fuzzy rough set based active learning determining the informativeness of unlabeled samples is a key issue in active learning one solution to this is using the sample s in consistency between conditional features and decision labels in this paper a fuzzy rough set based active learning model is proposed to tackle this problem first the consistence degree of a labeled sample is computed by the lower approximations in fuzzy rough set which reflects its minimum membership in the decision class then the concept of sample covering is proposed to measure the relationship between labeled samples and un labeled samples afterward the memberships of an unlabeled sample be longing to different decision classes are computed based on the covering degrees of labeled samples on it finally these memberships are used to form a sample selection criterion to measure the sample s inconsistency by applying gaussian kernel based similarity relation to the aforemen tioned processes a support vector machine svm based active learning scheme is developed experimental results demonstrate the effectiveness of the proposed model

approximate primal solutions and rate analysis for dual subgradient methods in this paper we study methods for generating approximate primal solutions as a byproduct of subgradient methods applied to the lagrangian dual of a primal convex possibly nondifferentiable constrained optimization problem our work is motivated by constrained primal problems with a favorable dual problem structure that leads to efficient implementation of dual subgradient methods such as the recent resource allocation problems in large scale networks for such problems we propose and analyze dual subgradient methods that use averaging schemes to generate approximate primal optimal solutions these algorithms use a constant stepsize in view of its simplicity and practical significance we provide estimates on the primal infeasibility and primal suboptimality of the generated approximate primal solutions these estimates are given per iteration thus providing a basis for analyzing the trade offs between the desired level of error and the selection of the stepsize value our analysis relies on the slater condition and the inherited boundedness properties of the dual problem under this condition it also relies on the boundedness of subgradients which is ensured by assuming the compactness of the constraint set

on the use of higher order projection methods for incompressible turbulent flow an important issue in the development of higher order methods for incompressible flow is how they perform when the flow is turbulent a useful diagnostic of a method for turbulent flow is the minimum resolution that is required to adequately resolve the turbulent energy cascade at a given reynolds number in this paper we present careful numerical experiments to assess the utility of higher order numerical methods based on this metric we first introduce a numerical method for the incompressible navier stokes equations based on fourth order discretizations in both space and time the method is based on an auxiliary variable formulation and combines fourth order finite volume differencing with a semi implicit spectral deferred correction temporal integration scheme we also introduce for comparison purposes versions based on second order spatial and or temporal discretizations we demonstrate that for smooth problems each of the methods exhibits the expected order of convergence in time and space we

an extension of the lax richtmyer theory we give a rigorous proof of the validity of the lax equivalence theorem when the true solution at timet and its approximation lie in different spaces thus modelling the practical situation where the former is a function of the space variables and the latter is only defined at grid points

submodular functions from discrete to continous domains submodular set functions have many applications in combinatorial optimization as they can be minimized and approximately maximized in polynomial time a key element in many of the algorithms and analyses is the possibility of extending the submodular set function to a convex function which opens up tools from convex optimization submodularity goes beyond set functions and has naturally been considered for problems with multiple labels or for functions defined on continuous domains where it corresponds essentially to cross second derivatives being nonpositive in this paper we show that most results relating submodularity and convexity for set functions can be extended to all submodular functions in particular a we naturally define a continuous extension in a set of probability measures b show that the extension is convex if and only if the original function is submodular c prove that the problem of minimizing a submodular function is equivalent to a typically non smooth convex optimization problem and d propose another convex optimization problem with better computational properties e g a smooth dual problem most of these extensions from the set function situation are obtained by drawing links with the theory of multi marginal optimal transport which provides also a new interpretation of existing results for set functions we then provide practical algorithms to minimize generic submodular functions on discrete domains with associated convergence rates

on total variation minimization and surface evolution using parametric maximum flows in a recent paper boykov et al lncs vol 3953 pp 409 422 2006 propose an approach for computing curve and surface evolution using a variational approach and the geo cuts method of boykov and kolmogorov international conference on computer vision pp 26 33 2003 we recall in this paper how this is related to well known approaches for mean curvature motion introduced by almgren et al siam journal on control and optimization 31 2 387 438 1993 and luckhaus and sturzenhecker calculus of variations and partial differential equations 3 2 253 271 1995 and show how the corresponding problems can be solved with sub pixel accuracy using parametric maximum flow techniques this provides interesting algorithms for computing crystalline curvature motion possibly with a forcing term

a stability result for sectorial operators in branch spaces in this paper the author considers approximations of holomorphic semigroups on general banach spaces based on a acceptable rational functions a general stability theorem is proved it covers rational functions r z with r infty 1 and can be applied even to variable step cases in particular the question of the text l p stability p ne 2 of crank nicolson schemes for parabolic initial value problems is settled

avoiding order reduction when integrating nonlinear schrodinger equation with strang method in this paper a technique is suggested to avoid order reduction when using strang method to integrate nonlinear schrodinger equation subject to time dependent dirichlet boundary conditions the computational cost of this technique is negligible compared to that of the method itself at least when the timestepsize is fixed moreover a thorough error analysis is given as well as a modification of the technique which allows to conserve the symmetry of the method while retaining its second order

galerkin runge kutta discretizations for parabolic equations with time dependent coefficients a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients unlike any classical counterpart this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction in support of this claim error estimates are proved and computational results are presented additionally since the time stepping equations involve coefficient matrices changing at each time step a preconditioned iterative technique is used to solve the linear systems only approximately nevertheless the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

three way group decisions with decision theoretic rough sets gdm based three way decisions as an extension model of dtrss is proposed gdm provides a new semantic interpretation for three way decisions our method explains the imprecise origin of the existing literatures on consideration of the effectiveness of group decision making gdm in practical complex decision problems we introduce gdm into three way decisions with decision theoretic rough sets dtrss and propose gdm based three way decisions gdm based three way decisions extend the range of applications of three way decisions with dtrss and provide a novel interpretation of the determination of loss functions based on dtrss we firstly focus on analyzing the determination for the loss functions under the gdm environment with the aid of the principle of justifiable granularity we adopt the important and majority suggestions of experts to measure each loss function which supports a coherent way of designing information granules in presence of numerics in this case the loss functions are determined in the form of interval valued information granule by using the interval comparison method we further deduce the three way decisions and design a corresponding decision procedure of gdm based three way decisions then an example of strategy supply selection is given to elaborate the gdm based three way decisions finally we validate the performance of our proposed method by experimental analysis

a novel risk decision making based on decision theoretic rough sets under hesitant fuzzy information decision theoretic rough sets dtrss play a crucial role in risk decision making problems with respect to the minimum expected risk dtrss deduce the rules of three way decisions considering the new expression of evaluation information with hesitant fuzzy sets hfss we introduce hfss into dtrss and explore their decision mechanisms more specifically we take into account the losses of dtrss with hesitant fuzzy elements and propose a new model of hesitant fuzzy decision theoretic rough sets hfdtrss some properties of the expected losses and their corresponding scores are carefully investigated under the hesitant fuzzy information three way decisions and the associated cost of each object are further derived with the above analysis a novel risk decision making method with the aid of hfdtrss is developed besides the three way decisions with dtrss the method investigates the ranking and resource allocation by utilizing the associated costs of alternatives and multiobjective 0 1 integer programming our study also offers a solution in the aspect of determining losses of dtrs and extends the range of applications

arbitrary order krylov deferred correction methods for differential algebraic equations in this paper a new framework for the construction of accurate and efficient numerical methods for differential algebraic equation dae initial value problems is presented the methods are based on applying spectral deferred correction techniques as preconditioners to a picard integral collocation formulation for the solution the resulting preconditioned nonlinear system is solved using newton krylov schemes such as the newton gmres method least squares based orthogonal polynomial approximations are computed using gaussian type quadratures and spectral integration is used to avoid the numerically unstable differentiation operator the resulting krylov deferred correction kdc methods are of arbitrary order of accuracy and very stable preliminary results show that these new methods are very competitive with existing dae solvers particularly when high precision is desired

active sample selection based incremental algorithm for attribute reduction with rough sets attribute reduction with rough sets is an effective technique for obtaining a compact and informative attribute set from a given dataset however traditional algorithms have no explicit provision for handling dynamic datasets where data present themselves in successive samples incremental algorithms for attribute reduction with rough sets have been recently introduced to handle dynamic datasets with large samples though they have high complexity in time and space to address the time space complexity issue of the algorithms this paper presents a novel incremental algorithm for attribute reduction with rough sets based on the adoption of an active sample selection process and an insight into the attribute reduction process this algorithm first decides whether each incoming sample is useful with respect to the current dataset by the active sample selection process a useless sample is discarded while a useful sample is selected to update a reduct at the arrival of a useful sample the attribute reduction process is then employed to guide how to add and or delete attributes in the current reduct the two processes thus constitute the theoretical framework of our algorithm the proposed algorithm is finally experimentally shown to be efficient in time and space

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

higher order temporal integration for the incompressible navier stokes equations in bounded domains author s minion ml saye ri abstract 2018 elsevier inc this paper compares and contrasts higher order semi implicit temporal integration strategies for the incompressible navier stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations particular focus is placed on the imposition of boundary conditions in the semi implicit formulation the accurate treatment of the pressure term and the smoothness of the numerical solution for different formulations the main result presented here is the formulation and numerical validation of a single step semi implicit method called spectral deferred pressure corrections which can in theory obtain arbitrary formal order of accuracy in time numerical results demonstrate up to eighth order accuracy scenarios where optimal temporal accuracy is attained and scenarios where order reduction is observed due to time dependent boundary conditions

semi implicit krylov deferred correction methods for differential algebraic equations in the recently developed krylov deferred correction kdc methods for differential algebraic equation initial value problems 33 a picardtype collocation formulation is preconditioned using low order time integration schemes based on spectral deferred correction sdc and the resulting system is solved efficiently using newton krylov methods kdc methods have the advantage that methods with arbitrarily high order of accuracy can be easily constructed which have similar computational complexity as lower order methods in this paper we investigate semi implicit kdc si kdc methods in which the stiff component of the preconditioner is treated implicitly and the non stiff parts explicitly for certain types of problems such a semi implicit treatment can significantly reduce the computational cost of the preconditioner compared to fully implicit kdc fi kdc methods preliminary analysis and numerical experiments show that the convergence of newton krylov iterations in the si kdc methods is similar to that in fi kdc and hence the si kdc methods offer a reduction in overall computational cost for such problems

rough set based conflict analysis model and method over two universes rough set theory as a new mathematical tool to handle uncertainty decision making problems was used to study conflict analysis decision making in late90 s and then the pawlak conflict analysis model was established the approach of rough set provides a new perspective and also gives an effective tool to deal with conflict analysis decision making both theoretically and practically in this paper we propose a new rough set model of conflict analysis a conflict analysis decision model based on rough set theory over two universes it is a natural extension of the pawlak conflict analysis model subsequently we define the conflict matrix and a consistent disagreement matrix on the basis of the model proposed in this paper then we can find the intrinsic reasons for conflict and attain a feasible strategy for a conflict situation furthermore we define a positive alliance matrix and a negative alliance matrix for a conflict situation these can help us analyze the opinions of different agents in the conflict situation moreover we develop a matrix approach for a conflict analysis model based on rough set theory over two universes which provides a convenient way to analyze and solve the conflict situation comparing to the pawlak conflict analysis model our proposed model not only could reveal the core causes for a conflict situation but also can find a possible optimal feasible consensus strategy to solve the conflict situation which satisfies the agents as much as possible finally we illustrate the idea and the basic principles established in this paper by analyzing a conflict decision making scenario

attribute reduction in decision theoretic rough set models rough set theory can be applied to rule induction there are two different types of classification rules positive and boundary rules leading to different decisions and consequences they can be distinguished not only from the syntax measures such as confidence coverage and generality but also the semantic measures such as decision monotocity cost and risk the classification rules can be evaluated locally for each individual rule or globally for a set of rules both the two types of classification rules can be generated from and interpreted by a decision theoretic model which is a probabilistic extension of the pawlak rough set model as an important concept of rough set theory an attribute reduct is a subset of attributes that are jointly sufficient and individually necessary for preserving a particular property of the given information table this paper addresses attribute reduction in decision theoretic rough set models regarding different classification properties such as decision monotocity confidence coverage generality and cost it is important to note that many of these properties can be truthfully reflected by a single measure c in the pawlak rough set model on the other hand they need to be considered separately in probabilistic models a straightforward extension of the c measure is unable to evaluate these properties this study provides a new insight into the problem of attribute reduction

fully discrete convergence estimates for vortex methods in bounded domains in this paper the authors study vortex method for 2 dimensionsal euler equations of incompressible flow in bounded domains to approximate the initial vortex field by a sum of vortex blobs with arbitrary high accuracy this field is extended smoothly to a small neighborhood of the boundary and the computation is carried out in the extended domain to construct the velocity field from vorticity a second order isoparametric finite element method is applied and to solve the characteristic equations the explicit euler s scheme is considered optimal error bounds for this fully discrete scheme are obtained

galerkin runge kutta discretizations for semilinear parabolic equations a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems unlike any classical counterpart this class offers arbitrarily high order convergence without suffering from what has been called order reduction in support of this claim error estimates are proved and computational results are presented furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

submodular spectral functions of principal submatrices of a hermitian matrix extensions and applications we extend the multiplicative submodularity of the principal determinants of a nonnegative definite hermitian matrix to other spectral functions we show that if f is the primitive of a function that is operator monotone on an interval containing the spectrum of a hermitian matrix a then the function i mapsto rm tr f a i is supermodular meaning that rm tr f a i rm tr f a j leq rm tr f a i cup j rm tr f a i cap j where a i denotes the i times i principal submatrix of a we discuss extensions to self adjoint operators on infinite dimensional hilbert space and to m matrices we discuss an application to cur approximation of nonnegative hermitian matrices

stability analyticity and almost best approximation in maximum norm for parabolic finite element equations we consider semidiscrete solutions in quasi uniform finite element spaces of order o hr of the initial boundary value problem with neumann boundary conditions for a second order parabolic differential equation with time independent coefficients in a bounded domain in r n we show that the semigroup on l defined by the semidiscrete solution of the homogeneous equation is bounded and analytic uniformly in h we also show that the semidiscrete solution of the inhomogeneous equation is bounded in the space time l norm modulo a logarithmic factor for r 2 and we give a corresponding almost best approximation property

combined effect of explicit time stepping and quadrature for curvature driven flows the flow of a closed surface of codimension 1 in bbb r n driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter varepsilon 0 conforming piecewise linear finite elements with mass lumping over a quasi uniform and weakly acute mesh of size h are further used for space discretization and combined with forward differences for time discretization with uniform time step tau the resulting explicit schemes are the basis for an efficient algorithm the so called dynamic mesh algorithm and exhibit finite speed of propagation and discrete ondegeneracy no iteration is required not even to handle the obstacle constraints the zero level set of the fully discrete solution is shown to converge past singularities to the true interface provided tau h 2 approx o varepsilon 4 and no fattening occurs if the more stringent relations tau h 2 approx o varepsilon 6 are enforced then an interface rate of convergence o varepsilon is derived in the vicinity of regular points along with a companion o varepsilon 1 2 for type i singularities for smooth flows an interface rate of convergence of o varepsilonq is proven provided tau h 2 approx o varepsilon 5 and exact integration is used for the potential term the analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature which bears some intrinsic interest lipschitz properties of viscosity solutions of the level set approach and discrete nondegeneracy these basic ingredients are also discussed

exact optimization for markov random fields with convex priors we introduce a method to solve exactly a first order markov random field optimization problem in more generality than was previously possible the mrf has a prior term that is convex in terms of a linearly ordered label set the method maps the problem into a minimum cut problem for a directed graph for which a globally optimal solution can be found in polynomial time the convexity of the prior function in the energy is shown to be necessary and sufficient for the applicability of the method

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

web based medical decision support systems for three way medical decision making with game theoretic rough sets the realization of the web as a common platform medium and interface for supporting human activities has attracted many researchers to the study of web based support systems wss an important branch of wss is web based decision support systems that provide intelligent support for making effective decisions in different domains we focus on decision making in web based medical decision support systems wmdss uncertainty is a critical factor that affects decision making and reasoning in the medical field a three way decision making approach is an effective and better choice to lessen the effects of uncertainty it provides the provision for delaying certain or definite decisions in situations that lack sufficient evidence or accurate information in reaching certain conclusions particularly the option of deferment decisions is added in this approach that provides the flexibility to further examine and investigate the uncertain and doubtful cases the game theoretic rough set gtrs model is a recent development in rough sets that can be used to determine the three rough set regions in the probabilistic rough sets framework by determining a pair of thresholds the three regions are used to obtain three way decision rules in the form of acceptance rejection and deferment rules in this paper we extend the gtrs model to analyze uncertainty involved in medical decision making experimental results with a gtrs based approach on different health care datasets suggest that the approach may improve the overall quality of decision making in the medical field as well as other fields it is hoped that the incorporation of a gtrs component in wmdss will enrich and enhance its decision making capabilities

on the generalization of fuzzy rough sets rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets the existing generalizations of fuzzy rough sets are all based on special fuzzy relations fuzzy similarity relations t similarity relations it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches in this paper from the viewpoint of constructive approach we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined in axiomatic approach we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms the lattice and topological structures of fuzzy rough sets are also proposed in order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets a special lower approximation operator is applied to a fuzzy reasoning system which coincides with the mamdani algorithm

duality between subgradient and conditional gradient methods given a convex optimization problem and its dual there are many possible first order algorithms in this paper we show the equivalence between mirror descent algorithms and algorithms generalizing the conditional gradient method this is done through convex duality and implies notably that for certain problems such as for supervised machine learning problems with nonsmooth losses or problems regularized by nonsmooth regularizers the primal subgradient method and the dual conditional gradient method are formally equivalent the dual interpretation leads to a form of line search for mirror descent as well as guarantees of convergence for primal dual certificates

notes on l m convex functions and the separation theorems the concepts of l convex function and m convex function have recently been introduced by murota as generalizations of submodular function and base polyhedron respectively and discrete separation theorems are established for l convex concave functions and for m convex concave functions as generalizations of frank s discrete separation theorem for submodular supermodular set functions and edmonds matroid intersection theorem this paper shows the equivalence between murota s l convex functions and favati and tardella s submodular integrally convex functions and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis

the correct formulation of intermediate boundary conditions for runge kutta time integration of initial boundary value problems a disturbing feature of applying explicit high order runge kutta rk time integrators to initial boundary value problems solved by the method of lines is the loss of accuracy that results from wrong specifications of intermediate stage boundary conditions the solution proposed here is to prescribe analytically those values that would result from applying the rk solver at the boundaries and hence maintain the order of the rk method without a reduction in the step size limit the procedure is described in detail for nonlinear hyperbolic equations and both scalar and vector examples are presented

interpretation of equilibria in game theoretic rough sets the gtrs provides a mechanism for determining a pair of thresholds in the probabilistic rough set model the equilibria play a critical role in determining thresholds in the gtrs model we address two equilibria related issues with the gtrs model the first issue is the interpretation of equilibria the second issue in the establishment of equilibria intelligent decision making models aim at improving the quality of decision making under uncertainty the fundamental issues that are generally encountered in these models are too many options to choose from and the involvement of contradictory decision making criteria the game theoretic rough set gtrs model provides an intelligent decision making mechanism that exploits a game theoretic environment for analyzing strategic situations between cooperative or conflicting decision making criteria in the probabilistic rough set framework the concept of equilibria is of central importance in the gtrs model which has not been sufficiently addressed in the current literature two key issues in this regard are the interpretation of equilibria and the establishment of their existence by reviewing examining and defining the basic game constructs in the gtrs model we are able to interpret an equilibrium in terms of the decision thresholds that control the rough sets based decision regions in particular an equilibrium is defined in terms of a pair of thresholds such that no player has a unilateral incentive to change these thresholds within the game an example game is considered to demonstrate the use of the interpretation in determining the thresholds the issue of existence of equilibria is addressed by considering a couple of typical two player games in the gtrs model the results suggest that the existence of equilibria may be established under certain limited conditions

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

on the stability of variable stepsize rational approximations of holomorphic semigroups we consider variable stepsize time approximations of holomorphic semigroups on general banach spaces for strongly a o acceptable rational functions a general stability theorem is proved which does not impose any constraint on the ratios between stepsizes

two algorithms for maximizing a separable concave function over a polymatroid feasible region abstract in this paper we present two new algorithms for maximizing a separable concave function on a polymatroid both algorithms apply to the discrete as well as the continuous version of the problem the application of these algorithms to several types of polymatroids is discussed and we show that the decomposition algorithm runs in polynomial time in the discrete version for network and generalized symmetric polymatroids and the bottom up algorithm in the discrete version runs in polynomial time when the polymatroid is given as an explicit list of constraints

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

convergence and order reduction of runge kutta schemes applied to evolutionary problems in partial differential equations we address the question of convergence of fully discrete runge kutta approximations we prove that under certain conditions the order in time of the fully discrete scheme equals the conventional order of the runge kutta formula being used however these conditions which are necessary for the result to hold are not natural as a result in many problems the order in time will be strictly smaller than the conventional one a phenomenon called order reduction this phenomenon is extensively discussed both analytically and numerically as distinct from earlier contributions we here treat explicit runge kutta schemes although our results are valid for both parabolic and hyperbolic problems the examples we present are therefore taken from the hyperbolic field as it is in this area that explicit discretizations are most appealing

the model of fuzzy variable precision rough sets the fuzzy rough set frs model has been introduced to handle databases with real values however frs was sensitive to misclassification and perturbation here misclassification means error or missing values in classification and perturbation means small changes of numerical data the variable precision rough sets vprss model was introduced to handle databases with misclassification however it could not effectively handle the real valued datasets now it is valuable from theoretical and practical aspects to combine frs and vprs so that a powerful tool which not only can handle numerical data but also is less sensitive to misclassification and perturbation can be developed in this paper we set up a model named fuzzy vprss fvprss by combining frs and vprs with the goal of making frs a special case first we study the knowledge representation ways of frs and vprs and then propose the set approximation operators of fvprs second we employ the discernibility matrix approach to investigate the structure of attribute reductions in fvprs and develop an algorithm to find all reductions third in order to overcome the np complete problem of finding all reductions we develop some fast heuristic algorithms to obtain one near optimal attribute reduction finally we compare fvprs with rs frs and several flexible rs based approaches with respect to misclassification and perturbation the experimental comparisons show the feasibility and effectiveness of fvprs

a linear programming approach to max sum problem a review the max sum labeling problem defined as maximizing a sum of binary i e pairwise functions of discrete variables is a general np hard optimization problem with many applications such as computing the map configuration of a markov random field we review a not widely known approach to the problem developed by ukrainian researchers schlesinger et al in 1976 and show how it contributes to recent results most importantly those on the convex combination of trees and tree reweighted max product in particular we review schlesinger et al s upper bound on the max sum criterion its minimization by equivalent transformations its relation to the constraint satisfaction problem the fact that this minimization is dual to a linear programming relaxation of the original problem and the three kinds of consistency necessary for optimality of the upper bound we revisit problems with boolean variables and supermodular problems we describe two algorithms for decreasing the upper bound we present an example application for structural image analysis

maximizing non monotone submodular functions submodular maximization generalizes many important problems including max cut in directed and undirected graphs and hypergraphs certain constraint satisfaction problems and maximum facility location problems unlike the problem of minimizing submodular functions the problem of maximizing submodular functions is np hard in this paper we design the first constant factor approximation algorithms for maximizing nonnegative non monotone submodular functions in particular we give a deterministic local search frac 1 3 approximation and a randomized frac 2 5 approximation algorithm for maximizing nonnegative submodular functions we also show that a uniformly random set gives a frac 1 4 approximation for symmetric submodular functions we show that a random set gives a frac 1 2 approximation which can also be achieved by deterministic local search these algorithms work in the value oracle model where the submodular function is accessible through a black box returning f s for a given set s we show that in this model a frac 1 2 epsilon approximation for symmetric submodular functions would require an exponential number of queries for any fixed epsilon 0 in the model where f is given explicitly as a sum of nonnegative submodular functions each depending only on a constant number of elements we prove np hardness of frac 5 6 epsilon approximation in the symmetric case and np hardness of frac 3 4 epsilon approximation in the general case

a unified framework of dynamic three way probabilistic rough sets abstract the incremental learning technology has been widely applied in efficient and effective data mining with big data based on granular computing rough sets and three way approaches in real life applications the information systems will evolve over time with four levels of variational situations which can be described by the combination of the variations of attributes objects condition attributes values and decision attributes values considering updating knowledge with multilevel variations of data this paper proposes a unified dynamic framework of decision theoretic rough sets for incrementally updating three way probabilistic regions namely positive region boundary region and negative region through improving the representation of three way regions based on the well established bayesian decision procedure a novel matrix approach is introduced by the construction of boolean matrix and specific definition of matrix operation subsequently at the variations of level 1 the fundamental updating propositions which can induce the corresponding propositions with the variations of level 2 level 3 level 4 respectively are presented by the matrix updating strategies finally experiments with four incremental algorithms developed for the verification of feasibility and efficiency under multilevel variations of data are conducted by comparison with non incremental algorithm

avoiding the order reduction of runge kutta methods for linear initial boundary value problems a new strategy to avoid the order reduction of runge kutta methods when integrating linear autonomous nonhomogeneous initial boundary value problems is presented the solution is decomposed into two parts one of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction a numerical illustration is given this idea applies to practical problems where spatial discretization is also required leading to the full order both in space and time

third order implicit explicit runge kutta local discontinuous galerkin methods with suitable boundary treatment for convection diffusion problems with dirichlet boundary conditions abstract to avoid the order reduction when third order implicit explicit runge kutta time discretization is used together with the local discontinuous galerkin ldg spatial discretization for solving convection diffusion problems with time dependent dirichlet boundary conditions we propose a strategy of boundary treatment at each intermediate stage in this paper the proposed strategy can achieve optimal order of accuracy by numerical verification also by suitably setting numerical flux on the boundary in the ldg methods and by establishing an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient and the given boundary conditions we build up the unconditional stability of the corresponding scheme in the sense that the time step is only required to be upper bounded by a suitable positive constant which is independent of the mesh size

active set algorithms for isotonic regression a unifying framework in this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods the active set approach provides a unifying framework for studying algorithms for isotonic regression simplifies the exposition of existing algorithms and leads to several new efficient algorithms we also investigate the computational complexity of several algorithms

monotone comparative statics the authors derive a necessary and sufficient condition for the solution set of an optimization problem to be monotonic in the parameters of the problem in addition they develop practical methods for checking the condition and demonstrate its applications to the classical theories of the competitive firm the monopolist the bertrand oligopolist consumer and growth theory game theory and general equilibrium analysis copyright 1994 by the econometric society

detecting and refining overlapping regions in complex networks with three way decisions the three way representation of communities as a triple of positive boundary and negative regions brings more insight into interpretation of communities the overlapping regions are further refined into 4 macro types and 8 micro types the three way decision strategies are introduced to detect the overlapping communities based on the introduced relation graph a new fitness function is devised to adapt to processing networks with overlapping the refinement of overlapping regions can reveal the different significance to impact forming or changing of communities the identification of communities is crucial to an understanding of the structural and functional properties of a complex network many methods and algorithms have been developed to detect overlapping communities a problem that has not been addressed satisfactorily is the relationship or difference between vertices in overlapping regions during the formation and growth of communities this paper investigates methods that not only detect the overlapping communities but also refine the overlapping regions we give a three way representation of a community by using interval sets and re formalize the problem of community detection as three way clustering we suggest four macro types and eight micro types of vertices to characterize members in overlapping regions for their refinement we propose an overlapping community detection algorithm by classifying the vertices into core vertices bone vertices and trivial vertices the main strategy of this algorithm is to find an initial cluster core to expand the core to a preliminary community according to a new fitness function and to merge trivial vertices based on three way decision strategies the experimental results on both real world social networks and computer generated artificial networks show the effectiveness and efficiency of the proposed methods

applications of discrete maximal l p regularity for finite element operators in this paper we present applications of discrete maximal lp regularity for finite element operators more precisely we show error estimates of order h2 for linear and certain semilinear problems in various lp norms discrete maximal regularity allows us to prove error estimates in a very easy and efficient way moreover we also develop interpolation theory for fractional powers of finite element operators and extend the results on discrete maximal lp regularity formerly proved by the author

maximum norm analysis of completely discrete finite element methods for parabolic problems we consider full discretizations of linear parabolic initial value problems in any space dimension finite elements are used for the discretization in space the resulting semidiscrete problem is integrated in time by means of a strongly a theta acceptable rational method stability estimates are derived and error bounds are established in the maximum norm for both smooth and rough initial data

a combinatorial strongly polynomial algorithm for minimizing submodular functions this paper presents a combinatorial polynomial time algorithm for minimizing submodular functions answering an open question posed in 1981 by grotschel lovasz and schrijver the algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter the resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value the paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set independent of the function values

three way decisions with probabilistic rough sets the rough set theory approximates a concept by three regions namely the positive boundary and negative regions rules constructed from the three regions are associated with different actions and decisions which immediately leads to the notion of three way decision rules a positive rule makes a decision of acceptance a negative rule makes a decision of rejection and a boundary rule makes a decision of abstaining this paper provides an analysis of three way decision rules in the classical rough set model and the decision theoretic rough set model the results enrich the rough set theory by ideas from bayesian decision theory and hypothesis testing in statistics the connections established between the levels of tolerance for errors and costs of incorrect decisions make the rough set theory practical in applications

on the removal of boundary errors caused by runge kutta integration of non linear partial differential equations the temporal integration of hyperbolic partial differential equations pdes has been shown to lead sometimes to the deterioration of accuracy of the solution because of boundary conditions a procedure for removal of this error in the linear case has been established previously in this paper we consider hyperbolic pdes linear and nonlinear whose boundary treatment is accomplished via the simultaneous approximation term sat procedure a methodology is presented for recovery of the full order of accuracy and has been applied to the case of a fourth order explicit finite difference scheme

the superiority of three way decisions in probabilistic rough set models three way decisions provide a means for trading off different types of classification error in order to obtain a minimum cost ternary classifier this paper compares probabilistic three way decisions probabilistic two way decisions and qualitative three way decisions of the standard rough set model it is shown that under certain conditions when considering the costs of different types of miss classifications probabilistic three way decisions are superior to the other two

avoiding the order reduction when solving second order in time pdes with fractional step runge kutta nystrom methods we study some of the main features of fractional step runge kutta nystrom methods when they are used to integrate initial boundary value problems of second order in time in combination with a suitable spatial discretization we focus our attention on the order reduction phenomenon which appears if classical boundary conditions are taken at the internal stages this drawback is specially hard when time dependent boundary conditions are considered in this paper we present an efficient technique very simple and computationally cheap which allows us to avoid the order reduction such technique consists in modifying the boundary conditions for the internal stages of the method

communication between information systems using fuzzy rough sets communication between information systems is a basic problem in granular computing and the concept of homomorphism is a useful mathematical tool to study this problem in this paper some properties of communication between information systems based on fuzzy rough sets are investigated the concepts of fuzzy relation mappings between universes are first proposed in order to construct a fuzzy relation of one universe according to the given fuzzy relation on the other universe the main properties of the mappings are studied the notions of homomorphism of information systems based on fuzzy rough sets are then proposed and it is proved that properties of relation operations in the original information system and structural features of the system such as approximations of arbitrary fuzzy sets and attribute reductions are guaranteed in its image system under the condition of homomorphism

three way decision approaches to conflict analysis using decision theoretic rough set theory social progress normally occurs through a sequence of struggles and conflicts and there has been relatively little progress in developing effective methods for conflict analysis decision theoretic rough set theory is a powerful mathematical tool for depicting ambiguous information and it can provide constructive advice for decision making in this paper we first present the concepts of probabilistic conflict neutral and allied sets of conflicts and then discuss the mechanism for computing the thresholds and for conflict analysis using decision theoretic rough set theory then we describe incremental algorithms for constructing the probabilistic conflict neutral and allied sets in dynamic information systems and their effectiveness is illustrated by experimental results in light of the relationship between maximal coalitions and allied sets we finally provide efficient approaches to help a government adjust various policies according to changes in the present international situation to calculate the maximal coalitions in dynamic information systems

three way decisions based on semi three way decision spaces abstract decision evaluation functions in three way decision spaces must meet three axioms the minimum element axiom the monotonicity axiom and the complement axiom maintaining the complement axiom of decision evaluation functions is crucial to three way decisions to simplify the decision rules based only on the conditional probability and the loss functions however some handy functions do not satisfy the complement axiom this paper introduces the notion of semi decision evaluation functions not necessarily satisfying the complement axiom but the minimum element axiom and the monotonicity axiom and presents some transformation methods from semi decision evaluation functions to decision evaluation functions through numerous examples this paper demonstrates the existence of semi decision evaluation functions and significance of the transformation methods

spectral fractional step runge kutta discretizations for initial boundary value problems with time dependent boundary conditions in this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting alternating direction or more generally fractional step methods such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive runge kutta method which is called the fractional step runge kutta method and a standard space discretization which can be of type finite differences finite elements or spectral methods among others spectral methods have been chosen here to complete the analysis of convergence of a totally discrete scheme of this type of improved fractionary steps the numerical experiences performed also show the increase of accuracy that this technique provides

the theoretical accuracy of runge kutta time discretizations for the initial boundary value problem a study of the boundary error the conventional method of imposing time dependent boundary conditions for runge kutta rk time advancement reduces the formal accuracy of the space time method to first order locally and second order globally independently of the spatial operator this counter intuitive result is analyzed in this paper two methods of eliminating this problem are proposed for the linear constant coefficient case 1 impose the exact boundary condition only at the end of the complete rk cycle 2 impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives the first method while retaining the rk accuracy in all cases results in a scheme with much reduced cfl condition rendering the rk scheme less attractive the second method retains the same allowable time step as the periodic problem however it is a general remedy only for the linear case for non linear hyperbolic equations the second method is effective only for for rk schemes of third order accuracy or less numerical studies are presented to verify the efficacy of each approach

a faster strongly polynomial time algorithm for submodular function minimization we consider the problem of minimizing a submodular function f defined on a set v with n elements we give a combinatorial algorithm that runs in o n 5eo n 6 time where eo is the time to evaluate f s for some s subseteq v this improves the previous best strongly polynomial running time by more than a factor of n we also extend our result to ring families

the trotter kato theorem and approximation of pdes we present formulations of the trotter kato theorem for approximation of linear c 0 semigroups which provide very useful framework when convergence of numerical approximations to solutions of pdes are studied applicability of our results is demonstrated using a first order hyperbolic equation a wave equation and stokes equation as illustrative examples

evaluation functions and decision conditions of three way decisions with game theoretic rough sets abstract three way decisions have been used over the years in many application areas the rough sets and its extensions provide useful approaches for three way decisions typically these approaches employ a single evaluation function or criterion to induce three way decisions when extending the rough set based three way decisions to multiple criteria decision making mcdm two issues are encountered the first issue is related to the construction and definition of aggregation mechanisms for dealing with differences in results of evaluation functions the second issue is related to the setting of choice structure for selecting the three types of decision choices in this article we consider the role and use of game theoretic rough set gtrs model to resolve and address these two issues the issue related to differences in evaluation functions is addressed with gtrs by implementing a game that considers multiple evaluation functions as game players the game theoretic analysis in the gtrs is employed to resolve the differences by determining a tradeoff between evaluation functions the issue related to choice structure is addressed by considering the conditions under which different game outcomes could constitute a game solution in particular the equilibrium analysis within games is used to construct the rules for three way decisions a demonstrative example is used to explain the use of the proposed approach the relationship between the proposed approach and the probabilistic rough sets is also discussed

error analysis of multipoint flux domain decomposition methods for evolutionary diffusion problems abstract we study space and time discretizations for mixed formulations of parabolic problems the spatial approximation is based on the multipoint flux mixed finite element method which reduces to an efficient cell centered pressure system on general grids including triangles quadrilaterals tetrahedra and hexahedra the time integration is performed by using a domain decomposition time splitting technique combined with multiterm fractional step diagonally implicit runge kutta methods the resulting scheme is unconditionally stable and computationally efficient as it reduces the global system to a collection of uncoupled subdomain problems that can be solved in parallel without the need for schwarz type iteration convergence analysis for both the semidiscrete and fully discrete schemes is presented

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis