convergence analysis without regularity assumptions for multigrid algorithms based on sor smoothing convergence estimates for the standard multigrid algorithms based on sor smoothing are established without regularity assumptions for symmetric and positive definite variational problems the results are applied to second order elliptic equations discretized by using finite element methods it is shown that the convergence is deteriorated no worse than 1 c sigma j 1 for the multigrid algorithms with sor nonsymmetric smoothing where j is the number of levels and sigma the relaxation parameter the results can be regarded as a generalization of the estimates in 10 where only the jacobi smoothing was considered this extension is not trivial since sor smoothing yields a nonsymmetric operator while the jacobi gives a symmetric smoothing operator the estimates are well suited to difficult problems with rough solutions as no regularity is assumed in the analysis

analysis of multilevel methods for eddy current problems in papers by arnold falk and winther and by hiptmair novel multigrid methods for discrete h curl elliptic boundary value problems have been proposed such problems frequently occur in computational electromagnetism particularly in the context of eddy current simulation this paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods it provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains in particular asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems the principal idea is to use approximate helmholtz decompositions of the function space h curl into an h1 regular subspace and gradients the main results of standard multilevel theory for h1 elliptic problems can then be applied to both subspaces this yields preliminary decompositions still outside the edge element spaces judicious alterations can cure this

designing structured tight frames via an alternating projection method tight frames also known as general welch bound equality sequences generalize orthonormal systems numerous applications including communications coding and sparse approximation require finite dimensional tight frames that possess additional structural properties this paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems ieps which includes the frame design problem to apply this method one needs only to solve a matrix nearness problem that arises naturally from the design specifications therefore it is the fast and easy to develop versions of the algorithm that target new design problems alternating projection will often succeed even if algebraic constructions are unavailable to demonstrate that alternating projection is an effective tool for frame design the paper studies some important structural properties in detail first it addresses the most basic design problem constructing tight frames with prescribed vector norms then it discusses equiangular tight frames which are natural dictionaries for sparse approximation finally it examines tight frames whose individual vectors have low peak to average power ratio par which is a valuable property for code division multiple access cdma applications numerical experiments show that the proposed algorithm succeeds in each of these three cases the appendices investigate the convergence properties of the algorithm

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

multigrid methods for saddle point problems stokes and lame systems we develop new multigrid methods for a class of saddle point problems that include the stokes system in fluid flow and the lame system in linear elasticity as special cases the new smoothers in the multigrid methods involve optimal preconditioners for the discrete laplace operator we prove uniform convergence of the w w cycle algorithm in the energy norm and present numerical results for w w cycle and v v cycle algorithms

iterative techniques for time dependent stokes problems abstract in this paper we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent stokes equations using a mixed finite element spatial discretization at each time step a two by two block system corresponding to a perturbed stokes problem must be solved although there are a number of techniques for iteratively solving this type of block system to be effective they require a good preconditioner for the resulting pressure operator schur complement in contrast to the time independent stokes equations where the pressure operator is well conditioned the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced and or the reynolds number is increased in this paper we shall describe and analyze preconditioners for the resulting pressure systems these preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and reynolds number parameter k

robust solution of singularly perturbed problems using multigrid methods we consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction diffusion type standard discretization techniques are not suitable for such problems and so specially tailored methods are required usually involving adapted or fitted meshes that resolve important features such as boundary and or interior layers in this study we consider classical finite difference schemes on the layer adapted meshes of shishkin and bakhvalov we show that standard direct solvers exhibit poor scaling behavior with respect to the perturbation parameter when solving the resulting linear systems we propose and prove optimality of a new block structured preconditioning approach that is robust for small values of the perturbation parameter and compares favorably with standard robust multigrid preconditioners for these linear systems we also derive stopping criteria which ensure that the potential accuracy

an application of multigrid methods for a discrete elastic model for epitaxial systems we apply an efficient and fast algorithm to simulate the atomistic strain model for epitaxial systems recently introduced by schindler et al phys rev b 67 075316 2003 the discrete effects in this lattice statics model are crucial for proper simulation of the influence of strain for thin film epitaxial growth but the size of the atomistic systems of interest is in general quite large and hence the solution of the discrete elastic equations is a considerable numerical challenge in this paper we construct an algebraic multigrid method suitable for efficient solution of the large scale discrete strain model using this method simulations are performed for several representative physical problems including an infinite periodic step train a layered nanocrystal and a system of quantum dots the results demonstrate the effectiveness and robustness of the method and show that the method attains optimal convergence properties regardless of the problem size the geometry and the physical parameters the effects of substrate depth and of invariance due to traction free boundary conditions are assessed for a system of quantum dots the simulated strain energy density supports the observations that trench formation near the dots provides strain relief

a sampling kaczmarz motzkin algorithm for linear feasibility we combine two iterative algorithms for solving large scale systems of linear inequalities the relaxation method of agmon motzkin et al and the randomized kaczmarz method we obtain a family of algorithms that generalize and extend both projection based techniques we prove several convergence results and our computational experiments show our algorithms often outperform the original methods

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

relationship between the impedance matrix and the transfer matrix with specific reference to symmetrical reciprocal and conservative systems impedance matrix and transfer matrix methods are often used in the analysis of linear dynamical systems in this paper general relationships between these matrices are derived the properties of the impedance matrix and the transfer matrix of symmetrical systems reciprocal systems and conservative systems are investigated in the process the following observations are made a symmetrical systems are not a subset of reciprocal systems as is often misunderstood b the cascading of reciprocal systems again results in a reciprocal system whereas cascading of symmetrical systems does not necessarily result in a symmetrical system c the determinant of the transfer matrix being 1 is a property of both symmetrical systems and reciprocal systems but this condition however is not sufficient to establish either the reciprocity or the symmetry of the system d the impedance matrix of a conservative system is skew hermitian

multilevel methods for elliptic problems with highly varying coefficients on nonaligned coarse grids in this paper we generalize the analysis of classical multigrid and two level overlapping schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition the theoretical results provide a recipe for designing hierarchies of standard piecewise linear coarse spaces such that the multigrid convergence rate and the condition number of the schwarz preconditioned system do not depend on the coefficient variation or on any mesh parameters an assumption we have to make is that the coarse grids are sufficiently fine in the vicinity of cross points or where regions with large diffusion coefficients are separated by a narrow region where the coefficient is small we do not need to align them with possible discontinuities in the coefficients the proofs make use of novel stable splittings based on weighted quasi interpolants and weighted poincare type inequalities numerical experiments are

robust multigrid method for the planar linear elasticity problems we consider the solution of the system of equations that arise from the higher order conforming finite element scott vogelius element discretizations of the boundary value problems associated with the differential operator 2 2 div where and are nonzero parameters robust multigrid method is constructed i e the convergence rate of multigrid method is optimal with respect to the mesh size the number of levels and weights on the two terms in the aforementioned differential operator

deflation and projection methods applied to symmetric positive semi definite systems linear systems with a singular symmetric positive semi definite matrix appear frequently in practice this usually does not lead to difficulties for cg methods as long as these systems are consistent however the construction of a preconditioner especially the construction of two level and multilevel methods becomes more complicated since singular coarse grid matrices or galerkin matrices may occur here we continue the work started in 21 22 where deflation is used for some special singular coefficient matrices here we show that deflation and other projection type preconditioners can be applied to arbitrary singular problems without any difficulties in each of these methods a two level preconditioner is involved where coarse grid systems based on a singular galerkin matrix should be solved we prove that each projection operator consisting of a singular galerkin matrix can be written as an operator with a nonsingular galerkin matrix therefore many results that hold for nonsingular galerkin matrices are also valid for problems with singular galerkin matrices

on the finite element solution of the pure neumann problem this paper considers the finite element approximation and algebraic solution of the pure neumann problem our goal is to present a concise variational framework for the finite element solution of the neumann problem that focuses on the interplay between the algebraic and variational problems while many of the results that stem from our analysis are known by some experts they are seldom derived in a rigorous fashion and remain part of numerical folklore as a result this knowledge is not accessible or appreciated by many practitioners both novices and experts in one source our paper contributes a simple yet insightful link between the continuous and algebraic variational forms that will prove useful

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

multilayer dielectric filter design using a multiobjective evolutionary algorithm design of multilayer dielectric filters involve the identification of suitable dielectric material and appropriate thicknesses of the layers that best satisfies the desired frequency response for the application such problems like any other practical design optimization problem require simultaneous consideration of multiple objectives and constraints in this paper we introduce a multiobjective evolutionary algorithm that is capable of handling unconstrained and constrained single and multiobjective problems without any restriction on the number and nature of variables constraints and objectives the algorithm handles constraints and objectives separately using two fitness measures derived out of nondominance unlike most of its counterparts which use a single fitness measure unlike most evolutionary algorithms where only the good parents participate in mating our algorithm ensures that all solutions participate in mating which is useful for exploring highly nonlinear search spaces the diversity of the solutions is controlled by the partner selection scheme that prefers elites with distant neighbors as mating partners the results of two multiobjective test problems three multilayer dielectric filter designs low pass bandpass and stopband and one variable layer low pass filter design are presented in this paper to highlight the benefits offered by our algorithm in terms of modeling flexibility computational efficiency and its ability to arrive at competitive nondominated designs a comparison of our results with those obtained using a single objective aggregated formulation for the stopband filter design is also presented we have also compared the performance of our algorithm with nondominated sorting genetic algorithm nsga ii for the low pass filter design where our results are better

algebraic multilevel iteration method for lowest order raviart thomas space and applications an optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems that have their weak formulation in the space h div is presented the algorithm is developed for the discrete problem obtained by using the lowest order raviart thomas space the method is theoretically analyzed and supporting numerical examples are presented furthermore as a particular application the algorithm is used for the solution of the discrete minimization problem which arises in the functional type a posteriori error estimates for the discontinuous galerkin approximation of elliptic problems copyright 2011 john wiley sons ltd

finite element approximation of multi scale elliptic problems using patches of elements in this paper we present a method for the numerical solution of elliptic problems with multi scale data using multiple levels of not necessarily nested grids the method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming this paper provides proofs of the results published earlier see c r acad sci paris ser i 337 2003 679 684 gives a generalization of the latter to more than two domains and contains extensive numerical illustrations new results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened cauchy buniakowski schwarz inequality are presented

transfer matrix analysis of the elastostatics of one dimensional repetitive structures transfer matrices are used widely for the dynamic analysis of engineering structures increasingly so for static analysis and are particularly useful in the treatment of repetitive structures for which in general the behaviour of a complete structure can be determined through the analysis of a single repeating cell together with boundary conditions if the structure is not of infinite extent for elastostatic analyses non unity eigenvalues of the transfer matrix of a repeating cell are the rates of decay of selfequilibrated loading as anticipated by saint venant s principle multiple unity eigenvalues pertain to the transmission of load e g tension or bending moment and equivalent homogenized continuum properties such as cross sectional area second moment of area and poisson s ratio can be determined from the associated eigen and principal vectors various disparate results the majority new others drawn from diverse sources are presented these include calculation of principal vectors using the moore penrose inverse bi and symplectic orthogonality and relationship with the reciprocal theorem restrictions on complex unity eigenvalues effect of cell left to right symmetry on both the stiffness and transfer matrices eigenvalue veering in the absence of translational symmetry and limitations on possible jordan canonical forms it is shown that only a repeating unity eigenvalue can lead to a non trivial jordan block form so degenerate decay modes cannot exist the present elastostatic analysis complements langley s langley 1996 proc r soc a 452 1631 1648 transfer matrix analysis of wave motion energetics

pareto optimal microwave filter design using multiobjective differential evolution microwave filters play an important role in modern wireless communications a novel method for the design of multilayer dielectric and open loop ring resonator olrr filters under constraints is presented the proposed design method is based on generalized differential evolution gde3 which is a multiobjective extension of differential evolution de gde3 algorithm can be applied for global optimization to any engineering problem with an arbitrary number of objective and constraint functions gde3 is compared against other evolutionary multiobjective algorithms like nondominated sorting genetic algorithm ii nsga ii multiobjective particle swarm optimization mopso and multiobjective particle swarm optimization with fitness sharing mopso fs for a number of microwave filter design cases in the multilayer dielectric filter design case a predefined database of low loss dielectric materials is used the results indicate the advantages of this approach and the applicability of this design method

a two level algorithm for the weak galerkin discretization of diffusion problems this paper analyzes a two level algorithm for the weak galerkin wg finite element methods based on local raviart thomas rt and brezzi douglas marini bdm mixed elements for two and three dimensional diffusion problems with dirichlet condition we first show the condition numbers of the stiffness matrices arising from the wg methods are of o h 2 we use an extended version of the xu zikatanov xz identity to derive the convergence of the algorithm without any regularity assumption finally we provide some numerical results

randomized block kaczmarz method with projection for solving least squares abstract the kaczmarz method is an iterative method for solving overcomplete linear systems of equations a x b the randomized version of the kaczmarz method put forth by strohmer and vershynin iteratively projects onto a randomly chosen solution space given by a single row of the matrix a and converges linearly 1 in expectation to the solution of a consistent system in this paper we analyze two block versions of the method each with a randomized projection designed to converge in expectation to the least squares solution often faster than the standard variants our approach utilizes both a row and column paving of the matrix a to guarantee linear convergence when the matrix has consistent row norms called nearly standardized and a single column paving when the row norms are unbounded the proposed methods are an extension of the block kaczmarz method analyzed by needell and tropp and the randomized extended kaczmarz method of zouzias and freris the contribution is thus two fold unlike the standard kaczmarz method our results demonstrate convergence to the least squares solution of inconsistent systems both methods in the nearly standardized case and the second method in other cases by using appropriate blocks of the matrix this convergence can be significantly accelerated as is demonstrated by numerical experiments

an algebraic multigrid method with interpolation reproducing rigid body modes for semi definite problems in two dimensional linear elasticity based on the geometric grid information as geometric coordinates an algebraic multigrid amg method with the interpolation reproducing the rigid body modes namely the kernel elements of semi definite operator arising from linear elasticity is constructed and such method is applied to the linear elasticity problems with a traction free boundary condition and crystal problems with free boundary conditions as well the results of various numerical experiments in two dimensions are presented it is shown from the numerical results that the constructed amg method is robust and efficient for such semi definite problems and the convergence is uniformly bounded away from one independent of the problem size furthermore the amg method proposed in this paper has better convergence rate than the commonly used amg methods simultaneously an amg method that can preserve the quotient space which means that if the exact solution of original problem belongs to the quotient space of discrete operator considered then the numerical solution of amg method is convergent in the same quotient space is obtained using the technique of orthogonal decomposition

a note on properties of splittings of singular symmetric positive semidefinite matrices recently benzi and szyld have published an important paper 1 concerning the existence and uniqueness of splittings for singular matrices however the assertion in theorem 3 9 on the inheriting property of p regular splitting for singular symmetric positive semidefinite matrices seems to be incorrect as a complement of paper 1 in this short note we point out that if a matrix t is resulted from a p regular splitting of a symmetric positive semidefinite matrix a then splittings induced by t are not all p regular

boundary condition identification using condensation and inversion application to operating piping network this paper deals with the experimental analysis of piping systems under operating conditions in the field of vibration and acoustic analysis a new approach to identify the boundary conditions of a part of a curvilinear structure is presented the basic concept consists of solving an inverse problem where the measured response of the system tested is combined with an incomplete analytical model in order to identify the boundary dynamical state in the frequency domain as in finite element methods the tested network is described using elements and nodes an original technique using a transfer matrix of continuous elements provides a small size analytical model in addition condensation procedure is used to eliminate all degrees of freedom d o f having a modelled boundary condition and to reduce the size of the solved inverse problem since identification of boundary conditions is performed the analysis of dynamic response of the tested network may be performed without further matrix computation the validity and the feasibility of the approach are shown using actual test results examples concerning real applications are also presented

the finite element approximation of the nonlinear poisson boltzmann equation a widely used electrostatics model in the biomolecular modeling community the nonlinear poisson boltzmann equation along with its finite element approximation are analyzed in this paper a regularized poisson boltzmann equation is introduced as an auxiliary problem making it possible to study the original nonlinear equation with delta distribution sources a priori error estimates for the finite element approximation are obtained for the regularized poisson boltzmann equation based on certain quasi uniform grids in two and three dimensions adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge the poisson boltzmann equation does not appear to have been previously studied in detail theoretically and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the poisson boltzmann equation note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear poisson boltzmann equation with delta distribution sources and it also introduces the first provably convergent adaptive method for the equation this last result is currently one of only a handful of existing convergence results of this type for nonlinear problems

preconditioning a class of fourth order problems by operator splitting we develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space we consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two discrete linear second order elliptic operators and exploit this property in designing the preconditioners the underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available we propose symmetric and non symmetric preconditioners along with theory and numerical experiments they both document crucial properties of the preconditioners as well as their practical performance it is important to note that we neither need h s regularity s 1 of the continuous problem nor quasi uniform grids

spectral multiscale finite element for nonlinear flows in highly heterogeneous media a reduced basis approach in this paper we study multiscale finite element methods for richards equation a mathematical model to describe fluid flow in unsaturated and highly heterogeneous porous media in order to compute solutions of richard s equation one can use numerical homogenization or multiscale methods that use two grid procedures a fine grid that resolves the heterogeneities and a coarse grid where computations are done the idea is that the coarse solution procedure captures the fine grid variations of the solution since the media has complicated variations inside of coarse grid blocks a large error can be generated during the computation of coarse scale solutions in this paper we consider the case of highly varying coefficients where variations can occur within coarse regions we develop accurate multiscale methods in order to obtain accurate coarse scale numerical solutions for richards equation we design an effective multiscale method that is able to capture the multiscale features of the solution without discarding the small scale details with a careful choice of the coarse basis functions for multiscale finite element methods we can significantly reduce errors we use coarse basis functions construction that combines local spectral problems and a reduced basis rb approach this is an extension to the nonlinear case of the method proposed by efendiev et al 2012 that combines spectral constructions of coarse spaces with rb procedures to efficiently solve linear parameter dependent flow problems the construction of coarse spaces begins with an initial choice of multiscale basis functions supported in coarse regions these basis functions are complemented using a local parameter dependent weighted eigenvalue problem the obtained basis functions can capture the small scale features of the solution within a coarse grid block and give us an accurate coarse scale solution the rb procedures are used to efficiently solve for all possible flow scenarios encountered in every single iteration of a fixed point iterative method we present representative numerical experiments

regularity and multigrid analysis for laplace type axisymmetric equations consider axisymmetric equations associated with laplace type operators we establish full regularity estimates in high order kondrat vetype spaces for possible singular solutions due to the non smoothness of the domain and to the singular coefficients in the operator then we show suitable graded meshes can be used in high order finite element methods to achieve the optimal convergence rate even when the solution is singular using these results we further propose multigrid v cycle algorithms solving the system from linear finite element discretizations on optimal graded meshes we prove the multigrid algorithm is a contraction with the contraction number independent of the mesh level numerical tests are provided to verify the theorem

a sharp convergence estimate for the method of subspace corrections for singular systems of equations this paper is devoted to the convergence rate estimate for the method of successive subspace corrections applied to symmetric and positive semidefinite singular problems in a general hilbert space setting a convergence rate identity is obtained for the method of subspace corrections in terms of the subspace solvers as an illustration the new abstract theory is used to show uniform convergence of a multigrid method applied to the solution of the laplace equation with pure neumann boundary conditions

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

a robust multigrid method for discontinuous galerkin discretizations of stokes and linear elasticity equations we consider multigrid methods for discontinuous galerkin h text div omega h div conforming discretizations of the stokes and linear elasticity equations we analyze variable v cycle and w cycle multigrid methods with nonnested bilinear forms we prove that these algorithms are optimal and robust i e their convergence rates are independent of the mesh size and also of the material parameters such as the poisson ratio numerical experiments are conducted that further confirm the theoretical results

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

multilevel preconditioners for discontinuous galerkin approximations of elliptic problems with jump coefficients in this article we develop and analyze two level and multi level methods for the family of interior penalty ip discontinuous galerkin dg discretizations of second order elliptic problems with rough coecients exhibiting large jumps across interfaces in the domain these methods are based on a decomposition of the dg nite element space that inherently hinges on the diusion coecient of the problem our analysis of the proposed preconditioners is presented for both symmetric and non symmetric ip schemes and we establish both robustness with respect to the jump in the coecient and near optimality with respect to the mesh size following the analysis we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods

an additive analysis of multiplicative schwarz methods we present an analysis of multiplicative schwarz methods for symmetric positive definite problems that is based on the theory of additive schwarz preconditioners and discuss applications to multigrid methods and domain decomposition methods

convergence analysis of variational and non variational multigrid algorithms for the laplace beltrami operator we design and analyze variational and non variational multigrid algorithms for the laplace beltrami operator on a smooth and closed surface in both cases a uniform convergence for the v cycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid the main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface in addition the vanishing mean value constraint is imposed on each level thereby avoiding singular quadratic forms without adding additional computational cost numerical results supporting our analysis are reported in particular the algorithms perform well even when applied to surfaces with a large aspect ratio

parallel subspace correction methods for nearly singular systems abstract in this paper we consider the parallel subspace correction psc methods for the nearly singular systems we apply the psc methods as the preconditioners when we solve the nearly singular systems by the conjugate gradient methods our focus is to estimate the condition number of the preconditioned systems we deduce the parameter independent estimates on the psc preconditioners for the nearly singular systems under appropriate assumptions on subspace decomposition the main assumption is that the kernel of the singular part of the system can be decomposed into a sum of local kernel subspaces our estimates can be applied into actual problems and two examples are analyzed in this paper one is the elliptic problem with large jumps in the coefficients the other is the planar nearly incompressible elasticity problem with the scott vogelius finite element discretization we prove that the related parallel multilevel methods for both examples are convergent uniformly with respect to the coefficients and the mesh size

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

uniform convergence of a nonlinear energy based multilevel quantization scheme a popular vector quantization scheme can be constructed by centroidal voronoi tessellations cvts which also have many other applications in diverse areas of science and engineering the development of efficient algorithms for their construction is a key to the successful applications of cvts in practice this paper studies the details of a new optimization based multilevel algorithm for the numerical computation of cvts the rigorous proof of its uniform convergence in one space dimension and the results of computational simulations are provided they substantiate recent claims on the significant speedup demonstrated by the new scheme in comparison with traditional methods

multigrid methods for saddle point systems using constrained smoothers the constrained smoother for solving the saddle point system arising from the constrained minimization problem is a relaxation scheme such that the iteration remains in the constrained subspace a multigrid method using constrained smoothers for saddle point systems is analyzed in this paper uniform convergence of two level and w cycle multigrid methods with sufficient many smoothing steps and full regularity assumptions are obtained for some stable finite element discretization of stokes equations for braess sarazin smoother a convergence theory using only partial regularity assumption is also developed

a posteriori error estimates of finite element methods by preconditioning abstract we present a framework that relates preconditioning with a posteriori error estimates in finite element methods in particular we use standard tools in subspace correction methods to obtain reliable and efficient error estimators as a simple example we recover the classical residual error estimators for the second order elliptic equations

angles between infinite dimensional subspaces with applications to the rayleigh ritz and alternating projectors methods we define angles for infinite dimensional subspaces of hilbert spaces inspired by the work of e j hannan 1961 1962 the angles of dixmier and friedrichs and the gaps are characterized we establish connections between the angles corresponding to orthogonal complements the sensitivity of angles with respect to subspaces is estimated we show that the squared cosines of the angles from one subspace to another can be interpreted as ritz values in the rayleigh ritz method the hausdorff distance between the ritz values corresponding to different trial subspaces is shown to be bounded by a constant times the gap between the subspaces we prove a similar eigenvalue perturbation bound that involves the gap squared an ultimate acceleration of the classical alternating projectors method is proposed its convergence rate is estimated in terms of the angles we illustrate the acceleration for a domain decomposition method with a small overlap for the 1d diffusion equation

on additive schwarz preconditioners for sparse grid discretizations based on the framework of subspace splitting and the additive schwarz scheme we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems for a bxp like preconditioner we derive an estimate of the optimal ordero 1 and for a hb like variant we obtain an estimate of the ordero k 2 2 k 2 wherek denotes the number of levels employed furthermore we confirm these results by numerically computed condition numbers

comparison theorems for the convergence factor of iterative methods for singular matrices abstract iterative methods for the solution of consistent singular systems of linear equations are governed by the convergence factor of the iteration matrix t i e by the quantity t max t 1 where t is the spectrum of t theorems are presented comparing the convergence factor of two iterative methods the comparison is based on the relationship between the matrices of the splittings a cone other than the usual nonnegative hyperoctant is used to define the order used in this comparison although this cone is based on the unknown projection onto the null space of a matrix the characterization provided in the paper allows in specific instances the cone to be readily computable

some nonoverlapping domain decomposition methods the purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second order elliptic problems in two and three dimensions the methods under scrutiny fall into two major categories the substructuring type methods and the neumann neumann type methods the basic framework used for analysis is the parallel subspace correction method or additive schwarz method and other technical tools include local global and global local techniques the analyses for both two and three dimensional cases are carried out simultaneously some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived

optimal multilevel methods for graded bisection grids we design and analyze optimal additive and multiplicative multilevel methods for solving h 1 problems on graded grids obtained by bisection we deal with economical local smoothers after a global smoothing in the finest mesh local smoothing for each added node during the refinement needs to be performed only for three vertices the new vertex and its two parent vertices we show that our methods lead to optimal complexity for any dimensions and polynomial degree the theory hinges on a new decomposition of bisection grids in any dimension which is of independent interest and yields a corresponding decomposition of spaces we use the latter to bridge the gap between graded and quasi uniform grids for which the multilevel theory is well established

on the abstract theory of additive and multiplicative schwarz algorithms in recent years it has been shown that many modern iterative algorithms multigrid schemes multilevel preconditioners domain decomposition methods etc for solving problems resulting from the discretization of pdes can be interpreted as additive jacobi like or multiplicative gauss seidel like subspace correction methods the key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space v into a sum of subspaces v j and the splitting of the variational problem on v into auxiliary problems on these subspaces in this paper we propose a modification of the abstract convergence theory of the additive and multiplicative schwarz methods that makes the relation to traditional iteration methods more explicit the analysis of the additive and multiplicative schwarz iterations can be carried out in almost the same spirit as in the traditional block matrix situation making convergence proofs of multilevel and domain decomposition methods clearer or at least more classical in addition we present a new bound for the convergence rate of the appropriately scaled multiplicative schwarz method directly in terms of the condition number of the corresponding additive schwarz operator these results may be viewed as an appendix to the recent surveys x ys

a randomized approximation algorithm for the minimal norm static output feedback problem a new randomized algorithm is suggested for extracting static output stabilizing feedbacks with approximately minimal norm for lti systems the algorithm has two similar stages where in the first one the feasibility problem is solved and in the second one the optimization problem is solved the formulation is unified for the feasibility and for the optimization problems as well as for continuous time or discrete time systems the method is demonstrated by applying it to the hard conjectured to be np hard problem of the minimal gain static output stabilizing feedback and to the hard conjectured to be np hard problem of regional pole placement via static output feedback in non convex or unconnected regions a proof of convergence in probability that captures the two rounds of the algorithm is given and complexity analysis is provided under some mild assumptions

optimum design of yagi uda antennas using computational intelligence optimization of yagi uda antennas is a challenging design problem since the antenna characteristics such as gain input impedance maximum sidelobe level etc are known to be extremely sensitive to the design variables viz element lengths and their spacings this corresponds to a highly nonlinear and multimodal function space with functional and slope discontinuities that limit the use of conventional gradient based optimization approaches although stochastic zeroth order methods like genetic algorithm and evolutionary algorithm are attractive choices for such classes of problems their successful application requires scaling and aggregating parameters to handle constraints and objectives that may not be easy to provide in this paper we introduce a stochastic zeroth order optimization algorithm that handles constraints and objectives separately via pareto ranking that eliminates the problem of scaling and aggregation the algorithm is based on principles of learning and is embedded with three key learning strategies that control whom to learn from i e leader identification and leader selection and what to learn i e information acquisition in order to better guide the search the leader identification mechanism partitions the individuals into a set of leaders and a set of followers the followers interact with the leader and move toward the better performing leaders in search for better solutions as the algorithm does not require parameters for scaling or aggregation it provides the designer the true flexibility that is necessary to handle various forms of the design problem effectively and at a computational cost that is comparable to existing stochastic optimization methods results of three single objective antenna design examples a four element a 15 element and a fixed boom length 22 element design are presented and compared with published results to illustrate the behavior of the proposed algorithm and highlight its benefits in solving a wide variety of antenna design problems a new set of results are presented for a multiobjective formulation of the design problem

a posteriori error estimation for generalized finite element methods in this paper we address the problem of a posteriori error estimation for generalized finite element methods based on the partition of unity method the computational results focus on the question of the reliability of the error estimators and its assessment

accelerating the convergence of the method of alternating projections the powerful von neumann halperin method of alternating projections map is an algorithm for determining the best approximation to any given point in a hilbert space from the intersection of a finite number of subspaces it achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the it individual subspaces which make up the intersection the main practical drawback of this algorithm at least for some applications is that the method is slowly convergent in this paper we consider a general class of iterative methods which includes the map as a special case for such methods we study an accelerated version of this algorithm which was considered earlier by gubin polyak and raik 14 and by gearhart and koshy 13 we show that the accelerated algorithm converges faster than the map in the case of two subspaces but is in general it not faster than the map for more than two subspaces however for a symmetric version of the map the accelerated algorithm always converges faster for any number of subspaces our proof seems to require the use of the spectral theorem for self adjoint mappings

high order extended finite element methods for solving interface problems abstract in this paper we study arbitrary order extended finite element xfe methods based on two discontinuous galerkin dg schemes in order to solve elliptic interface problems in two and three dimensions optimal error estimates in the piecewise h 1 norm and l 2 norm are rigorously proved for both schemes in particular we have devised a new parameter friendly dg xfem method which means that no sufficiently large parameters are needed to ensure the optimal convergence of the scheme to prove the stability of bilinear forms we derive non standard trace and inverse inequalities for high order polynomials on curved sub elements divided by the interface this paper is adapted from the work originally post on arxiv com by the same authors arxiv 1604 06171 new ingredients are an optimal multigrid solver for the generated linear system and its analysis this multigrid method converges uniformly with respect to the mesh size and is independent of the location of the interface relative to the meshes just like all the other estimates in this paper numerical examples are given to support the theoretical results

on stability accuracy and fast solvers for finite element approximations of the axisymmetric stokes problem by hood taylor elements we provide a proof of both the stability and the approximation property for the finite element approximations of the axisymmetric stokes problem by continuous piecewise polynomials of degree kappa 1 for the velocity and continuous piecewise polynomials of degree kappa for the pressure with any kappa geq1 new techniques are designed so that in this perspective by a simple transformation the existing theory developed in three dimensional cartesian coordinates can be effectively exploited in fact this perspective provides a new way of developing theories for the axisymmetric stokes problems and it can be applied potentially to other problems as well a simple illustration is provided for the application in the development and analysis of fast solvers for the resulting discrete saddle point problems sample numerical experiments have been presented as well to confirm the theoretical results

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations

a nearly optimal multigrid method for general unstructured grids in this paper we develop a multigrid method on unstructured shape regular grids for a general shape regular unstructured grid of mathcal o n o n elements we present a construction of an auxiliary coarse grid hierarchy on which a geometric multigrid method can be applied together with a smoothing on the original grid by using the auxiliary space preconditioning technique such a construction is realized by a cluster tree which can be obtained in mathcal o n log n o nlogn operations for a grid of n elements this tree structure in turn is used for the definition of the grid hierarchy from coarse to fine for the constructed grid hierarchy we prove that the convergence rate of the multigrid preconditioned cg for an elliptic pde is 1 mathcal o 1 log n 1 o 1 logn numerical experiments confirm the theoretical bounds and show that the total complexity is in mathcal o n log n o nlogn

mimo radar waveform design via alternating projection waveform design is essential to unleash the performance advantages promised by multiple input multiple output mimo radar and this topic has attracted a lot of attention in the recent years revisiting an earlier examined mimo radar waveform design problem that optimizes both minimum mean square error estimation mmse and mutual information mi in this correspondence we formulate a new waveform design problem and provide some further results which complement the previous study more specifically we present an iterative optimization algorithm based on the alternating projection method to determine waveform solutions that can simultaneously satisfy a structure constraint and optimize the design criteria numerical examples are provided which illustrate the effectiveness of the proposed approach in particular we find that the waveform solutions obtained through our proposed algorithm can achieve very close and virtually indistinguishable performance from that predicted in the previous study

numerical analysis of vanka type solvers for steady stokes and navier stokes flows we consider vanka type smoothers for solving stokes and navier stokes problems in each iteration step this smoother requires the solution of several small local subproblems over finite element blocks it is shown that for particular choices for the blocks the algorithm always converges to the solution of the stokes problem and under suitable conditions to the solution of the navier stokes problem the convergence properties are analyzed and numerical examples are presented

discriminative common vector method with kernels in some pattern recognition tasks the dimension of the sample space is larger than the number of samples in the training set this is known as the small sample size problem linear discriminant analysis lda techniques cannot be applied directly to the small sample size case the small sample size problem is also encountered when kernel approaches are used for recognition in this paper we attempt to answer the question of how should one choose the optimal projection vectors for feature extraction in the small sample size case based on our findings we propose a new method called the kernel discriminative common vector method in this method we first nonlinearly map the original input space to an implicit higher dimensional feature space in which the data are hoped to be linearly separable then the optimal projection vectors are computed in this transformed space the proposed method yields an optimal solution for maximizing a modified fisher s linear discriminant criterion discussed in the paper thus under certain conditions a 100 recognition rate is guaranteed for the training set samples experiments on test data also show that in many situations the generalization performance of the proposed method compares favorably with other kernel approaches

accelerating the alternating projection algorithm for the case of affine subspaces using supporting hyperplanes abstract the von neumann halperin method of alternating projections converges strongly to the projection of a given point onto the intersection of finitely many closed affine subspaces we propose acceleration schemes making use of two ideas firstly each projection onto an affine subspace identifies a hyperplane of codimension 1 containing the intersection and secondly it is easy to project onto a finite intersection of such hyperplanes we give conditions for which our accelerations converge strongly finally we perform numerical experiments to show that these accelerations perform well for a matrix model updating problem

the method of subspace corrections abstract this paper gives an overview for the method of subspace corrections the method is first motivated by a discussion on the local behavior of high frequency components in a solution to an elliptic problem a simple domain decomposition method is discussed as an illustrative example and multigrid methods are discussed in more detail brief discussions are also given to some non linear examples including eigenvalue problems obstacle problems and liquid crystal modelings the relationship between the method of subspace correction and the method of alternating projects is observed and discussed

subspace correction methods for total variation and ell_1 minimization this paper is concerned with the numerical minimization of energy functionals in hilbert spaces involving convex constraints coinciding with a seminorm for a subspace the optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces on each subspace an iterative proximity map algorithm is implemented via oblique thresholding which is the main new tool introduced in this work we provide convergence conditions for the algorithm in order to compute minimizers of the target energy analogous results are derived for a parallel variant of the algorithm applications are presented in domain decomposition methods for degenerate elliptic pdes arising in total variation minimization and in accelerated sparse recovery algorithms based on ell 1 minimization we include numerical examples which show efficient solutions to classical problems in signal and image processing

algebraic schwarz methods for the numerical solution of markov chains the convergence of additive and multiplicative schwarz methods for computing certain characteristics of markov chains such as stationary probability vectors and mean first passage matrices is studied the main result is a convergence theorem for multiplicative schwarz iterations when applied to singular systems as a byproduct a convergence result for alternating iterations is also obtained it is also shown that when the markov chain is ergodic additive and multiplicative schwarz methods can be applied to the nonsingular systems that result from reducing the equations the so called coarse grid corrections are also studied