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

forced vibrations of superquadratic hamiltonian systems on etudie l existence des solutions t periodiques ter t 0 du systeme hamiltonien suivant z gh z f t ou g est la matrice asymetrique standard

a parallel relaxation method for quadratic programming problems with interval constraints abstract optimization problems with interval constraints are encountered in various fields such as network flows and computer tomography as these problems are usually very large they are not easy to solve without taking their sparsity into account recently row action methods which originate from the classical hildreth s method for quadratic programming problems have drawn much attention since they are particularly useful for large and sparse problems various row action methods have already been proposed for optimization problems with interval constraints but they mostly belong to the class of sequential methods based on the gauss seidel and sor methods in this paper we propose a highly parallelizable method for solving those problems which may be regarded as an application of the jacobi method to the dual of the original problems we prove convergence of the algorithm and report some computational results to demonstrate its effectiveness

subgrid modeling in a galerkin method for the helmholtz equation abstract we improve the accuracy of finite element acoustic wave solutions without introducing additional degrees of freedom with physically inspired subgrid models the models explored here will not adversely impact system matrix storage or bandwidth can be used for structured or unstructured meshes do not require the solution of any auxiliary value problems and have minimal marginal overall computational cost one interpretation of the new models is as momentum transferred to and from modes at length scales below the local mesh dimension we show a method to approximate the effects of all subgrid modes on the grid scale solution and the significant improvements in local accuracy that the new models provide the new subgrid models also improve global or pollution accuracy through a reduction in the element dispersion error in addition we draw comparisons between this proposed method and much of the alternative and related work in this area

improved convergence of scattering calculations in the oscillator representation the schrodinger equation for two and tree body problems is solved for scattering states in a hybrid representation where solutions are expanded in the eigenstates of the harmonic oscillator in the interaction region and on a finite difference grid in the near and far field the two representations are coupled through a high order asymptotic formula that takes into account the function values and the third derivative in the classical turning points for various examples the convergence is analyzed for various physics problems that use an expansion in a large number of oscillator states the results show significant improvement over the jm ecs method y bidasyuk w vanroose j broeckhove f arickx v vasilevsky hybrid method jm ecs combining the j matrix and exterior complex scaling methods for scattering calculations phys rev c 82 6 2010 064603

error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem consider the affine variational inequality problem it is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point this bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem this latter result improves upon a recent result of luo and tseng that further assumes the problem to be monotone

massively parallel row action algorithms for some nonlinear transportation problems row action iterative algorithms are developed for two classes of nonlinear optimization problems with transportation constraints entropy problems where the objective function is a sum of x ln x a 1 terms and quadratic problems where the objective function is a sum of frac 1 2 wx 2 cx terms the algorithms are specialized to take advantage of both the structure of the transportation constraints and the special forms of the objective functions both generalized and pure networks are dealt with in this paper the algorithms are well suited for parallel computing implementations are developed on a massively parallel connection machine cm 2 with up to 32k processing elements the algorithms solve test problems with 4000 nodes and 4 million arcs in less than one minute of computer time on a maximally configured machine with 64k processing elements they achieve a peak computing rate of 3 gflops

on the convergence of sor and jor type methods for convex linear complementarity problems abstract we consider sor and jor type iterative methods for solving linear complementarity problems if the solution set is not discrete weak convergence proofs are usually obtained for these methods i e every accumulation point of the generated sequence is a solution we prove that for the convex case the whole sequence converges and if the limit point is nondegenerate convergence is linear

a note on the existence of multiple solutions for a class of systems of second order odes abstract we prove the existence of multiple solutions for some systems of second order odes with dirichlet boundary conditions such systems are obtained by coupling scalar odes with different growth conditions the proof relies on a global continuation technique

data parallel quadratic programming on box constrained problems we develop designs for the data parallel solution of quadratic programming problems subject to box constraints in particular we consider the class of algorithms that iterate between projection steps that identify candidate active sets and conjugate gradient steps that explore the working space using the algorithm of more and toraldo report mcs p77 05 89 argonne national laboratory illinois 1989 as a specific instance of this class of algorithms we show how its components can be implemented efficiently on a data parallel simd computer architecture alternative designs are developed for both arbitrary unstructured hessian matrices and for structured problems implementations are carried out on a connection machine cm 2 they are shown to be very efficient achieving a peak computing rate over 2 gflops problems with several hundred thousand variables are solved within one minute of solution time on the 8k cm 2 extremely large test problems with up to 2 89 million variables are also solved efficie

a derivative algorithm for inexact quadratic program application to environmental decision making under uncertainty abstract inexact quadratic programming iqp is an extension of conventional quadratic programming for handling both nonlinearities in cost objectives and uncertainties with modeling parameters it has been a useful tool for environmental systems analysis however inefficiency in its solution method has existed leading to difficulties in its practical application in this study a derivative algorithm dam is proposed for solving the iqp it improves upon the existing method through provision of a quantitative expression for uncertain relationships between the quadratic objective function and the decision variables the dam requires much lower computational efforts than the existing algorithm which is especially meaningful for the iqp s application to large scale problems the developed dam is applied to a hypothetical problem of municipal solid waste management and planning detailed solution steps are provided to clearly demonstrate the method s advantages

a galerkin least squares finite element method for the two dimensional helmholtz equation in this paper a galerkin least squares gls finite element method in which residuals in least squares form are added to the standard galerkin variational equation is developed to solve the helmholtz equation in two dimensions an important feature of gls methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes previous work has accomplished this for the one dimensional helmholtz equation using dispersion analysis in this paper the selection of the gls mesh parameter for two dimensions is considered and leads to elements that exhibit improved phase accuracy for any given direction of wave propagation an optimal gls mesh parameter is determined using two dimensional fourier analysis in general problems the direction of wave propagation will not be known a priori in this case an optimal gls parameter is found which reduces phase error for all possible wave vector orientations over elements the optimal gls parameters are derived for both consistent and lumped mass approximations several numerical examples are given and the results compared with those obtained from the galerkin method the extension of gls to higher order quadratic interpolations is also presented

on the convergence of iterative methods for symmetric linear complementarity problems we prove convergence of the whole sequence generated by any of a large class of iterative algorithms for the symmetric linear complementarity problem lcp under the only hypothesis that a quadratic form associated with the lcp is bounded below on the nonnegative orthant this hypothesis holds when the matrix is strictly copositive and also when the matrix is copositive plus and the lcp is feasible the proof is based upon the linear convergence rate of the sequence of functional values of the quadratic form as a by product we obtain a decomposition result for copositive plus matrices finally we prove that the distance from the generated sequence to the solution set and the sequence itself if its limit is a locally unique solution have a linear rate of r convergence

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz problem a new technique to solve elliptic linear pdes called ultra weak variational formulation uwvf in this paper is introduced in b despres c r acad sci paris 318 1994 pp 939 944 this paper is devoted to an evaluation of the potentialities of this technique it is applied to a model wave problem the two dimensional helmholtz problem the new method is presented in three parts following the same style of presentation as the classical one of the finite elements method even though they are definitely conceptually different methods the first part is committed to the variational formulation and to the continuous problem the second part defines the discretization process using a galerkin procedure the third part actually studies the efficiency of the technique from the order of convergence point of view this is achieved using theoretical proofs and a series of numerical experiments in particular it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom an application to scattering problems is presented in a fourth part

algorithms for bound constrained quadratic programming problems we present an algorithm which combines standard active set strategies with the gradient projection method for the solution of quadratic programming problems subject to bounds we show in particular that if the quadratic is bounded below on the feasible set then termination occurs at a stationary point in a finite number of iterations moreover if all stationary points are nondegenerate termination occurs at a local minimizer a numerical comparison of the algorithm based on the gradient projection algorithm with a standard active set strategy shows that on mildly degenerate problems the gradient projection algorithm requires considerable less iterations and time than the active set strategy on nondegenerate problems the number of iterations typically decreases by at least a factor of 10 for strongly degenerate problems the performance of the gradient projection algorithm deteriorates but it still performs better than the active set method

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

a residual based finite element method for the helmholtz equation a new residual based finite element method for the scalar helmholtz equation is developed this method is obtained from the galerkin approximation by appending terms that are proportional to residuals on element interiors and inter element boundaries the inclusion of residuals on inter element boundaries distinguishes this method from the well known galerkin least squares method and is crucial to the resulting accuracy of this method in two dimensions and for regular bilinear quadrilateral finite elements it is shown via a dispersion analysis that this method has minimal phase error numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods it is found that even for unstructured meshes this method retains a high level of accuracy copyright 2000 john wiley sons ltd

on the existence of many solutions for a class of superlinear elliptic systems abstract we seek multiple solutions to systems of semilinear elliptic equations of the type u x h x u x x u x 0 x where r n n 1 is a bounded regular domain and u u1 u2 r 2 for a class of potentials where the variables are strongly coupled as in the model h x u1 u2 u1 1 u2 2 with i 1 we provide the existence of nine nontrivial solutions characterized by sign properties of each component

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

a simultaneous iterative method for computing projections on polyhedra a simultaneous version of hildreth s iterative algorithm for norm minimization over linear inequalities is presented proofs are given showing that the algorithm converges from any starting point to its projection on the linear constraints set in the feasible case and to the nearest least squares solution in the general case

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

the second generation feti methods and their application to the parallel solution of large scale linear and geometrically non linear structural analysis problems the feti algorithms are a family of numerically scalable domain decomposition methods they have been designed in the early 1990s for solving iteratively and on parallel machines large scale systems of equations arising from the finite element discretization of solid mechanics structural engineering structural dynamics and acoustic scattering problems and for analyzing complex structures obtained from the assembly of substructures with incompatible discrete interfaces in this paper we present the second generation of these methods that operate more efficiently on large numbers of subdomains offer greater robustness better performance and more flexibility for implementation on a wider variety of computational platforms we also report on the application and performance of these methods for the solution of geometrically non linear structural analysis problems we discuss key aspects of their implementation on shared and distributed memory parallel processors benchmark them against optimized direct sparse solvers and highlight their potential with the solution of large scale structural mechanics problems with several million degrees of freedom

existence of forced oscillations for some nonlinear differential equations abstract this article studies the existence of t periodic solutions for systems of nonlinear second order ordinary differential equations assuming function v to be superquadratic it is shown that this system possesses infinitely many t periodic solutions the proof of this result rests on showing that certain homotopy groups of level sets of the functional associated with the system are not trivial some more general results concerning systems of another type are also presented

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

a two level domain decomposition method for the iterative solution of high frequency exterior helmholtz problems summary we present a lagrange multiplier based two level domain decomposition method for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed method is essentially an extension of the regularized feti finite element tearing and interconnecting method to indefinite problems its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves we show numerically that the proposed method is scalable with respect to the mesh size the subdomain size and the wavenumber we report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

modified j matrix method for scattering we modify the j matrix technique for scattering so that problems with long range interactions are easily solved this is done by introducing additional terms in the asymptotic three term recurrence relation that take into account asymptotic effects of the potential the solutions of this modified recurrence relation are a very good approximation of the exact scattering solution only a small number of residual coefficients need to be calculated as a result the numerical effort to solve the scattering problem is seriously reduced the technique is illustrated with a yukawa potential