an extended kuhn tucker approach for linear bilevel programming kuhn tucker approach has been applied with remarkable success in linear bilevel programming blp however it still has some extent unsatisfactory and incomplete one principle challenges is that it could not well handle a linear blp problem when the constraint functions at the upper level are of arbitrary linear form this paper describes theoretical foundation of kuhn tucker approach and proposes an extended kuhn tucker approach to deal with the problem the results have demonstrated that the extended kuhn tucker approach can solve a wider class of linear blp problems can than current capabilities permit

numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media a numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented the method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield upon appropriate averaging of the fine scale velocity field the coarse scale or equivalent grid block permeability when the region over which this coarse scale permeability is computed constitutes a representative elementary volume rev the resulting equivalent permeability may be interpreted as the effective permeability of the region solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle based finite element numerical procedure which allows for the modeling of geometrically complex features the specification of periodic boundary conditions is shown to yield symmetric positive definite equivalent permeability tensors in all cases the method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields the applicability and limitations of the method for these more general heterogeneity fields are discussed

two level linear programming decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which often exist in practice multilevel programming models partition control over decision variables among ordered levels within a hierarchical planning structure a planner at one level of the hierarchy may have his objective function and set of feasible decisions determined in part by other levels however his control instruments may allow him to influence the policies at other levels and thereby improve his own objective function this paper examines the special case of the two level linear programming problem geometric characterizations and algorithms are presented with some examples the goal is to demonstrate the tractability of such problems and motivate a wider interest in their study

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

on multiscale methods in petrov galerkin formulation in this work we investigate the advantages of multiscale methods in petrov galerkin pg formulation in a general framework the framework is based on a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space with good approximation properties and a high dimensional remainder space which only contains negligible fine scale information the multiscale space can then be used to obtain accurate galerkin approximations as a model problem we consider the poisson equation we prove that a petrov galerkin formulation does not suffer from a significant loss of accuracy and still preserve the convergence order of the original multiscale method we also prove inf sup stability of a pg continuous and a discontinuous galerkin finite element multiscale method furthermore we demonstrate that the petrov galerkin method can decrease the computational complexity significantly allowing for more efficient solution algorithms as another application of the framework we show how the petrov galerkin framework can be used to construct a locally mass conservative solver for two phase flow simulation that employs the buckley leverett equation to achieve this we couple a pg discontinuous galerkin finite element method with an upwind scheme for a hyperbolic conservation law

a hybrid simplex search and particle swarm optimization for unconstrained optimization this paper proposes the hybrid nm pso algorithm based on the nelder mead nm simplex search method and particle swarm optimization pso for unconstrained optimization nm pso is very easy to implement in practice since it does not require gradient computation the modification of both the nelder mead simplex search method and particle swarm optimization intends to produce faster and more accurate convergence the main purpose of the paper is to demonstrate how the standard particle swarm optimizers can be improved by incorporating a hybridization strategy in a suite of 20 test function problems taken from the literature computational results via a comprehensive experimental study preceded by the investigation of parameter selection show that the hybrid nm pso approach outperforms other three relevant search techniques i e the original nm simplex search method the original pso and the guaranteed convergence particle swarm optimization gcpso in terms of solution quality and convergence rate in a later part of the comparative experiment the nm pso algorithm is compared to various most up to date cooperative pso cpso procedures appearing in the literature the comparison report still largely favors the nm pso algorithm in the performance of accuracy robustness and function evaluation as evidenced by the overall assessment based on two kinds of computational experience the new algorithm has demonstrated to be extremely effective and efficient at locating best practice optimal solutions for unconstrained optimization

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

a new certification framework for the port reduced static condensation reduced basis element method abstract in this paper we introduce a new certification framework for the port reduced static condensation reduced basis element pr scrbe method which has been developed for the simulation of large component based applications such as bridges or acoustic waveguides in an offline computational stage we construct a library of interoperable parametrized reference components in the subsequent online stage we instantiate and connect the components at the interfaces ports to form a system of components to compute a truth finite element approximation of the say coercive elliptic partial differential equation on the component based system we use a domain decomposition approach for an efficient simulation we employ two different types of model reduction a reduced basis rb approximation within the interior of the component huynh et al 2013 and empirical port reduction eftang and patera 2013 on the ports where the components connect we demonstrate the well posedness of the pr scrbe approximation and introduce a new certification framework to assess the quality of the port reduction we use conservative fluxes we adapt the standard estimators from rb methods to the scrbe setting to derive an a posteriori error estimator for the rb error contribution in order to combine the a posteriori estimators for both error contributions and derive a rigorous a posteriori error estimator for pr scrbe we adapt techniques from multi scale methods and component mode synthesis finally we prove that the effectivity of the derived estimator can be bounded we provide numerical experiments for heat conduction and linear elasticity to show that the derived a posteriori error estimator provides an effective estimator moreover we demonstrate the applicability of the introduced certification framework by analyzing the computational online costs

a mixed multiscale finite element method for elliptic problems with oscillating coefficients the recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large scale structure of the solutions without resolving all the fine scale structures motivated by the numerical simulation of flow transport in highly heterogeneous porous media we propose a mixed multiscale finite element method with an over sampling technique for solving second order elliptic equations with rapidly oscillating coefficients the multiscale finite element bases are constructed by locally solving neumann boundary value problems we provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain the asymptotic structure of the solutions numerical experiments are carried out for flow transport in a porous medium with a random log normal relative permeability to demonstrate the efficiency and accuracy of the proposed method

efficient implementation of the localized orthogonal decomposition method abstract in this paper we present algorithms for an efficient implementation of the localized orthogonal decomposition method lod the lod is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales we show how the method can be implemented in a fairly standard finite element framework and discuss its realization for different types of problems such as linear elliptic problems with rough coefficients and linear eigenvalue problems

total positivity properties of absolute value multinormal variables with applications to confidence interval estimates and related probabilistic inequalities total positivity properties of multivariate densities are useful in deducing positive dependence of random vector components and related probability inequalities in this paper we determine necessary and sufficient conditions for total positivity of absolute value multinormal variables the results are applied to obtain positive dependence and associated inequalities for the multinormal and related distributions e g the multivariate t and wishart distributions inequalities of this type yield bounds for multivariate confidence set probabilities

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

a reduced basis localized orthogonal decomposition in this work we combine the framework of the reduced basis method rb with the framework of the localized orthogonal decomposition lod in order to solve parametrized elliptic multiscale problems the idea of the lod is to split a high dimensional finite element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information the low dimensional space is spanned by locally supported basis functions associated with the node of a coarse mesh obtained by solving decoupled local problems however for parameter dependent multiscale problems the local basis has to be computed repeatedly for each choice of the parameter to overcome this issue we propose an rb approach to compute in an offline stage lod for suitable representative parameters the online solution of the multiscale problems can then be obtained in a coarse space thanks to the lod decomposition and for an arbitrary value of the parameters thanks to a suitable interpolation of the selected rb the online rb lod has a basis with local support and leads to sparse systems applications of the strategy to both linear and nonlinear problems are given

first order corrections to the homogenised eigenvalues of a periodic composite medium a convergence proof let e be a dirichlet eigenvalue of the periodically rapidly oscillating elliptic operator a x e and let be a corresponding simple eigenvalue of the homogenised operator a we characterise the possible limit points of the ratio e e as e 0 our characterisation is quite explicit when the underlying domain is a planar convex classical polygon with sides of rational or infinite slopes in particular in this case it implies that there is often a continuum of such limit points

a multiscale reduced basis method for parametrized elliptic partial differential equations with multiple scales we present a technique for solving parametrized elliptic partial differential equations with multiple scales the technique is based on the combination of the reduced basis method c prud homme d rovas k veroy y maday a t patera g turinici reliable real time solution of parametrized partial differential equations reduced basis output bound methods journal of fluids engineering 124 1 2002 70 80 and the multiscale finite element method t y hou x h wu a multiscale finite element method for elliptic problems in composite materials and porous media journal of computational physics 134 1 1997 169 189 to treat problems in which the differential coefficient is characterized by a large number of independent parameters for the multiscale finite element method a large number of cell problems has to be solved at the fine local mesh for each new configuration of the differential coefficient in order to improve the computational efficiency of this method we construct reduced basis spaces that are adapted to the local parameter dependence of the differential operator the approximate solutions of the cell problems are computed accurately and efficiently via performing galekin projection onto the reduced basis spaces and implementing the offline online computational procedure therefore a large number of similar computations at the fine local mesh can be carried out with lower computational cost for each new configuration of the differential coefficient numerical results are provided to demonstrate the accuracy and efficiency of the proposed approach

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

adaptive iterative multiscale finite volume method the multiscale finite volume msfv method is a computationally efficient numerical method for the solution of elliptic and parabolic problems with heterogeneous coefficients it has been shown for a wide range of test cases that the msfv results are in close agreement with those obtained with a classical computationally expensive technique the method however fails to give accurate results for highly anisotropic heterogeneous problems due to weak localization assumptions recently a convergent iterative msfv i msfv method was developed to enhance the quality of the multiscale results by improving the localization conditions although the i msfv method proved to be efficient for most practical problems it is still favorable to improve the localization condition adaptively i e only for a sub domain where the original msfv localization conditions are not acceptable e g near shale layers and long coherent structures with high permeability contrasts in this paper a space time adaptive i msfv ai msfv method is introduced it is shown how to improve the msfv results adaptively in space and simulation time the fine scale smoother which is necessary for convergence of the i msfv method is also applied locally finally for multiphase flow problems two criteria are investigated for adaptively updating the msfv interpolation functions 1 a criterion based on the total mobility change for the transient coefficients and 2 a criterion based on the pressure equation residual for the accuracy of the results for various challenging test cases it is demonstrated that iterations in order to obtain accurate results even for highly anisotropic heterogeneous problems are required only in small sub domains and not everywhere the findings show that the error introduced in the msfv framework can be controlled and improved very efficiently with very little additional computational cost compared to the original non iterative msfv method

blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition 1 this paper offers a blueprint for extending perturbative ensemble moment and potentially other e g gaussian or perturbative karhunen loeve solutions of stochastic flow and transport problems in composite randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large the proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition it considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram i e a truncated fractal this consideration is supported either exactly or approximately by a growing amount of observational data it allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales combining this fractal decomposition with a variational multiscale decomposition formalism motivated by the multiscale nature of turbulence allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was the problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales the proposed approach does not require introducing effective parameters on any scale instead it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution the approach nevertheless allows subgrid closure another idea inspired by the literature on turbulence through subgrid refinement or the use of effective subgrid parameters it additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements details are presented in the context of steady state single phase flow in a composite medium

particle method approximation of oscillatory solutions to hyperbolic differential equations particle methods approximating hyperbolic partial diferential equations with oscillatory solutions are studied convergence is proved for approximations for which the continuous solution is not well resolved on the computational grid highly oscillatory solutions to the broadwell and variable coefficient carleman models are considered homogenization results are given and the approximations of more general systems are discussed numerical examples are presented

an error analysis for the finite element method applied to convection diffusion problems abstract this paper analyzes the finite element method applied to a convection diffusion model problem linear elements are used for the trial space the error is measured in a norm closely related to the l sup p norm when the test space is composed of linear elements with parabolic upwinding the method is shown to be optimal when the input data is piecewise smooth a condition which is usually observed in practice without these smoothness assumptions the method is shown to be non optimal even if the class of test spaces is extended to include any elements which have a shape independent of the mesh size author

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

adaptive reduced basis finite element heterogeneous multiscale method an adaptive reduced basis finite element heterogeneous multiscale method rb fe hmm is proposed for elliptic problems with multiple scales the multiscale method is based on the rb fe hmm introduced in a abdulle y bai reduced basis finite element heterogeneous multiscale method for high order discretizations of elliptic homogenization problems j comput phys 231 21 2012 7014 7036 it couples a macroscopic solver with effective data recovered from the solution of micro problems solved on sampling domains unlike classical numerical homogenization methods the micro problems are computed in a finite dimensional space spanned by a small number of accurately computed representative micro solutions the reduced basis obtained by a greedy algorithm in an offline stage in this paper we present a residual based a posteriori error analysis in the energy norm as well as an a posteriori error analysis in quantities of interest for both type of adaptive strategies rigorous a posteriori error estimates are derived and corresponding error estimators are proposed in contrast to the adaptive finite element heterogeneous multiscale method fe hmm there is no need to adapt the micro mesh simultaneously to the macroscopic mesh refinement up to an offline preliminary stage the rb fe hmm has the same computational complexity as a standard adaptive fem for the effective problem two and three dimensional numerical experiments confirm the efficiency of the rb fe hmm and illustrate the improvements compared to the adaptive fe hmm c 2013 elsevier b v all rights reserved

robust multilevel methods for general symmetric positive definite operators a general robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed the term robust refers to the convergence rate of the method being independent of discretization parameters i e the problem size and problem parameters important instances of such problem parameters are in particular highly varying coefficients the method belongs to the class of nonlinear algebraic multilevel iterations the crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems the method is analyzed in a rather general setting applying to the scalar elliptic equation the equations of linear elasticity and equations arising in the solution of maxwell s equations numerical results for the scalar elliptic equation are presented showing its robust convergence behavior

localized orthogonal decomposition techniques for boundary value problems in this paper we propose a local orthogonal decomposition method lod for elliptic partial differential equations with inhomogeneous dirichlet and neumann boundary conditions for this purpose we present new boundary correctors which preserve the common convergence rates of the lod even if the boundary condition has a rapidly oscillating fine scale structure we prove a corresponding a priori error estimate and present numerical experiments we also demonstrate numerically that the method is reliable with respect to thin conductivity channels in the diffusion matrix accurate results are obtained without resolving these channels by the coarse grid and without using patches that contain the channels

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

fast and robust numerical solution of the richards equation in homogeneous soil we derive and analyze a solver friendly finite element discretization of a time discrete richards equation based on kirchhoff transformation it can be interpreted as a classical finite element discretization in physical variables with nonstandard quadrature points our approach allows for nonlinear outflow or seepage boundary conditions of signorini type we show convergence of the saturation and in the nondegenerate case of the discrete physical pressure the associated discrete algebraic problems can be formulated as discrete convex minimization problems and therefore can be solved efficiently by monotone multigrid methods in numerical examples for two and three space dimensions we observe l 2 convergence rates of order mathcal o h 2 and h 1 convergence rates of order mathcal o h as well as robust convergence behavior of the multigrid method with respect to extreme choices of soil parameters

a multiscale a posteriori error estimate we introduce a hierarchic a posteriori error estimate for singularly perturbed reaction diffusion problems the estimator is based on a petrov galerkin method in which the trial space is enriched with nonpolynomial functions or multiscale functions we study the equivalence between the a posteriori estimate and the exact error in the energy norm moreover we prove a relationship between the hierarchic estimator and an explicit residual estimator the approach provides accurate estimates for the boundary layer regions which is confirmed by numerical experiments 2004 elsevier b v all rights reserved

constraint qualifications and kkt conditions for bilevel programming problems in this paper we consider the bilevel programming problem blpp which is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem we extend well known constraint qualifications for nonlinear programming problems such as the abadie constraint qualification the kuhn tucker constraint qualification the zangwill constraint qualification the arrow hurwicz uzawa constraint qualification and the weak reverse convex constraint qualification to blpps and derive a karash kuhn tucker kkt type necessary optimality condition under these constraint qualifications without assuming the lower level problem satisfying the mangasarian fromovitz constraint qualification relationships among various constraint qualifications are also given

the localized reduced basis multiscale method for two phase flows in porous media summary in this work we propose a novel model order reduction approach for two phase flow in porous media by introducing a global pressure formulation in which the total mobility which realizes the coupling between saturation and pressure is regarded as a parameter to the pressure equation in our approach the mobility itself is an optimal fit of mobility profiles that are precomputed using the time of flight for the initial saturation using this formulation and the localized reduced basis multiscale method we obtain a low dimensional surrogate of the high dimensional pressure equation by applying ideas from model order reduction for parametrized partial differential equations we are able to split the computational effort for solving the pressure equation into a costly offline step that is performed only once and an inexpensive online step that is carried out in every time step of the two phase flow simulation which is thereby largely accelerated usage of elements from numerical multiscale methods allows us to displace the computational intensity between the offline and online steps to reach an ideal runtime at acceptable error increase for the two phase flow simulation this is achieved by constructing reduced dimensional local spaces that lead to a non conforming global approximation space as one example for a coupling on the global space we introduce a discontinuous galerkin scheme copyright 2014 john wiley sons ltd

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

characterizing an optimal solution to the linear bilevel programming problem abstract this note presents a theorem which characterizes the geometry of an optimal solution of a linear bilevel programming blpp in its feasible region

reduced basis finite element heterogeneous multiscale method for high order discretizations of elliptic homogenization problems a new finite element method for the efficient discretization of elliptic homogenization problems is proposed these problems characterized by data varying over a wide range of scales cannot be easily solved by classical numerical methods that need mesh resolution down to the finest scales and multiscale methods capable of capturing the large scale components of the solution on macroscopic meshes are needed recently the finite element heterogeneous multiscale method fe hmm has been proposed for such problems based on a macroscopic solver with effective data recovered from the solution of micro problems on sampling domains at quadrature points of a macroscopic mesh departing from the approach used in the fe hmm we show that interpolation techniques based on the reduced basis methodology an offline online strategy allow one to design an efficient numerical method relying only on a small number of accurately computed micro solutions this new method called the reduced basis finite element heterogeneous multiscale method rb fe hmm is significantly more efficient than the fe hmm for high order macroscopic discretizations and for three dimensional problems when the repeated computation of micro problems over the whole computational domain is expensive a priori error estimates of the rb fe hmm are derived numerical computations for two and three dimensional problems illustrate the applicability and efficiency of the numerical method

common random numbers in multivariate simulations abstract it is shown how common random numbers can be used in computer simulations where the aim is to compare two systems on several parameters conditions are derived under which the use of common random numbers leads to a reduction in the size of the confidence region for the difference between the parameter vectors theoretical and experimental results suggest that this reduction is greatest when the covariance structure is the same for both systems

application of particle swarm optimization algorithm for solving bi level linear programming problem bi level linear programming is a technique for modeling decentralized decision it consists of the upper level and lower level objectives this paper attempts to develop an efficient method based on particle swarm optimization pso algorithm with swarm intelligence the performance of the proposed method is ascertained by comparing the results with genetic algorithm ga using four problems in the literature and an example of supply chain model the results illustrate that the pso algorithm outperforms ga in accuracy

zonal multiscale finite volume framework in this work the zonal multiscale finite volume zmsfv framework is presented the framework generalizes previous extensions of the multiscale finite volume msfv method and allows to integrate additional models in a unified way the new approach is based on splitting the domain of interest into a classical msfv domain and additional zones within each zone the solution space is spanned by so called zonal functions coupling of the zonal solutions to the surrounding domain is achieved via extended zonal functions the construction of basis zonal and extended zonal functions can be performed such that the functions form a partition of unity the zmsfv framework allows to include the influence of the zonal functions directly in the coarse scale problem similar to previous msfv method formulations the zmsfv framework can ensure conservative coarse scale solutions and allows for the construction of conservative fine scale solutions the main features of the zmsfv framework are used to derive robust preconditioners for iterative solutions of high contrast problems as encountered in subsurface flow and transport simulations it is shown that the obtained convergence rates are independent of the contrast

an offline online homogenization strategy to solve quasilinear two scale problems at the cost of one scale problems inspired by recent analyses of the finite element heterogeneous multiscale method and the reduced basis technique for nonlinear problems we present a simple and concise finite element algorithm for the reliable and efficient resolution of elliptic or parabolic multiscale problems of nonmonotone type solutions of appropriate cell problems on sampling domains are selected by a greedy algorithm in an offline stage and assembled in a reduced basis rb this rb is then used in an online stage to solve two scale problems at a computational cost comparable to the single scale case both the offline and the online cost are independent of the smallest scale in the physical problem the performance and accuracy of the algorithm are illustrated on 2d and 3d stationary and evolutionary nonlinear multiscale problems

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

first order corrections to the homogenized eigenvalues of a periodic composite medium this paper studies the eigenvalue problem associated with the vibration of a com posite medium assumed to have a periodic microstructure the investigation concentrates on the first order correction to the homogenized eigenvalues a major difficulty arises due to the interaction of the periodic microstructure with the boundary of the medium the way in which the boundary intersects the underlying periodic structure strongly influences the value of the correction a rigorous derivation is given of the eigenvalue correction formula the properties of the eigenvalue correction are studied in one and two dimensional numerical calculations

two level discretization techniques for ground state computations of bose einstein condensates this work presents a new methodology for computing ground states of bose einstein condensates based on finite element discretizations on two different scales of numerical resolution in a preprocessing step a low dimensional coarse generalized finite element space is constructed it is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties the nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low dimensional space without significant loss of accuracy when compared with the solution of the full fine scale problem the preprocessing step is independent of the types and numbers of bosons a postprocessing step further improves the accuracy of the method we present rigorous a priori error estimates that predict convergence rates h 3 for the ground state eigenfunction and h 4 for the corresponding eigenvalue without pre asymptotic effects h being the coarse scale discretization parameter numerical experiments indicate that these high rates may still be pessimistic

error control for the localized reduced basis multiscale method with adaptive on line enrichment in this contribution we consider localized robust and efficient a posteriori error estimation of the localized reduced basis multiscale lrbms method for parametric elliptic problems with possibly heterogeneous diffusion coefficient the numerical treatment of such parametric multiscale problems is characterized by a high computational complexity arising from the multiscale character of the underlying differential equation and the additional parameter dependence the lrbms method can be seen as a combination of numerical multiscale methods and model reduction using reduced basis rb methods to efficiently reduce the computational complexity with respect to the multiscale as well as the parametric aspect of the problem simultaneously in contrast to the classical residual based error estimators currently used in rb methods we are considering error estimators that are based on conservative flux reconstruction and provide an efficient and rigorous bound on the full error with respect to the weak solut

a reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks abstract in this paper we propose a reduced basis hybrid method rbhm for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent the rbhm is applied to stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled the rbhm is built upon the reduced basis element method rbem and it takes advantage from both the reduced basis methods rb and the domain decomposition method we move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes on the latter representative solutions corresponding to the same governing partial differential equations are computed for different values of some parameters of interest representing for example the deformation of the blocks a generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration the desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across subdomain interfaces by suitable coupling conditions the geometrical parametrization of the domain by transfinite mapping induces non affine parameter dependence an empirical interpolation technique is used to recover an approximate affine parameter dependence and a subsequent offline online decomposition of the reduced basis procedure this computational decomposition yields a considerable reduction of the problem complexity results computed on some combinations of 2d and 3d geometries representing cardiovascular networks show the advantage of the method in terms of reduced computational costs and the quality of the coupling to guarantee continuity of both stresses pressure and velocity at subdomain interfaces

homogenization and convergence of the vortex method for 2 d euler equations with oscillatory vorticity fields the 2 d incompressible euler equations with oscillatory vorticity fields are studied a homogenization result for 2 d euler equations in velocity vorticity formulation is obtained and weak continuity of the equations is proved convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles numerical results are given comparisons are made with the corresponding finite difference approximation

modeling multiscale diffusion processes in random heterogeneous media a simple stochastic up scaling strategy is presented for incorporating the effects of micro scale variations of thermal conductivity in the thermal analysis of macro scale systems incomplete characterization of the fine scale variation necessitates posing the thermal conductivity as a stochastic multiscale random field a data driven approach coupled with a non linear dimension reduction algorithm is used to construct a low dimensional stochastic descriptor of the fine scale variation of the stochastic conductivity a representative volume element rve argument is utilized to construct a simple model for up scaling the stochastic thermal conductivity in this model the effective stochastic thermal conductivity is estimated by virtually interrogating the stochastic rv a realistic example of the proposed stochastic multiscale diffusion strategy is illustrated utilizing limited experimental data

stochastic comparison of random vectors with a common copula we consider two random vectorsx andy such that the components of x are dominated in the convex order by the corresponding components of y we want to find conditions under which this implies that any positive linear combination of the components of x is dominated in the convex order by the same positive linear combination of the components of y this problem has a motivation in the comparison of portfolios in terms of risk the conditions for the above dominance will concern the dependence structure of the two random vectorsx andy namely the two random vectors will have a common copula and will be conditionally increasing this new concept of dependence is strictly related to the idea of conditionally increasing in sequence but in addition it is invariant under permutation we will actually prove that under the above conditions x will be dominated byy in the directionally convex order which yields as a corollary the dominance for positive linear combinations this result will be applied to a portfolio optimization problem

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

the reduced basis element method application to a thermal fin problem the reduced basis element method is a new approach for the approximation of partial differential equations that takes its roots in the domain decomposition method and in the reduced basis discretization the basic idea is to decompose the domain of computation into a series of subdomains the elements that are similar to a few reference domains these reference domains are actually filled with reduced basis functional spaces that are mapped to each subdomain together with the geometry the discrete approximation space is then composed of functions with the property that a function restricted to a subdomain belongs to the mapped reduced spaces finally a mortar type method is applied to glue the various local functions in this paper we focus on the definition of the reference shapes and together with theoretical and numerical justifications of the method we provide a posteriori error analysis tools that allow us to certify the computational results

a simple tabu search method to solve the mixed integer linear bilevel programming problem abstract multilevel programming is characterized as mathematical programming to solve decentralized planning problems the models partition control over decision variables among ordered levels within a hierarchical planning structure of which the linear bilevel form is a special case of a multilevel programming problem in a system with such a hierarchical structure the high level decision making situations generally require inclusion of zero one variables representing yes no decisions we provide a mixed integer linear bilevel programming formulation in which zero one decision variables are controlled by a high level decision maker and real value decision variables are controlled by a low level decision maker an algorithm based on the short term memory component of tabu search called simple tabu search is developed to solve the problem and two supplementary procedures are proposed that provide variations of the algorithm computational results disclose that our approach is effective in terms of both solution quality and efficiency

a multiscale method for linear elasticity reducing poisson locking we propose a generalized finite element method for linear elasticity equations with highly varying and oscillating coefficients the method is formulated in the framework of localized orthogonal decomposition techniques introduced by malqvist and peterseim 2014 assuming only l infinity coefficients we prove linear convergence in the h 1 norm also for materials with large lame parameter a the theoretical a priori error estimate is confirmed by numerical examples

an immersed weak galerkin method for elliptic interface problems abstract in this paper we present an immersed weak galerkin method for solving second order elliptic interface problems the proposed method does not require solution meshes to be aligned with the interface consequently uniform cartesian meshes can be used for nontrivial interfacial geometry we show the existence and uniqueness of the numerical algorithm and provide the error analysis in the energy norm numerical results are reported to demonstrate the performance of the method

some guidelines and guarantees for common random numbers common random numbers crn is a widely used technique for reducing variance in comparing stochastic systems through simulation its popularity derives from its intuitive appeal and ease of implementation however though crn has been observed to work well with a broad range of models the class of systems for which it is provably advantageous has remained rather limited this paper has two purposes we first discuss the effectiveness and optimality of crn in a general setting stressing the roles played by monotonicity and continuity properties we then present specific new classes of systems and comparisons for which crn is beneficial and even optimal our conclusions for these systems are largely consistent with simulation practice and lend further theoretical support to folklore our results differ from those of previous analyses primarily because we put conditions on the timing of events rather than the sequence of states in a discrete event simulation we formulate our results in three settings corresponding to three applications of crn distributional comparisons structural comparisons and sensitivity analysis in each case we make use of conditions that simultaneously ensure monotonicity and continuity in the timing of events these properties are established through explicit recursions for event epochs in terms of increasing continuous functions

localization of elliptic multiscale problems this paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients the basis functions are solutions of local problems on vertex patches the error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches hence on a uniform mesh of size h patches of diameter h log 1 h are sufficient to preserve a linear rate of convergence in h without pre asymptotic or resonance effects the analysis does not rely on regularity of the solution or scale separation in the coefficient this result motivates new and justifies old classes of variational multiscale methods see more at http www ams org journals mcom 2014 83 290 s0025 5718 2014 02868 8 sthash z2ccfxig dpuf

solving bi level linear programming problem through hybrid of immune genetic algorithm and particle swarm optimization algorithm bi level linear programming consisting of upper level and lower level objectives is a technique for modeling decentralized decision this study presents a hybrid of immune genetic algorithm and vector controlled particle swarm optimization igvpso to solve the bi level linear programming problem blpp it is applied to a supply chain model that is a blpp using four problems from the literature and the supply chain distribution models the computational results indicate that the proposed method is superior to some algorithms

analysis of the multiscale finite element method for nonlinear and random homogenization problems in this paper we study the convergence of the multiscale finite element method for nonlinear and random homogenization problems error estimates similar to those for linear homogenization problems are obtained here

a multiscale probabilistic collocation method for subsurface flow in heterogeneous media 1 owing to the spatial variability of the media properties uncertainty quantification for subsurface flow and solute transport usually requires high resolution simulations in this work a multiscale probabilistic collocation method mspcm is developed for solving such problems in a computationally efficient manner the subsurface flow problem is cast in a stochastic framework and probabilistic collocation strategy is used to represent the original stochastic differential equation the resulting equations are a set of decoupled deterministic equations with respect to collocation points a multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh coarse scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the fine scale heterogeneity the fine scale solution is reconstructed after the coarse scale solution is available since the pcm and multiscale finite element method are implemented at different levels the mspcm inherits their respective advantages in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid the performance of the proposed method is demonstrated with numerical examples the capability of mspcm in reproducing the probability density functions pdfs of head and velocity is investigated the numerical results show that the mspcm with proper coarsening level is able to capture small scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity

a multiscale two point flux approximation method a large number of multiscale finite volume methods have been developed over the past decade to compute conservative approximations to multiphase flow problems in heterogeneous porous media in particular several iterative and algebraic multiscale frameworks that seek to reduce the fine scale residual towards machine precision have been presented common for all such methods is that they rely on a compatible primal dual coarse partition which makes it challenging to extend them to stratigraphic and unstructured grids herein we propose a general idea for how one can formulate multiscale finite volume methods using only a primal coarse partition to this end we use two key ingredients that are computed numerically i elementary functions that correspond to flow solutions used in transmissibility upscaling and ii partition of unity functions used to combine elementary functions into basis functions we exemplify the idea by deriving a multiscale two point flux approximation mstpfa method which is robust with regards to strong heterogeneities in the permeability field and can easily handle general grids with unstructured fine and coarse scale connections the method can easily be adapted to arbitrary levels of coarsening and can be used both as a standalone solver and as a preconditioner several numerical experiments are presented to demonstrate that the mstpfa method can be used to solve elliptic pressure problems on a wide variety of geological models in a robust and efficient manner

a multiscale finite element linearization scheme for the unsaturated flow problems in heterogeneous porous media 1 numerical solution of unsaturated water flow in heterogeneous porous media is difficult not only because of the spatial variability in the parameters used to characterize the relevant physical properties of the natural porous media but also because of the mathematical problems that arise in dealing with the nonlinearities in this study a multiscale finite element linearization scheme is presented for effectively simulating unsaturated flow in heterogeneous porous media spanning over many scales the central goal of this method is to obtain the large scale solution of richards equation with heterogeneous coefficients accurately and efficiently on a coarse grid without resolving all the small scale details the underlying idea is to employ the slodicka linear relaxation approximation scheme to address the nonlinear characteristics of the equation and to use the multiscale finite element base functions to account for the spatial variability in the equation coefficients we describe the principle for constructing such a method and give an algorithm for implementing it numerical experiments are carried out for the unsaturated flow equation with both periodic and randomly generated lognormal hydraulic conductivity to demonstrate the efficiency and accuracy of the proposed method

application of a conservative generalized multiscale finite element method to flow models in this paper we propose a method for the construction of locally conservative flux fields from generalized multiscale finite element method gmsfem pressure solutions the flux values are obtained from an element based postprocessing procedure in which an independent set of 4x4 linear systems need to be solved to test the performance of the method we consider two heterogeneous permeability coefficients and couple the resulting fluxes to a two phase flow model the increase in accuracy associated with the computation of the gmsfem pressure solutions is inherited by the postprocessed flux fields and saturation solutions and is closely correlated to the size of the reduced order systems in particular the addition of more basis functions to the enriched coarse space yields solutions that more accurately capture the behavior of the fine scale model a number of numerical examples are offered to validate the performance of the method

an analysis of the nlmc upscaling method for high contrast problems abstract in this paper we propose simple multiscale basis functions with constraint energy minimization to solve elliptic problems with high contrast medium our methodology is based on the recently developed non local multicontinuum method nlmc the main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non local effects in our method each coarse block is decomposed into various regions according to the contrast ratio and we require that the contrast ratio should be relatively small within each region the basis functions are constructed by solving a local problem defined on the oversampling domains and they have mean value one on the chosen region and zero mean otherwise numerical analysis shows that the resulting basis functions can be localizable and have a decay property the convergence of the multiscale solution is also proved finally some numerical experiments are carried out to illustrate the performances of the proposed method they show that the proposed method can solve problem with high contrast medium efficiently in particular if the oversampling size is large enough then we can achieve the desired error

multi scale finite volume method for elliptic problems in subsurface flow simulation in this paper we present a multi scale finite volume msfv method to solve elliptic problems with many spatial scales arising from flow in porous media the method efficiently captures the effects of small scales on a coarse grid is conservative and treats tensor permeabilities correctly the underlying idea is to construct transmissibilities that capture the local properties of the differential operator this leads to a multi point discretization scheme for the finite volume solution algorithm transmissibilities for the msfv have to be constructed only once as a preprocessing step and can be computed locally therefore this step is perfectly suited for massively parallel computers furthermore a conservative fine scale velocity field can be constructed from the coarse scale pressure solution two sets of locally computed basis functions are employed the first set of basis functions captures the small scale heterogeneity of the underlying permeability field and it is computed in order to construct the effective coarse scale transmissibilities a second set of basis functions is required to construct a conservative fine scale velocity field the accuracy and efficiency of our method is demonstrated by various numerical experiments

discontinuous galerkin approximation of two phase flows in heterogeneous porous media with discontinuous capillary pressures we design and investigate a sequential discontinuous galerkin method to approximate two phase immiscible incompressible flows in heterogeneous porous media with discontinuous capillary pressures the nonlinear interface conditions are enforced weakly through an adequate design of the penalties on interelement jumps of the pressure and the saturation an accurate reconstruction of the total velocity is considered in the raviart thomas nedelec finite element spaces together with diffusivity dependent weighted averages to cope with degeneracies in the saturation equation and with media heterogeneities the proposed method is assessed on one dimensional test cases exhibiting rough solutions degeneracies and capillary barriers stable and accurate solutions are obtained without limiters

antithetic variates multivariate dependence and simulation of stochastic systems some theoretical and practical aspects antithetic and common random numbers for variance reduction in simulation of stochastic systems with dependent elements are considered a proof of their optimality in estimating the expected value of the response sum or the response difference of a pair of functions of vector arguments with dependent components is presented the efficiency of antithetic and common random numbers for variance reduction under different assumptions for the response functions is discussed applications to reliability networks and queueing systems are given

fast computation of multiphase flow in porous media by implicit discontinuous galerkin schemes with optimal ordering of elements we present a family of implicit discontinuous galerkin schemes for purely advective multiphase flow in porous media in the absence of gravity and capillary forces to advance the solution one time step one must solve a discrete system of nonlinear equations by reordering the grid cells the nonlinear system can be shown to have a lower triangular block structure where each block corresponds to the degrees of freedom in a single or a small number of cells to reorder the system we view the grid cells and the fluxes over cell interfaces as vertices and edges in a directed graph and use a standard topological sorting algorithm then the global system can be computed by processing the blocks sequentially using a standard newton raphson algorithm for the degrees of freedom in each block decoupling the system offers greater control over the nonlinear solution procedure and reduces the computational costs memory requirements and complexity of the scheme significantly in particular the first order version of the method may be at least as efficient as modern streamline methods when accuracy requirements or the dynamics of the flow allow for large implicit time steps

high performance model reduction techniques in computational multiscale homogenization a novel model order reduction technique for the solution of the fine scale equilibrium problem appearing in computational homogenization is presented the reduced set of empirical shape functions is obtained using a partitioned version that accounts for the elastic inelastic character of the solution of the proper orthogonal decomposition pod on the other hand it is shown that the standard approach of replacing the nonaffine termby an interpolant constructed using only pod modes leads to ill posed formulations we demonstrate that this ill posedness can be avoided by enriching the approximation space with the span of the gradient of the empirical shape functions furthermore interpolation points are chosen guided not only by accuracy requirements but also by stability considerations the approach is assessed in the homogenization of a highly complex porous metal material computed results show that computational complexity is independent of the size and geometrical complexity of the representative volume element the speedup factor is over three orders of magnitude as compared with finite element analysis whereas the maximum error in stresses is less than 10