rate distortion optimal fast thresholding with complete jpeg mpeg decoder compatibility we show a rate distortion optimal way to threshold or drop the dct coefficients of the jpeg and mpeg compression standards our optimal algorithm uses a fast dynamic programming recursive structure the primary advantage of our approach lies in its complete compatibility with standard jpeg and mpeg decoders

a superconvergent ldg hybridizable galerkin method for second order elliptic problems we identify and study an ldg hybridizable galerkin method which is not an ldg method for second order elliptic problems in several space dimensions with remarkable convergence properties unlike all other known discontinuous galerkin methods using polynomials of degree k 0 for both the potential as well as the flux the order of convergence in l 2 of both unknowns is k 1 moreover both the approximate potential as well as its numerical trace superconverge in l 2 like norms to suitably chosen projections of the potential with order k 2 this allows the application of element by element postprocessing of the approximate solution which provides an approximation of the potential converging with order k 2 in l 2 the method can be thought to be in between the hybridized version of the raviart thomas and that of the brezzi douglas marini mixed methods

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

accurate multiscale finite element methods for two phase flow simulations in this paper we propose a modified multiscale finite element method for two phase flow simulations in heterogeneous porous media the main idea of the method is to use the global fine scale solution at initial time to determine the boundary conditions of the basis functions this method provides a significant improvement in two phase flow simulations in porous media where the long range effects are important this is typical for some recent benchmark tests such as the spe comparative solution project m christie m blunt tenth spe comparative solution project a comparison of upscaling techniques spe reser eval eng 4 2001 308 317 where porous media have a channelized structure the use of global information allows us to capture the long range effects more accurately compared to the multiscale finite element methods that use only local information to construct the basis functions we present some analysis of the proposed method to illustrate that the method can indeed capture the long range effect in channelized media

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

weak galerkin method for the biot s consolidation model abstract we develop a weak galerkin wg finite element method for the biot s consolidation model in the classical displacement pressure two field formulation weak galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations backward euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme we study the well posedness of the linear system at each time step and also derive the overall optimal order convergence of the wg formulation such wg scheme is designed on general shape regular polytopal meshes and provides stable and oscillation free approximation for the pressure without special treatment numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak galerkin finite element method

joint optimization of run length coding huffman coding and quantization table with complete baseline jpeg decoder compatibility to maximize rate distortion performance while remaining faithful to the jpeg syntax the joint optimization of the huffman tables quantization step sizes and dct indices of a jpeg encoder is investigated given huffman tables and quantization step sizes an efficient graph based algorithm is first proposed to find the optimal dct indices in the form of run size pairs based on this graph based algorithm an iterative algorithm is then presented to jointly optimize run length coding huffman coding and quantization table selection the proposed iterative algorithm not only results in a compressed bitstream completely compatible with existing jpeg and mpeg decoders but is also computationally efficient furthermore when tested over standard test images it achieves the best jpeg compression results to the extent that its own jpeg compression performance even exceeds the quoted psnr results of some state of the art wavelet based image coders such as shapiro s embedded zerotree wavelet algorithm at the common bit rates under comparison both the graph based algorithm and the iterative algorithm can be applied to application areas such as web image acceleration digital camera image compression mpeg frame optimization and transcoding etc

weak galerkin finite element methods for darcy flow anisotropy and heterogeneity this paper presents a family of weak galerkin finite element methods wgfems for darcy flow computation the wgfems are new numerical methods that rely on the novel concept of discrete weak gradients the wgfems solve for pressure unknowns both in element interiors and on the mesh skeleton the numerical velocity is then obtained from the discrete weak gradient of the numerical pressure the new methods are quite different than many existing numerical methods in that they are locally conservative by design the resulting discrete linear systems are symmetric and positive definite and there is no need for tuning problem dependent penalty factors we test the wgfems on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in darcy flow

an efficient algorithm for optimizing dct quantization we describe the rd opt algorithm for dct quantization optimization which can be used as an efficient tool for near optimal rate control in dct based compression techniques such as jpeg and mpeg rd opt measures dct coefficient statistics for the given image data to construct rate distortion specific quantization tables with nearly optimal tradeoffs

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

a weak galerkin finite element method for a coupled stokes darcy problem on general meshes abstract in this paper we introduce and analyze a weak galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow flows are governed by the stokes equations in primal velocity pressure formulation and darcy equation in the second order primary formulation respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law by using the weak galerkin approach we consider the two dimensional problem with the usual polynomials of degree k 1 for the velocity and hydraulic head while polynomials of degree k 1 for the pressure the velocity and hydraulic head is enhanced by polynomials of degree k 1 on the edge of a finite element partition this new method has a lot of attractive computational features more general finite element partitions of arbitrary polygons with certain shape regularity fewer degrees of freedom and parameter free stability and error estimates of optimal order are obtained moreover numerical experiences are presented to illustrate the good performance confirm the optimal order of convergence and verify the efficiency of the proposed weak galerkin method in this paper we also deal with finite element partitions consisting of general meshes such as triangular mesh quadrilateral mesh hexagonal dominant mesh and voronoi mesh for the numerical weak galerkin approximation

an analysis of a weak galerkin finite element method for stationary navier stokes problems abstract in this article we present and analyze a weak galerkin finite element method for stationary navier stokes problems this weak galerkin finite element scheme is based on a shape regular partition consisting of arbitrary polygons polyhedra we first establish a discrete embedding inequality that is useful in weak galerkin finite element analysis for nonlinear problems then the stability and unique existence are proved for the discrete velocity and pressure by means of a discrete inf sup condition furthermore we derive the optimal error estimates for velocity approximation in the discrete h 1 norm and pressure approximation in the l 2 norm respectively numerical examples are provided that corroborate the optimal convergence of the proposed method

a chebyshev expansion method for solving nonlinear optimal control problems this paper presents a numerical technique for solving the controlled duffing oscillator the control and state variables are approximated by chebyshev series a technique is provided for approximating the system dynamics boundary conditions and performance index this technique is based on using an explicit formula for the chebyshev polynomials in terms of arbitrary order of their derivatives the system dynamics and the performance index are converted into some algebraic equations then the optimal control problem is reduced to constrained optimization problem results and comparisons are given at the end of this paper

a discontinuous petrov galerkin method with lagrangian multipliers for second order elliptic problems we present a discontinuous petrov galerkin dpg method for the finite element discretization of second order elliptic boundary value problems the novel approach emanates from a one element weak formulation of the differential problem this procedure which is typical of discontinuous galerkin dg methods is based on introducing variables defined in the interior and on the boundary of the element the interface variables are suitable lagrangian multipliers that enforce interelement continuity of the solution and of its normal derivative thus providing the proper connection between neighboring elements the internal variables can be eliminated in favor of the interface variables using static condensation to end up with a system of reduced size in the sole lagrangian multipliers a stability and convergence analysis of the novel formulation is carried out and its connection with mixed hybrid and dg methods is explored numerical tests on several benchmark problems are included to validate the convergence performance and the flux conservation properties of the dpg method

multigrid for the mortar finite element method a multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed these problems are posed on domains partitioned into subdomains each of which is independently triangulated in a multilevel fashion the multilevel mortar finite element spaces based on such triangulations which need not align across subdomain interfaces are in general not nested suitable grid transfer operators and smoothers are developed which lead to a variable v cycle preconditioner resulting in a uniformly preconditioned algebraic system computational results illustrating the theory are also presented

the arbitrary order mimetic finite difference method for a diffusion equation with a non symmetric diffusion tensor abstract we present the arbitrary order mimetic finite difference mfd discretization for the diffusion equation with non symmetric tensorial diffusion coefficient in a mixed formulation on general polygonal meshes the diffusion tensor is assumed to be positive definite the asymmetry of the diffusion tensor requires changes to the standard mfd construction we present new approach for the construction that guarantees positive definiteness of the non symmetric mass matrix in the space of discrete velocities the numerically observed convergence rate for the scalar quantity matches the predicted one in the case of the lowest order mimetic scheme for higher orders schemes we observed super convergence by one order for the scalar variable which is consistent with the previously published result for a symmetric diffusion tensor the new scheme was also tested on a time dependent problem modeling the hall effect in the resistive magnetohydrodynamics

brief numerical treatment of multiobjective optimal control problems we present a numerical procedure for solving optimal control problems with both linear terminal constraints and multiple criteria using a chebyshev spectral procedure the problem reduces to a constrained optimization problem which can be solved using hybrid penalty partial quadratic interpolation hppqi technique the proposed procedure compares quite favorably with other methods on a sample of well known examples

the hellan herrmann johnson method some new error estimates and postprocessing analyse du comportement de la methode mixte de hellan herrmann johnson pour la resolution du probleme biharmonique 2 f etablissement d un resultat de superconvergence

a chebyshev technique for solving nonlinear optimal control problems a numerical technique for solving nonlinear optimal control problems is introduced the state and control variables are expanded in the chebyshev series and an algorithm is provided for approximating the system dynamics boundary conditions and performance index application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the chebyshev coefficients the optimum condition is obtained by applying the method of constrained extremum for linear quadratic optimal control problems the state and control variables are determined by solving a set of linear equations in the chebyshev coefficients applicability is illustrated with the minimum time and maximum radius orbit transfer problems

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

accuracy of mixed and control volume finite element approximations to darcy velocity and related quantities the accuracy of the darcy velocity flux and stream function computed from lowest order triangle based control volume and mixed finite element approximations to the two dimensional pressure equation is considered the control volume finite element method similar to integrated finite difference methods and analogous to the interpolation of galerkin finite element results over control volumes is shown to yield a conservative velocity field and smooth streamlines the streamlines and fluxes through the system computed with the control volume finite element approach are compared to those computed from the mixed finite element method which approximates the pressure and velocity variables separately it is shown that for systems with only moderate degrees of heterogeneity the control volume finite element method is the more computationally efficient alternative i e it provides more accurate flow results for a given number of unknowns for more variable or discontinuous permeability fields by contrast such as sand shale systems the mixed finite element method is shown to approximate flow variables more accurately and more realistically than the control volume method with the same number of unknowns

a quasi positive family of continuous darcy flux finite volume schemes with full pressure support a new family of flux continuous locally conservative finite volume schemes is presented for solving the general tensor pressure equation of subsurface flow in porous media the new schemes have full pressure continuity imposed across control volume faces previous families of flux continuous schemes are point wise continuous in pressure and flux when applying the earlier point wise flux continuous schemes to strongly anisotropic full tensor fields their failure to satisfy a maximum principle as with other fem and finite volume methods can result in loss of local stability for high anisotropy ratios which can cause strong spurious oscillations in the numerical pressure solution an m matrix analysis reveals the upper limits for guaranteeing a maximum principle for general 9 point schemes and aids in the design of schemes that minimize the occurrence of spurious oscillations in the discrete pressure field the full pressure continuity schemes are shown to possess a larger range of flux continuous schemes than the previous point wise counter parts for strongly anisotropic full tensor cases it is shown that the full quadrature range possessed by the new schemes permits these schemes to exploit quadrature points previously out of range that are shown to minimize spurious oscillations in discrete pressure solutions the new formulation leads to a more robust quasi positive family of flux continuous schemes applicable to general discontinuous full tensor fields

an effective implementation for stokes equation by the weak galerkin finite element method abstract in this paper we introduce and analyze the schur complement technique to a weak galerkin wg for short finite element method for solving stokes equation due to the special structure of the weak functions the main idea of schur decomposition method is to express the interior velocity functions defined inside partition elements in terms of the velocity functions defined on the partition boundary and the pressure functions this can largely reduce the degree of freedom of the derived discrete linear system we investigate the well posedness of the solution generated from schur complement and its coincidence with the standard wg method finally we exhibit some numerical experiments to illustrate the theoretical analysis

the weak galerkin finite element method for the transport reaction equation abstract we present and analyze a weak galerkin finite element method for solving the transport reaction equation in d space dimensions this method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon polyhedra we derive the l 2 error estimate of o h k 1 2 order for the discrete solution when the kth order polynomials are used for k 0 moreover for a special class of meshes we also obtain the optimal error estimate of o h k 1 order in the l 2 norm a derivative recovery formula is presented to approximate the convection directional derivative and the corresponding superconvergence estimate is given numerical examples on compatible and non compatible meshes are provided to show the effectiveness of this weak galerkin method

joint thresholding and quantizer selection for transform image coding entropy constrained analysis and applications to baseline jpeg striving to maximize baseline joint photographers expert group jpeg image quality without compromising compatibility of current jpeg decoders we develop an image adaptive jpeg encoding algorithm that jointly optimizes quantizer selection coefficient thresholding and huffman coding within a rate distortion r d framework practically speaking our algorithm unifies two previous approaches to image adaptive jpeg encoding r d optimized quantizer selection and r d optimal thresholding conceptually speaking our algorithm is a logical consequence of entropy constrained vector quantization ecvq design principles in the severely constrained instance of jpeg compatible encoding we explore both viewpoints the practical to concretely derive our algorithm and the conceptual to justify the claim that our algorithm approaches the best performance that a jpeg encoder can achieve this performance includes significant objective peak signal to noise ratio psnr improvement over previous work and at high rates gives results comparable to state of the art image coders for example coding the lena image at 1 0 b pixel our jpeg encoder achieves a psnr performance of 39 6 db that slightly exceeds the quoted psnr results of shapiro s wavelet based zero tree coder using a visually based distortion metric we can achieve noticeable subjective improvement as well furthermore our algorithm may be applied to other systems that use run length encoding including intraframe mpeg and subband or wavelet coding

superconvergent discontinuous galerkin methods for second order elliptic problems we identify discontinuous galerkin methods for second order elliptic problems in several space dimensions having superconvergence properties similar to those of the raviart thomas and the brezzi douglas marini mixed methods these methods use polynomials of degree k 0 for both the potential as well as the flux we show that the approximate flux converges in l2 with the optimal order of k 1 and that the approximate potential and its numerical trace superconverge in l2 like norms to suitably chosen projections of the potential with order k 2 we also apply element by element postprocessing of the approximate solution to obtain new approximations of the flux and the potential the new approximate flux is proven to have normal components continuous across inter element boundaries to converge in l2 with order k 1 and to have a divergence converging in l2 also with order k 1 the new approximate potential is proven to converge with order k 2 in l2 numerical experiments validating these theoretical results are presented

weak galerkin finite element method for viscoelastic wave equations abstract in this article we consider a weak galerkin finite element method wg fem for solving one type of viscoelastic wave equation a discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme we introduce both the semi and fully discrete wg fems and establish the corresponding stability estimates optimal order error estimate in h 1 norm is derived numerical experiments are preformed to verify the theoretical results

a class of stabilized extended one step methods for the numerical solution of odes abstract to overcome the order barrier imposed by a stability on linear multistep methods lmms usmani and agarwal 1 had constructed a third order a stabilized extended one step method by coupling two lmms in the present paper we introduce a class of extended one step methods generalizing the method of usmani and agarwal methods of this class of orders three and four which are a and or l stable have been given previously in 2 3 it is natural to ask the maximum attainable order for methods of this class which are also a stable or l stable the purpose of this paper is to show that the maximum attainable order of a method in this class is five we derive fifth order extended one step methods and show the existence of sub families of these methods which are a stable or l stable these methods are illustrated by considering two numerical examples

coupling discontinuous galerkin and mixed finite element discretizations using mortar finite elements discontinuous galerkin dg and mixed finite element mfe methods are two popular methods that possess local mass conservation in this paper we investigate dg dg and dg mfe domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface the subdomain grids need not match and the mortar grid may be much coarser giving a two scale method convergence results in terms of the fine subdomain scale h and the coarse mortar scale h are established for both types of couplings in addition a nonoverlapping parallel domain decomposition algorithm is developed which reduces the coupled system to an interface mortar problem the properties of the interface operator are analyzed

machine learning and transport simulations for groundwater anomaly detection abstract this paper presents studies on modeling and algorithms for groundwater anomaly detection specifically conductivity along with four other surrogates are used for identifying anomaly in groundwater the one class support vector machine 1 svm technique is utilized for model training and real data from colorado water watch is used for testing the model and algorithms design of code modules in python is briefly discussed since groundwater contamination rarely happens in reality we also use synthetic data from numerical simulations of flow and transport in porous media to test this model for anomaly detection

the weak galerkin finite element method for incompressible flow abstract we study the weak galerkin finite element method for stationary navier stokes problem we propose a weak finite element velocity pressure space pair that satisfies the discrete inf sup condition this space pair is then employed to construct a stable weak galerkin finite element scheme without adding any stabilizing term or penalty term we prove a discrete embedding inequality on the weak finite element space which together with the discrete inf sup condition enables us to establish the unique existence and stability estimates of the discrete velocity and pressure then we derive the optimal error estimates for velocity and pressure approximations in the h 1 norm and l 2 norm respectively numerical experiments are provided to illustrate the theoretical analysis

the weak galerkin finite element method for the symmetric hyperbolic systems abstract in this paper we present and analyze a weak galerkin finite element wg method for solving the symmetric hyperbolic systems this method is highly flexible by allowing the use of discontinuous finite elements on element and its boundary independently of each other by introducing special weak derivative we construct a stable weak galerkin scheme and derive the optimal l 2 error estimate of o h k 1 2 order for the discrete solution when the k order polynomials are used for k 0 as application we discuss this wg method for solving the singularly perturbed convection diffusion reaction equation and derive an e uniform error estimate of order k 1 2 numerical examples are provided to show the effectiveness of the proposed wg method

a stable weak galerkin finite element method for stokes problem abstract we study the weak galerkin finite element method for stokes problem a new weak galerkin finite element velocity pressure space pair is presented which satisfies the discrete inf sup condition based on this space pair we establish a stable weak galerkin approximation scheme without adding any stability term or penalty term then we further derive the optimal error estimates for velocity and pressure approximations respectively numerical experiments are provided to illustrate the theoretical analysis

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

a locally conservative finite element method based on piecewise constant enrichment of the continuous galerkin method this paper presents a locally conservative finite element method based on enriching the approximation space of the continuous galerkin method with elementwise constant functions the proposed method has a smaller number of degrees of freedom than the discontinuous galerkin method numerical examples on coupled flow and transport in porous media are provided to illustrate the advantages of this method we also present a theoretical analysis of the method and establish optimal convergence of numerical solutions

the lowest order weak galerkin finite element method for the darcy equation on quadrilateral and hybrid meshes abstract this paper investigates the lowest order weak galerkin finite element method for solving the darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles in this approach the pressure is approximated by constants in element interiors and on edges the discrete weak gradients of these constant basis functions are specified in local raviart thomas spaces specifically r t 0 for triangles and unmapped r t 0 for quadrilaterals these discrete weak gradients are used to approximate the classical gradient when solving the darcy equation the method produces continuous normal fluxes and is locally mass conservative regardless of mesh quality and has optimal order convergence in pressure velocity and normal flux when the quadrilaterals are asymptotically parallelograms implementation is straightforward and results in symmetric positive definite discrete linear systems we present numerical experiments and comparisons with other existing methods

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

arbitrary order nodal mimetic discretizations of elliptic problems on polygonal meshes we develop and analyze a new family of mimetic methods on unstructured polygonal meshes for the diffusion problem in primal form these methods are derived from the local consistency condition that is exact for polynomials of any degree m geq1 the degrees of freedom are a solution values at the quadrature nodes of the gauss lobatto formulas on each mesh edge and b solution moments inside polygons the convergence of the method is proven theoretically and an optimal error estimate is derived in a mesh dependent norm that mimics the energy norm numerical experiments confirm the convergence rate that is expected from the theory

state control spectral chebyshev parameterization for linearly constrained quadratic optimal control problems in this paper we propose a computationally attractive numerical method for determining the optimal control of constrained linear dynamic systems with a quadratic performance the method is based upon constructing the mth degree interpolating polynomials using chebyshev nodes to approximate the control and the state vectors the system dynamics are collocated at chebyshev nodes the performance index is discretized by a cell averaging method the state and control inequality constraints are converted into algebraic inequalities through collocation at the nodes the linear quadratic optimal control problem is thereby transformed into a quadratic programming one simulation studies demonstrate computational advantages relative to a standard riccati method a classical chebyshev based method fourier based method and other methods in the literature

optimal control approach for system of ordinary differential equations a method is proposed for solving a system of ordinary differential equation using parametric optimal control solution the boundary conditions of the problem are used as control variables and the objective functional is considered as a certain experimental function necessary conditions of the investigated problem are presented the examples illustrate the robustness accuracy and efficiency of the proposed numerical technique

a new weak galerkin finite element scheme for general second order elliptic problems abstract a new weak galerkin wg finite element scheme is presented for general second order elliptic problems in this paper in this new scheme a skew symmetric form has been used for handling the convection term the advantage of the new scheme is that the system of linear equations from the scheme is positive definite and one might easily get the well posedness of the system in this scheme the wg elements are designed to have the form of p k t p k 1 e that is we choose the polynomials of degree k 1 on each element and the polynomials of degree k 1 on the edge face of each element as a result fewer degrees of freedom are generated in the new wg finite element scheme it is also worth pointing out that the wg finite element scheme is established on finite element partitions consisting of arbitrary shape of polygons polyhedra which are shape regular optimal order error estimates are presented for the corresponding numerical approximation in various norms some numerical results are reported to confirm the theory

hybridized globally divergence free ldg methods part i the stokes problem we devise and analyze a new local discontinuous galerkin ldg method for the stokes equations of incompressible fluid flow this optimally convergent method is obtained by using an ldg method to discretize a vorticity velocity formulation of the stokes equations and by applying a new hybridization to the resulting discretization one of the main features of the hybridized method is that it provides a globally divergence free approximate velocity without having to construct globally divergence free finite dimensional spaces only elementwise divergence free basis functions are used another important feature is that it has significantly less degrees of freedom than all other ldg methods in the current literature in particular the approximation to the pressure is only defined on the faces of the elements on the other hand we show that as expected the condition number of the schur complement matrix for this approximate pressure is of order h 2 in the mesh size h finally we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates

double families of quasi positive darcy flux approximations with highly anisotropic tensors on structured and unstructured grids this paper focuses on flux continuous pressure equation approximation for strongly anisotropic media previous work on families of flux continuous schemes for solving the general geometry permeability tensor pressure equation has focused on single parameter families these schemes have been shown to remove the o 1 errors introduced by standard two point flux reservoir simulation schemes when applied to full tensor flow approximation improved convergence of the schemes has also been established for specific quadrature points however these schemes have conditional m matrices depending on the strength of the off diagonal tensor coefficients when applied to cases involving full tensors arising from strongly anisotropic media the point wise continuous schemes can fail to satisfy the maximum principle and induce severe spurious oscillations in the numerical pressure solution new double family flux continuous locally conservative schemes are presented for the general geometry permeability tensor pressure equation the new double family formulation is shown to expand on the current single parameter range of existing conditional m matrix schemes the conditional m matrix bounds on a double family formulation are identified for both quadrilateral and triangle cell grids a quasi positive qm matrix analysis is presented that classifies the behaviour of the new schemes with respect to double family quadrature in regions beyond the m matrix bounds the extension to double family quadrature is shown to be beneficial resulting in novel optimal anisotropic quadrature schemes the new methods are applied to strongly anisotropic full tensor field problems and yield results with sharp resolution with only minor or practically zero spurious oscillations

a weak galerkin generalized multiscale finite element method in this paper we propose a general framework for weak galerkin generalized multiscale wg gms finite element method for the elliptic problems with rapidly oscillating or high contrast coefficients this general wg gms method features in high order accuracy on general meshes and can work with multiscale basis derived by different numerical schemes a special case is studied under this wg gms framework in which the multiscale basis functions are obtained by solving local problem with the weak galerkin finite element method convergence analysis and numerical experiments are obtained for the special case

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

a lowest order weak galerkin finite element method for stokes flow on polygonal meshes abstract this paper presents a lowest order weak galerkin wg finite element method for solving the stokes equations on convex polygonal meshes constant vectors are used separately in element interiors and on edges to approximate fluid velocity whereas constant scalars are used on elements to approximate the pressure for the constant vector basis functions their discrete weak gradients are established in a matrix space that is based on the c w 0 space chen and wang 2017 whereas their discrete weak divergences are calculated as elementwise constants to circumvent the saddle point problem a reduced scheme for velocity is established by using three types of basis functions for the discretely divergence free subspace a procedure for subsequent pressure recovery is also developed error analysis along with numerical experiments on benchmarks are presented to demonstrate accuracy and efficiency of the proposed new method

a weak galerkin finite element method for high dimensional time fractional diffusion equation abstract this paper is concerned with the application of a weak galerkin finite element method wg fem to the time fractional diffusion equation the wg fem with l1 formula and the fast evaluation scheme based on wg fem are designed the optimal convergence rates for both semi discrete and fully discrete wg fem schemes are obtained and the stability analysis for the semi discrete wg fem is derived numerical experiments are implemented to verify the theoretical results

a weak galerkin least squares finite element method for div curl systems abstract in this paper we introduce a weak galerkin least squares method for solving div curl problem this finite element method leads to a symmetric positive definite system and has the flexibility to work with general meshes such as hybrid mesh polytopal mesh and mesh with hanging nodes error estimates of the finite element solution are derived the numerical examples demonstrate the robustness and flexibility of the proposed method

a local post processing technique for improving the accuracy in mixed finite element approximations a general simple post processing technique is given in this paper under certain conditions if an algorithm approximating a function u also provides a good approximation for its gradient and for some locally defined projection pu then by this method a better approximation for u can be easily obtained essentially the leading term of the error between u and its original approximation can be computed it is shown how the method may be applied to various mixed type finite element methods

note on global convergence of ode methods for unconstrained optimization in this paper we present a proof of global convergence of ode methods for unconstrained optimization based on point monotoneity of maps and the theory of dynamic systems

an sqp method for the optimal control of large scale dynamical systems we propose a sequential quadratic programming sqp method for the optimal control of large scale dynamical systems the method uses modified multiple shooting to discretize the dynamical constraints when these systems have relatively few parameters the computational complexity of the modified method is much less than that of standard multiple shooting moreover the proposed method is demonstrably more robust than single shooting in the context of the sqp method the use of modified multiple shooting involves a transformation of the constraint jacobian the affected rows are those associated with the continuity constraints and any path constraints applied within the shooting intervals path constraints enforced at the shooting points and other constraints involving only discretized states are not transformed the transformation is cast almost entirely at the user level and requires minimal changes to the optimization software we show that the modified quadratic subproblem yields a descent direction for the l1 penalty function numerical experiments verify the efficiency of the modified method

a coupling of weak galerkin and mixed finite element methods for poroelasticity abstract in this paper we present a coupling of a weak galerkin method for the displacement of the solid phase with a standard mixed finite method for the pressure and velocity of the fluid phase in poroelasticity equation because our method provides a stable element combination for the mixed linear elasticity problem it can avoid the poroelasticity locking mathematically indeed uniform in time error estimates of all the unknowns are obtained for both semidiscrete scheme and fully discrete scheme without assuming that the constrained specific storage coefficient is uniformly positive compared with the method proposed by jeonghun j lee our method can get the arbitrary convergence order by changing degree of the polynomial as for the computational cost we can use the modified weak galerkin method to eliminate the boundary term and reduce the number of unknowns in addition we do not use gronwall inequality in the error analysis finally the numerical experiments verify the theoretical analysis and show the effectiveness to overcome spurious pressure oscillations

numerical methods based on extended one step methods for solving optimal control problems in this article we develop extended one step methods for solving the dynamic system of optimal control problems governed by ordinary differential equations the control variables are approximated by polynomial functions the proposed problem is reduced to either a constrained or unconstrained minimization problem according to the nature of the dynamic system and the given conditions numerical results and comparisons with other methods are presented

rate distortion optimization for h 264 interframe coding a general framework and algorithms rate distortion rd optimization for h 264 interframe coding with complete baseline decoding compatibility is investigated on a frame basis using soft decision quantization sdq rather than the standard hard decision quantization we first establish a general framework in which motion estimation quantization and entropy coding in h 264 for the current frame can be jointly designed to minimize a true rd cost given previously coded reference frames we then propose three rd optimization algorithms a graph based algorithm for near optimal sdq in h 264 baseline encoding given motion estimation and quantization step sizes an algorithm for near optimal residual coding in h 264 baseline encoding given motion estimation and an iterative overall algorithm to optimize h 264 baseline encoding for each individual frame given previously coded reference frames with them embedded in the indicated order the graph based algorithm for near optimal sdq is the core given motion estimation and quantization step sizes it is guaranteed to perform optimal sdq if the weak adjacent block dependency utilized in the context adaptive variable length coding of h 264 is ignored for optimization the proposed algorithms have been implemented based on the reference encoder jm82 of h 264 with complete compatibility to the baseline profile experiments show that for a set of typical video testing sequences the graph based algorithm for near optimal sdq the algorithm for near optimal residual coding and the overall algorithm achieve on average 6 8 and 12 respectively rate reduction at the same psnr ranging from 30 to 38 db when compared with the rd optimization method implemented in the h 264 reference software

a comparative study on the weak galerkin discontinuous galerkin and mixed finite element methods this paper presents a comparative study on the newly introduced weak galerkin finite element methods wgfems with the widely accepted discontinuous galerkin finite element methods dgfems and the classical mixed finite element methods mfems for solving second order elliptic boundary value problems we examine the differences similarities and connection among these methods in scheme formulations implementation strategies accuracy and computational cost the comparison and numerical experiments demonstrate that wgfems are viable alternatives to mfems and hold some advantages over dgfems due to their properties of local conservation normal flux continuity no need for penalty factor and definiteness of discrete linear systems

estimation of penalty parameters for symmetric interior penalty galerkin methods this paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty galerkin methods in particular we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements numerical examples in all dimensions and for different polynomial degrees are presented we show the numerical effects of loss of coercivity

an a stable extended trapezoidal rule for the integration of ordinary differential equations abstract we examine a single step implicit integration algorithm which is obtained by a modification of the well known trapezoidal rule the obtained new method is a third order numerical process and preserves the property of a stability of the trapezoidal rule numerical examples involving stiff linear systems of first order differential equations are also included to demonstrate the practical usefulness of this new integration procedure

locally conservative fluxes for the continuous galerkin method the standard continuous galerkin cg finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations a much needed quantity in many applications we show how to overcome this shortcoming by using a two step postprocessing the first step is the computation of a numerical flux trace defined on element interfaces and is motivated by the structure of the numerical traces of discontinuous galerkin methods this computation is nonlocal in that it requires the solution of a symmetric positive definite system but the system is well conditioned independently of mesh size so it can be solved at asymptotically optimal cost the second step is a local element by element postprocessing of the cg solution incorporating the result of the first step this leads to a conservative flux approximation with continuous normal components this postprocessing applies for the cg method in its standard form or for a hybridized version of it we present the hybridized version since it allows easy handling of variable degree polynomials and hanging nodes furthermore we provide an a priori analysis of the error in the postprocessed flux approximation and display numerical evidence suggesting that the approximation is competitive with the approximation provided by the raviart thomas mixed method of corresponding degree

image compression with adaptive local cosines a comparative study the goal of this work is twofold first we demonstrate that an advantage can be gained by using local cosine bases over wavelets to encode images that contain periodic textures we designed a coder that outperforms one of the best wavelet coders on a large number of images the coder finds the optimal segmentation of the image in terms of local cosine bases the coefficients are encoded using a scalar quantizer optimized for laplacian distributions this new coder constitutes the first concrete contribution of the paper second we used our coder to perform an extensive comparison of several optimized bells in terms of rate distortion and visual quality for a large collection of images this study provides for the first time a rigorous evaluation in realistic conditions of these bells our experiments show that bells that are designed to reproduce exactly polynomials of degree 1 resulted in the worst performance in terms of the psnr however a visual inspection of the compressed images indicates that these bells often provide reconstructed images with very few visual artifacts even at low bit rates the bell with the most narrow fourier transform gave the best results in terms of the psnr on most images this bell tends however to create annoying visual artifacts in very smooth regions at low bit rate

hierarchical mixed hybridized methods for elliptic problems abstract in this article we propose a computational procedure for the efficient implementation of dual mixed hybridized methods of arbitrary degree the procedure relies on the decomposition of the finite element spaces into a vertical p type hierarchy consisting of a lower order part and a defect correction coupled with an additional horizontal decomposition of the defect correction space for the vector variable based on the helmholtz principle an appropriate definition of the basis function set allows us to obtain a systematic substructuring of the block matrix system this property in turn naturally gives rise to an efficient implementation of the procedure through an approximate fixed point block iteration exploiting the equivalence between the principle of defect correction and the variational multiscale modeling theory we also devise and numerically validate a hierarchical a posteriori error estimator for dual mixed methods in hybridized form

an inexpensive method for the evaluation of the solution of the lowest order raviart thomas mixed method for the model problem operatorname div a nabla u f we show how the approximate solution produced by the raviart thomas lecture notes in mathematics 606 springer verlag berlin 1977 mixed method of lowest degree can be obtained from the solution of the p 1 nonconforming galerkin method modified in a virtually cost free manner