulae for the entrance laws to the excursion measure n z of z evans also showed that k is distributed like a linear time change of the inverse local time of x at zero in this work we basically answer three questions left open by evans first what happens when symmetry fails the only result of evans that is clearly based on symmetry is the coincidence of the laws of k and the inverse local time of x in zero also the class of processes possessing inverses is not restricted to those having a positive gaussian component in the symmetric setting this is evans s characterization result but those form again an important class we provide a further characterization in the bounded variation case and we also point

subordination in the wide sense for levy processes given t t 0 a transient convolution semigroup over rd we characterize the transient semi groups which are subordinate in the wide sense to the latter via some quotient relation between the two levy generators this is also equivalent to an inclusion between excessive cones thanks to a theorem of hirsch this allows us in big dimensions to give a probabilistic description of some measures satisfying the classical maximum principle

a simple approach to reliable and robust a posteriori error estimation for singularly perturbed problems abstract a simple flux reconstruction for finite element solutions of reaction diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approximation of singularly perturbed reaction diffusion problem the flux reconstruction is based on simple independent post processing operations over patches of elements in conjunction with standard raviart thomas vector fields we prove that the corresponding local error indicators are locally efficient and robust up to a nonstandard oscillation term with respect to any mesh size and any size of the reaction coefficient including the singularly perturbed limit

algorithms for linear programming problems with interval objective function coefficients this paper presents three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals which algorithm is applicable depends upon a the number of interval objective function coefficients b the number of nonzero objective function coefficients and c whether or not the feasible region is bounded the algorithms output all extreme points and unbounded edge directions that are multiparametrically optimal with respect to the ranges placed on the objective function coefficients the algorithms are most suitable to linear programs in which the objective function coefficients are deterministic but are likely to vary from time period to time period as for example in blending problems

robust error bounds for finite element approximation of reaction diffusion problems with non constant reaction coefficient in arbitrary space dimension abstract we present a fully computable a posteriori error estimator for piecewise linear finite element approximations of reaction diffusion problems with mixed boundary conditions and piecewise constant reaction coefficient formulated in arbitrary dimension the estimator provides a guaranteed upper bound on the energy norm of the error and it is robust for all values of the reaction coefficient including the singularly perturbed case the approach is based on robustly equilibrated boundary flux functions of ainsworth and oden 2000 and on subsequent robust and explicit flux reconstruction this paper simplifies and extends the applicability of the previous result of ainsworth and vejchodsk 2011 in three aspects i arbitrary dimension ii mixed boundary conditions and iii non constant reaction coefficient it is the first robust upper bound on the error with these properties an auxiliary result that is of independent interest is the derivation of new explicit constants for two types of trace inequalities on simplices

special cases of the tolerance approach in multiobjective linear programming abstract in this paper we deal with sensitivity analysis of the multiobjective linear problem we enlarge the range of meaningful regions of weights that can be handled easily from the tolerance approach point of view the regions that we propose are important because they represent the information that the decision maker is able to offer about the relationships between the importance of the different objectives in terms of marginal substitution rates particularly the well known interval order and nonhomogeneous order relations are included we also present an algorithm which reduces the computational effort in order to get the maximum tolerance percentage several examples are included illustrating the results

tolerance sensitivity and optimality bounds in linear programming traditional sensitivity analysis in linear programming usually focuses on variations of one coefficient or term at a time the tolerance approach was proposed to provide a decision maker with an effective and easy to use method to summarize the effects of simultaneous and independent changes in selected parameters in particular for variations of the objective function coefficients the approach gives a maximum tolerance percentage within which selected coefficients may vary from their estimated values within a priori limits while still retaining the same optimal basic feasible solution although an optimal solution may cease being optimal for variations beyond the maximum tolerance percentage it may still be close to optimal herein we characterize the potential loss of optimality for variations beyond the maximum tolerance percentage as a maximum regret function we consider theoretical properties of this function and propose a method to compute a relevant portion of it

sensitivity analysis in minisum facility location problems this paper considers tradeoffs between cost and perturbations of the weights in a single facility minisum location problem over a finite set of feasible points specifically the paper characterizes a tradeoff curve of variability in the weights versus the degree of optimality of a solution the general theoretical framework includes problems of location on a tree network and location using block norms special results are given for these cases

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

embedding processes in brownian motion in r n we give a potential theoretic characterization of the right continuous processes which can be embedded in brownian motion in r by means of an increasing family of standard stopping times in general it is necessary to use a brownian motion process whose filtration is richer than the natural one

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains this is the pre peer reviewed version of the following article pares n santos h diez p guaranteed energy error bounds for the poisson equation using a flux free approach solving the local problems in subdomains international journal for numerical methods in engineering 23 marc 2009 vol 79 num 10 p 1203 1244 which has been published in final form at http www3 interscience wiley com journal 122269073 abstract

multi parametric analysis of the maximum tolerance in a linear programming problem abstract this study intends to propose an alternative approach to the multi parametric sensitivity analysis of a linear programming problem by the concept of a maximum volume in the tolerance region for the purposes of management and control the study first proposes the necessary and sufficient conditions of the focal parameters then those nonfocal parameters that have unlimited variations can be identified and thus because of their low sensitivity in practice and simplicity in analysis they will be deleted for those focal parameters their different levels of sensitivity are investigated and conceived by the proposed maximum volume region so that they can be handled with the greatest flexibility simultaneously and independently this maximum volume region is bounded by a symmetrically rectangular parallelepiped it can be characterized by a maximization problem and solved by the existing technique dynamic programming besides an extension to the extended maximum volume region for the case of semi bounded region is considered theoretical proofs are provided with numerical illustrations the result has been compared to wendell s approach

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

a fresh view on the tolerance approach to sensitivity analysis in linear programming abstract the tolerance approach to sensitivity analysis in linear programming aims at finding a unique numerical value tolerance representing the maximum absolute perturbation which can be applied simultaneously and independently on each right hand side or objective coefficient without affecting the optimality of the given basis some extensions have been proposed in the literature which allow for individual tolerances for each coefficient thus enlarging the tolerance region in this paper we review the main results concerning the approach giving new and simpler proofs and we propose an efficient geometric algorithm returning a tolerance region that is maximal with respect to inclusion we compare our method with the existing ones on two examples showing how a priori information can be naturally exploited by our algorithm to further enlarge individual tolerances

robust a posteriori error estimators for a singularly perturbed reaction diffusion equation we derive robust a posteriori error estimators for a singularly perturbed reaction diffusion equation here robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter the estimators are based either on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

the tolerance approach to sensitivity analysis of matrix coefficients in linear programming the tolerance approach to sensitivity analysis allows for simultaneous and independent variations of the elements of a column or a row of the coefficient matrix in a standard linear programming problem in particular the approach yields a maximum tolerance percentage within which the elements of a column may all vary simultaneously and independently from their estimated values while still retaining the same set of basic variables in an optimal solution a similar result is also derived for the perturbations of the elements of a row

global sensitivity analysis in applications of operations research models decision makers must assess the sensitivity of outputs to imprecise values for some of the model s parameters existing analytic approaches for classic optimization models rely heavily on duality properties for assessing the impact of local parameter variations parametric programming for examining systematic variations in model coefficients or stochastic programming for ascertaining a robust solution this paper accommodates extensive simultaneous variations in any of an operations research model s parameters for constrained optimization models the paper demonstrates practical approaches for determining relative parameter sensitivity with respect to a model s optimal objective function value decision variables and other analytic functions of a solution relative sensitivity is assessed by assigning a portion of variation in an output value to each parameter that is imprecisely specified the computing steps encompass optimization monte carlo sampling

using bounds on the data in linear programming the tolerance approach to sensitivity analysis in contrast to traditional sensitivity analysis in linear programming the tolerance approach considers simultaneous and independent variations in a number of parameters a primary focus of this approach is to determine a maximum tolerance percentage for selected right hand side terms in which the same basis is optimal as long as each term is accurate to within that percentage of its estimated value similarly the approach yields a maximum tolerance percentage for selected objective function coefficients this paper shows how the tolerance approach can exploit information on the range of possible values over which terms and coefficients can vary to yield larger maximum tolerance percentages

sensitivity analysis in multiple objective linear programming the tolerance approach abstract we consider a multiple objective linear program solved by the weighted sum approach a method is proposed to determine the maximum percentage by which all weights can deviate simultaneously and independently from their estimated values while retaining the same optimal basis it is also shown how a priori bounds on the ranges of the weights can be exploited to yield a larger maximum tolerance percentage

a posteriori error estimates including algebraic error and stopping criteria for iterative solvers for the finite volume discretization of a second order elliptic model problem we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system we show that the algebraic error can be bounded by constructing an equilibrated raviart thomas nedelec discrete vector field whose divergence is given by a proper weighting of the residual vector next claiming that the discretization error and the algebraic one should be in balance we construct stopping criteria for iterative algebraic solvers an attention is paid in particular to the conjugate gradient method which minimizes the energy norm of the algebraic error using this convenient balance we also prove the efficiency of our a posteriori estimates i e we show that they also represent a lower bound up to a generic constant for the overall energy error a local version of this result is also stated this makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers

robust a posteriori error estimation for nonconforming finite element approximation the equilibrated residual method for a posteriori error estimation is extended to nonconforming finite element schemes for the approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media the estimator is shown to provide a computable upper bound on the error and up to a constant depending only on the geometry provides two sided bounds on the error the robustness of the estimator is also studied and the dependence of the constant on the jumps in permeability is given explicitly

fully computable robust a posteriori error bounds for singularly perturbed reaction diffusion problems a procedure for the construction of robust upper bounds for the error in the finite element approximation of singularly perturbed reaction diffusion problems was presented in ainsworth and babuska siam j numer anal 36 2 331 353 1999 which entailed the solution of an infinite dimensional local boundary value problem it is not possible to solve this problem exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a subsequent article we view the present work as fulfilling that promise and as completing the investigation begun in ainsworth and babuska siam j numer anal 36 2 331 353 1999 by removing the obligation to solve a local problem exactly the resulting new estimator is indeed fully computable and the first to provide fully computable robust upper bounds in the setting of singularly perturbed problems discretised by the finite element method

two sided bounds for eigenvalues of differential operators with applications to friedrichs poincare trace and similar constants we present a general numerical method for computing guaranteed two sided bounds for principal eigenvalues of symmetric linear elliptic differential operators the approach is based on the galerkin method on the method of a priori a posteriori inequalities and on a complementarity technique the two sided bounds are formulated in a general hilbert space setting and as a byproduct we prove an abstract inequality of friedrichs poincare type the abstract results are then applied to friedrichs poincare and trace inequalities and fully computable two sided bounds on the optimal constants in these inequalities are obtained accuracy of the method is illustrated in numerical examples

maximal and supremal tolerances in multiobjective linear programming abstract let a multiobjective linear programming problem and any efficient solution be given tolerance analysis aims to compute interval tolerances for possibly all objective function coefficients such that the efficient solution remains efficient for any perturbation of the coefficients within the computed intervals the known methods either yield tolerances that are not the maximal possible ones or they consider perturbations of weights of the weighted sum scalarization only we focus directly on perturbations of the objective function coefficients which makes the approach independent on a scalarization technique used in this paper we propose a method for calculating the supremal tolerance the maximal one need not exist the main disadvantage of the method is the exponential running time in the worst case nevertheless we show that the problem of determining the maximal supremal tolerance is np hard so an efficient polynomial time procedure is not likely to exist we illustrate our approach on examples and present an application in transportation problems since the maximal tolerance may be small we extend the notion to individual lower and upper tolerances for each objective function coefficient an algorithm for computing maximal individual tolerances is proposed

the tolerance approach to sensitivity analysis in linear programming in constrast to ordinary sensitivity analysis in linear programming the tolerance approach considers simultaneous and independent changes in the objective function coefficients and in the right hand side terms this approach yields a maximum tolerance percentage such that as long as selected coefficients or terms are accurate to within that percentage of their estimated values the same basis is optimal in particular if the objective function coefficients are accurate to within the maximum tolerance percentage of their specified values then the same solution is optimal

a posteriori error estimation for discontinuous galerkin finite element approximation it is shown that the interelement discontinuities in a discontinuous galerkin finite element approximation are subordinate to the error measured in the broken h 1 seminorm one consequence is that the dg norm of the error is equivalent to the broken energy seminorm computable a posteriori error bounds are obtained for the error measured in both the dg norm and the broken energy seminorm and are shown to be efficient in the sense that they also provide lower bounds up to a constant and higher order data oscillation terms the estimators are completely free of unknown constants and provide guaranteed numerical bounds for the error

enlarging the radius of stability and stability regions in data envelopment analysis abstract this paper extends fundamental results on metric sensitivity in data envelopment analysis specifically it shows how to obtain a larger radius of stability for a decision making unit dmu by exploiting knowledge about its variability and how to enlarge a dmu s region of stability to a nonsymmetric hyperbox

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

a new equilibrated residual method improving accuracy and efficiency of flux free error estimates this paper presents a new methodology to compute guaranteed upper bounds for the energy norm of the error in the context of linear finite element approximations of the reaction diffusion equation the new approach revisits the ideas in pares et al 2009 6 4 with the goal of substantially reducing the computational cost of the flux free method while retaining the good quality of the bounds the new methodology provides also a technique to compute equilibrated boundary tractions improving the quality of standard equilibration strategies the zeroth order equilibration conditions are imposed using an alternative less restrictive form of the first order equilibration conditions along with a new efficient minimization criterion this new equilibration strategy provides much more accurate upper bounds for the energy and requires only doubling the dimension of the local linear systems of equations to be solved

the self equilibration of residuals and complementary a posteriori error estimates in the finite element method it is demonstrated that the residual in a compatible displacement finite element solution can be partitioned into local self equilibrating systems on each element an a posteriori error analysis is then based on a complementary approach and examples indicate that the guaranteed upper bound on the energy of the error is preserved

construction of the largest sensitivity region for general linear programs this paper develops an alternative approach to post optimality analysis for general linear program models which provides the largest sensitivity region of any single or simultaneous change of right hand side of constraints and the coefficients of decision variables in the objective function the goal is a theoretical unification of various types of sensitivity analyses as well as advancement in the practical implementation of post optimality analysis we extend our proposed approach in the construction of sensitivity region to maintain the degenerate vertex for models with degenerate optimal solution as well as maintaining the multiple solutions for the models with non unique optimal solutions as a by product the paper resolves the paradoxical situation known as the more for less less for more situations the proposed method is based on the given optimal solution s therefore it is easy to understand easy to implement and provides useful information to the decision makers the methodology and their computational algorithms are presented and discussed in the context of some illustrative numerical examples

on the complexity of a class of combinatorial optimization problems with uncertainty we consider a robust minmax regret version of the problem of selecting p elements of minimum total weight out of a set of m elements with uncertainty in weights of the elements we present a polynomial algorithm with the order of complexity o min p m p 2m for the case where uncertainty is represented by means of interval estimates for the weights we show that the problem is np hard in the case of an arbitrary finite set of possible scenarios even if there are only two possible scenarios this is the first known example of a robust combinatorial optimization problem that is np hard in the case of scenario represented uncertainty but is polynomially solvable in the case of the interval representation of uncertainty