application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

analysis of the kleiser schumann method the kleiser schumann algorithm for the approximation of the stokes problem by fourier legendre polynomials is analized stability when the degree of the polynomials increases is established whereas error estimates in sobolev spaces are proven

a modified alternating direction method for convex quadratically constrained quadratic semidefinite programs we propose a modified alternating direction method for solving convex quadratically constrained quadratic semidefinite optimization problems the method is a first order method therefore requires much less computational effort per iteration than the second order approaches such as the interior point methods or the smoothing newton methods in fact only a single inexact metric projection onto the positive semidefinite cone is required at each iteration we prove global convergence and provide numerical evidence to show the effectiveness of this method

analysis of spectral projectors in one dimensional domains in this paper we analyze a class of projection operators with values in a subspace of polynomials these projection operators are related to the hubert spaces involved in the numerical analysis of spectral methods they are in the first part of the paper the standard sobolev spaces and in the second part some weighted sobolev spaces the weight of which is related to the orthogonality relation satisfied by the chebyshev polynomials these results are used to study the approximation of a model fourth order problem

asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions under the shizuta kawashima condition we show that these solutions approach constant equilibrium state in the lp norm at a rate o t m 2 1 1 p as t tends to infty for p in min m 2 infty moreover we can show that we can approximate with a faster order of convergence theconservative part of the solution in terms of the linearized hyperbolic operator for m 2 and by a parabolic equation in the spirit of chapman enskog expansion the main tool is given by a detailed analysis of the green function for the linearized problem

equilibrium schemes for scalar conservation laws with stiff sources we consider a simple model case of stiff source terms in hyperbolic conservation laws namely the case of scalar conservation laws with a zeroth order source with low regularity it is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced cfl condition and refined meshes required because of the lack of accuracy on equilibrium states the source term should be taken into account in the upwinding and discretized at the nodes of the grid in order to solve numerically the problem we introduce a so called equilibrium schemes with the properties that i the maximum principle holds true ii discrete entropy inequalities are satisfied iii steady state solutions of the problem are maintained one of the difficulties in studying the convergence is that there are no bv estimates for this problem we therefore introduce a kinetic interpretation of upwinding taking into account the source terms based on the kinetic formulation we give a new convergence proof that only uses property ii in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions the computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that in comparison with an usual upwind scheme the corresponding equilibrium version is far more accurate furthermore numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients

spectral tau approximation of the two dimensional stokes problem we analyse the spectral tau method for the approximation of the stokes system on a square with dirichlet boundary conditions we provide an error estimate in the norm of the sobolev spaceh s for the approximation of a divergence free vector field with polynomial divergence free vector fields we apply this result to prove some convergence estimates for the solution of the discrete stokes problem

a new inexact alternating directions method for monotone variational inequalities the alternating directions method adm is an effective method for solving a class of variational inequalities vi when the proximal and penalty parameters in sub vi problems are properly selected in this paper we propose a new adm method which needs to solve two strongly monotone sub vi problems in each iteration approximately and allows the parameters to vary from iteration to iteration the convergence of the proposed adm method is proved under quite mild assumptions and flexible parameter conditions

time asymptotic high order schemes for dissipative bgk hyperbolic systems we introduce a new class of finite differences schemes to approximate one dimensional dissipative semilinear hyperbolic systems with a bgk structure using precise analytical time decay estimates of the local truncation error it is possible to design schemes based on the standard upwind approximation which are increasingly accurate for large times when approximating small perturbations of constant asymptotic states numerical tests show their better performances with respect to those of other schemes

p version of mixed finite element methods for stokes like problems abstract we investigate various sample extremum saddle point and minimization problems from the point of view of stability of increasing order mixed finite element methods problems discussed include the stokes poisson and linear elasticity problems

linearized alternating direction method with adaptive penalty and warm starts for fast solving transform invariant low rank textures transform invariant low rank textures tilt is a novel and powerful tool that can effectively rectify a rich class of low rank textures in 3d scenes from 2d images despite significant deformation and corruption the existing algorithm for solving tilt is based on the alternating direction method it suffers from high computational cost and is not theoretically guaranteed to converge to a correct solution to the inner loop in this paper we propose a novel algorithm to speed up solving tilt with guaranteed convergence for the inner loop our method is based on the recently proposed linearized alternating direction method with adaptive penalty to further reduce computation warm starts are also introduced to initialize the variables better and cut the cost on singular value decomposition extensive experimental results on both synthetic and real data demonstrate that this new algorithm works much more efficiently and robustly than the existing algorithm it could be at least five times faster than the previous method

a reduced newton method for constrained linear least squares problems we propose an iterative method that solves constrained linear least squares problems by formulating them as nonlinear systems of equations and applying the newton scheme the method reduces the size of the linear system to be solved at each iteration by considering only a subset of the unknown variables hence the linear system can be solved more efficiently we prove that the method is locally quadratic convergent applications to image deblurring problems show that our method gives better restored images than those obtained by projecting or scaling the solution into the dynamic range

asymptotic numerical solution of nonlinear systems of one dimensional balance laws abstract an asymptotic numerical method is proposed to solve nonlinear scalar dissipative balance laws as well as systems of them in one space dimension namely hyperbolic conservation laws affected by a certain kind of source term time asymptotics allows to obtain a hierarchy of coupled ordinary differential equations which can be solved by accurate methods these provide first the long time stationary solution and then corrections to it to obtain an approximation valid at lower times no accumulation of errors as time grows affects this method on the contrary results are more accurate at larger times in the scalar case an important role is played by the auxiliary function k u s u f u where f is the flux function and s is the source a similar role is played by a certain matrix in case of systems comparison is made with the godunov method and with the ahop asymptotic high order numerical methods recently developed by natalini et al

maxwellian decay for well balanced approximations of a super characteristic chemotaxis model we focus on the numerical simulation of a one dimensional model of chemotaxis dynamics proposed by greenberg and alt trans amer math soc 300 1987 pp 235 258 in a bounded domain by means of a previously introduced well balanced wb and asymptotic preserving ap scheme l gosse j math anal appl 2011 we are especially interested in studying the decay onto numerical steady states for two reasons 1 conventional upwind schemes have been shown to stabilize onto spurious non maxwellian states with a very big mass flux see e g f guarguaglini et al discrete contin dyn syst ser b 12 2009 pp 39 76 and 2 the initial data lead to a dynamic which is mostly super characteristic in the sense of s jin and m katsoulakis siam j appl math 61 2000 pp 273 292 thus the stability results of gosse do not apply a reflecting boundary condition which is compatible with the well balanced character is presented a mass preservation property is proved and some results on super characteristic relaxation are recalled numerical experiments with coarse computational grids are presented in detail they illustrate the bifurcation diagrams of guarguaglini et al which relate the total initial mass of cells with the time asymptotic values of the chemoattractant substance on each side of the domain it is shown that the wb scheme stabilizes correctly onto zero mass flow rate hence maxwellian steady states which agree with the aforementioned bifurcation diagrams the evolution in time of residues is commented for every considered test case

recovering low rank and sparse components of matrices from incomplete and noisy observations many problems can be characterized by the task of recovering the low rank and sparse components of a given matrix recently it was discovered that this nondeterministic polynomial time hard np hard task can be well accomplished both theoretically and numerically via heuristically solving a convex relaxation problem where the widely acknowledged nuclear norm and l 1 norm are utilized to induce low rank and sparsity this paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and gaussian noise we show that the resulting model falls into the applicable scope of the classical augmented lagrangian method moreover the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented lagrangian function hence some splitting numerical algorithms are developed for solving the new recovery model some preliminary numerical experiments verify that these augmented lagrangian based splitting algorithms are easily implementable and surprisingly efficient for tackling the new recovery model

hyperbolic conservation laws with stiff relaxation terms and entropy we study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms reduced systems inviscid and viscous local conservation laws and weakly nonlinear limits are derived through asymptotic expansions an entropy condition is introduced for n n systems that ensures the hyperbolicity of the reduced inviscid system the resulting characteristic speeds are shown to be interlaced with those of the original system moreover the first correction to the reduced system is shown to be dissipative a partial converse is proved for 2 2 systems this structure is then applied to study the convergence to the reduced dynamics for the 2 2 case 1994 john wiley sons inc

asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes the following work is devoted to the construction and validation of a numerical scheme for the 2d shallow water system on unstructured meshes supplemented by topography and friction source terms approximate solutions of frictionless flows are obtained considering a suitable formulation of the conservation laws involving the water free surface and some fractions of water accounting for the topography variations the discretization of the friction source terms relies on the use of a modified riemann solver for the flux computation the resulting scheme is then corrected in order to achieve an asymptotic regime preservation a muscl reconstruction is also performed to increase the space order of accuracy the overall numerical approach is shown to be consistent well balanced and to satisfy a domain invariant principle these results are assessed through several benchmark tests involving complex geometry and varying bathymetry in the presence of dry areas special interest is given to the wave front speed computation highlighting the stability of the method even when implementing the asymptotic preserving correction

alternating direction method with gaussian back substitution for separable convex programming we consider the linearly constrained separable convex minimization problem whose objective function is separable into m individual convex functions with nonoverlapping variables a douglas rachford alternating direction method of multipliers adm has been well studied in the literature for the special case of m 2 but the convergence of extending adm to the general case of m ge 3 is still open in this paper we show that the straightforward extension of adm is valid for the general case of m ge 3 if it is combined with a gaussian back substitution procedure the resulting adm with gaussian back substitution is a novel approach towards the extension of adm from m 2 to m ge 3 and its algorithmic framework is new in the literature for the adm with gaussian back substitution we prove its convergence via the analytic framework of contractive type methods and we show its numerical efficiency by some application problems

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy we consider the cauchy problem for a general one dimensional n n hyperbolic symmetrizable system of balance laws it is well known that in many physical examples for instance for the isentropic euler system with damping the dissipation due to the source term may prevent the shock formation at least for smooth and small initial data our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions and possibly to investigate their asymptotic behavior for systems which are entropy dissipative a quite natural generalization of the kawashima condition for hyperbolic parabolic systems can be given in this paper we first propose a general framework for this kind of problem by using the so called entropy variables then we go on to prove some general statements about the global existence of smooth solutions under different sets of conditions in particular the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension our main tools will be some refined energy estimates and the use of a suitable version of the kawashima condition

error estimates for the spectral approximation of the nonstationary stokes problem this paper analyzes a discretization of the nonstationary stokes problem with periodic nonperiodic boundary conditions fourier legendre spectral approximations are considered with respect to the space variables while the time discretization is based on the second order crank nicolson scheme optimal convergence estimates are proved for the velocity field

a feasible method for optimization with orthogonality constraints minimization with orthogonality constraints e g x top x i and or spherical constraints e g vert x vert 2 1 has wide applications in polynomial optimization combinatorial optimization eigenvalue problems sparse pca p harmonic flows 1 bit compressive sensing matrix rank minimization etc these problems are difficult because the constraints are not only non convex but numerically expensive to preserve during iterations to deal with these difficulties we apply the cayley transform a crank nicolson like update scheme to preserve the constraints and based on it develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics the efficiency of the proposed algorithms is demonstrated on a variety of test problems in particular for the maxcut problem it exactly solves a decomposition formulation for the sdp relaxation for polynomial optimization nearest correlation matrix estimation and extreme eigenvalue problems the proposed algorithms run very fast and return solutions no worse than those from their state of the art algorithms for the quadratic assignment problem a gap 0 842 to the best known solution on the largest problem tai256c in qaplib can be reached in 5 min on a typical laptop

asymptotic preserving and well balanced schemes for the 1d cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes abstract the original well balanced wb framework of greenberg and leroux 1996 24 and gosse 2002 18 relying on non conservative nc products see lefoch and tzavaras 1999 41 is set up in order to efficiently treat the so called cattaneo model of chemotaxis in 1d see hillen and stevens 2000 29 it proceeds by concentrating the source terms onto dirac masses this allows to handle them by nc jump relations based on steady state equations which can be integrated explicitly a riemann solver is deduced and the corresponding wb godunov scheme completed with the standard hoff smoller theory see hillen and stevens 2000 29 for the diffusion reaction equation ruling the evolution of the chemoattractant concentration is studied in detail later following former results of gosse and toscani 2002 2003 21 22 a simple rewriting of the nc jump relations allows to generate another version of the same godunov scheme which is well adapted to the parabolic scaling involving a small parameter e the standard bv framework is used to study the uniform stability of this asymptotic preserving ap scheme with respect to e allows to pass to the limit and derive a simple centered discretization of the keller segel model finally results by filbet 2006 14 permit to pass to the complementary limit when the space step h is sent to zero numerical results are included to illustrate the feasibility and the efficiency of the method

asymptotic high order mass preserving schemes for a hyperbolic model of chemotaxis we introduce a new class of finite difference schemes for approximating the solutions to an initial boundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term which arises as a simple model of chemotaxis since the solutions to this problem may converge to nonconstant asymptotic states for large times standard schemes usually fail to yield a good approximation therefore we propose a new class of schemes which use an asymptotic higher order correction second and third order in our examples to balance the effects of the source term and the influence of the asymptotic solutions special care is needed to deal with boundary conditions to avoid harmful loss of mass convergence results are proved for these new schemes and several numerical tests are presented and discussed to verify the effectiveness of their behavior

computation of natural convection in two dimensional cavities with chebyshev polynomials abstract a semi implicit spectral method for the computation of buoyancy induced two dimensional flow of a viscous incompressible fluid is presented the incompressible navier stokes equations expressed in terms of the primitive variables velocity and pressure and an additional equation for the temperature are considered all the variables are expanded in double truncated series of chebyshev polynomials the pressure distribution at the boundary of the computational domain is determined by means of an influence matrix technique in order to satisfy the incompressibility condition everywhere in the field the algorithm is applied to the classical problem of natural convection of a boussinesq fluid within a differentially heated vertical square cavity results show that for a prandtl number of 0 71 corresponding to air and up to rayleigh numbers of 106 very accurate solutions are obtained with a spatial resolution of 33 polynomials in each direction new interesting phenomena are shown for higher values of the rayleigh number

nonlinear galerkin method using chebyshev and legendre polynomials i the one dimensional case a new strategy stemming from the nonlinear galerkin method m marion and r temam siam j numer anal 26 1989 pp 1139 1157 for solving linear elliptic and nonlinear dissipative evolution equations by using chebyshev and legendre polynomials is presented the essential idea is to decompose the solution into a low mode part and a high mode part and to treat them separately the robustness of the method over the classical galerkin method is substantiated by rigorous error estimates and preliminary numerical experiments

polynomial approximation of divergence free functions etude de la meilleure approximation d une fonction sans divergence par un polynome algebrique ou trigonometrique sans divergence etablissement d une estimation optimale

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

asymptotic high order schemes for 2 times2 dissipative hyperbolic systems we investigate finite difference schemes which approximate 2 times 2 one dimensional linear dissipative hyperbolic systems we show that it is possible to introduce some suitable modifications in standard upwinding schemes which keep into account the long time behavior of the solutions to yield numerical approximations which are increasingly accurate for large times when computing small perturbations of stable asymptotic states respectively around stationary solutions and in the diffusion chapman enskog limit

fast multiple splitting algorithms for convex optimization we present in this paper two different classes of general multiple splitting algorithms for solving finite dimensional convex optimization problems under the assumption that the function being minimized can be written as the sum of k convex functions each of which has a lipschitz continuous gradient we prove that the number of iterations needed by the first class of algorithms to obtain an epsilon optimal solution is o k 1 l epsilon where l is an upper bound on all of the lipschitz constants the algorithms in the second class are accelerated versions of those in the first class where the complexity result is improved to o sqrt k 1 l epsilon while the computational effort required at each iteration is almost unchanged to the best of our knowledge the complexity results presented in this paper are the first ones of this type that have been given for splitting and alternating direction type methods moreover all algorithms proposed in this paper are parallelizable which makes the

a parallel primal dual splitting method for image restoration we develop a parallel primal dual splitting method to solve large scale image restoration problems which involve the sum of several linear operator coupled nonsmooth but proximable terms with the proposed method the objective function is decomposed into pieces that can be processed individually no inverse operator is involved in our method and the highly parallel structure makes it preferable in distributed computation the convergence is proven and the convergence rate is analyzed besides its equivalence to the relaxed parallel linearized alternating direction method of multipliers pladmm is addressed applications to image restoration problems with compound l1 regularizer and comparisons with state of the art methods are detailed to show the superiority of the proposed method

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

a proximal based decomposition method for convex minimization problems this paper presents a decomposition method for solving convex minimization problems at each iteration the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables we derive this algorithm from rockafellar s proximal method of multipliers which involves an augmented lagrangian with an additional quadratic proximal term the algorithm preserves the good features of the proximal method of multipliers with the additional advantage that it leads to a decoupling of the constraints and is thus suitable for parallel implementation we allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem s data the method is globally convergent and at a linear rate the method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets

generalized inf sup conditions for chebyshev spectral approximation of the stokes problem a mixed problem and its approximation in an abstract framework are considered they are proved to be well posed if and only if several inf sup conditions are satisfied these results are applied to the stokes equations in a square formulated in chebyshev weighted sobolev spaces and their approximations two kinds of spectral discretizations are analyzed a galerkin method and a collocation method at the chebyshev nodes

on the linear convergence of the alternating direction method of multipliers we analyze the convergence rate of the alternating direction method of multipliers admm for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints previous analysis of the admm typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks moreover there has been no rate of convergence analysis for the admm without strong convexity in the objective function in this paper we establish the global r linear convergence of the admm for minimizing the sum of any number of convex separable functions assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping and a nonsmooth ell 1 l1 regularizer this result implies the linear convergence of the admm for contemporary applications such as lasso without assuming strong convexity of the objective function

decomposition through formalization in a product space when an optimization problem is posed in a product space it is classical to decompose this problem the goal of this paper is to show how such an approach can be used when the problem to be solved is not naturally posed in a product space by associating systematically to this problem an equivalent one posed in ann fold cartesian product space we obtain by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former applications to three rather classical mathematical programming problems are given

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

a variable penalty alternating directions method for convex optimization we study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints the method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both lagrangian and proximal terms the minimizers determine the new proximal terms and a simple update of the lagrangian terms follows we prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers another novelty is that we allow penalty matrices and these may vary per iteration this can be beneficial in applications since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties as an application we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems

the p and h p versions of some finite element methods for stokes problem abstract we investigate discretizations of stokes problem as a sample saddle point problem from the point of view of stability of increasing order mixed finite element methods for the cases of velocity pressure space pairs where the stability can be characterized in terms of the growth of a right inverse of the divergence operator we exhibit algebraic instabilities of varying size depending on boundary conditions and the kind of element employed