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

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

boundedness theorems for the relaxation method a classical theorem by block and levin block h d s a levin 1970 on the boundedness of an iterative procedure for solving a system of linear inequalities proc amer math soc 26 229 235 states that certain variants of the relaxation method for solving systems of linear inequalities generate bounded sequences of intermediate solutions even when applied to infeasible systems using a new approach we prove a more general version of this result and answer an old open problem of quantifying the bound as a function of the input data

recursion relations in u n integral in lattice qcd on the basis of the schwinger dyson variational method the recursion relations of the character expansion coefficients of the u n gauge action are derived three recursion relations that are associated with the casimir invariants are shown to be enough to determine the coefficients completely

normal and anomalous scaling of turbulence a non gaussian model of the probability density function pdf of ensuremath delta u r is proposed to study how the scaling exponents s p of the structure function ensuremath delta u r p of finite reynolds number turbulence depends upon the taylor microscale reynolds number r ensuremath lambda here ensuremath delta u r is the longitudinal velocity difference across a distance r and is the statistical average the model not only predicts anomalous scaling s p lp 3 for pg3 observed in experiments at a finite reynolds number but also predicts that s p approaches normal scaling p 3 while r ensuremath lambda is very high hence in contrast to the prevailing multiscaling models the non gaussian pdf model suggests a completely different picture of scaling of isotropic turbulence the anomalous scaling observed in experiments is a finite reynolds number effect and the normal scaling is valid in the real kolmogorov inertial range corresponding to an infinite reynolds number

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

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

a linearly convergent doubly stochastic gauss seidel algorithm for solving linear equations and a certain class of over parameterized optimization problems consider the classical problem of solving a general linear system of equations ax b it is well known that the successively over relaxed gauss seidel scheme and many of its variants may not converge when a is neither diagonally dominant nor symmetric positive definite can we have a linearly convergent g s type algorithm that works for anya in this paper we answer this question affirmatively by proposing a doubly stochastic g s algorithm that is provably linearly convergent in the mean square error sense for any feasible linear system of equations the key in the algorithm design is to introduce a nonuniform double stochastic scheme for picking the equation and the variable in each update step as well as a stepsize rule these techniques also generalize to certain iterative alternating projection algorithms for solving the linear feasibility problem a x le b with an arbitrary a as well as high dimensional minimization problems for training over parameterized models in machine learning our results demonstrate that a carefully designed randomization scheme can make an otherwise divergent g s algorithm converge

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

fiber dynamics in turbulent flows general modeling framework the paper at hand deals with the modeling of turbulence effects on the dynamics of a long slender elastic fiber independent of the choice of the drag model a general aerodynamic force concept is derived on the basis of the velocity field for the randomly fluctuating component of the flow its construction as a centered differentiable gaussian field complies thereby with the requirements of the stochastic k epsilon turbulence model and kolmogorov s universal equilibrium theory on local isotropy

closure approach to high order structure functions of turbulence let t p be the structure function of order p of turbulence first t2 is determined then t3 is derived from t2 and finally t p p 3 are derived from t2 and t3 this closure scheme is realized by a non gaussian statistical model we use it to study the scaling law of t p and we find that the available data on scaling exponents favor kolmogorov s 1941 theory rather than his 1962 theory we also predict the high order universal constants of inertial range scaling

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

recursive equation of character expansion coefficient in lattice gauge theory the strict mathematical derivation is given for the recursion relations of the coefficients of the character expansion of the single link action in the su 2 u 2 and su 3 groups this relies on the immediate schwinger dyson variation on the character it will be suggested that the independent recursion relations for the u n and su n groups are associated with the group invariants i e the casimir invariants and the unit generator for u n

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

spectral analysis of structure functions and their scaling exponents in forced isotropic turbulence the pseudospectral method in conjunction with a technique for obtaining scaling exponents n from the structure functions sn r is presented as an alternative to the extended self similarity ess method and the use of generalized structure functions we propose plotting the ratio sn r s3 r against the separation r in accordance with a standard technique for analyzing experimental data this method differs from the ess technique which plots sn r against s3 r with the assumption s3 r r using our method for the particular case of s2 r we obtain the result that the exponent 2 decreases as the taylor reynolds number increases with 2 0 679 0 013 as r this supports the idea of finite viscosity corrections to the k41 prediction for s2 and is the opposite of the result obtained by ess the pseudospectral method also permits the forcing to be taken into account exactly through the calculation of the energy input in real space from the work spectrum of the stirring forces

a hierarchy of maximum entropy closures for galerkin systems of incompressible flows abstract we propose a maximum entropy closure strategy for dissipative dynamical systems building on and generalizing earlier examples noack niven 2012 11 focus is placed on galerkin systems arising from a projection of the incompressible navier stokes equation onto orthonormal expansion modes the maximum entropy closure is motivated by a simple analytical example and elaborated to a hierarchical framework with sufficient conditions for the existence of solutions

on kaczmarz s projection iteration as a direct solver for linear least squares problems abstract in this paper we construct and theoretically analyze a class of direct projection algorithms for the numerical solution of linear least squares problems these algorithms are obtained by adding supplementary directions for projection constructed as linear combinations of the initial system rows and columns in kaczmarz and extended kaczmarz iterative methods the above ideas are extended to the block row and column versions of the previously mentioned methods the developed algorithms are then compared with other direct projection based methods by the application to problems arising in multibody elasticity

polynomial algorithms for a class of linear programs this paper describes several algorithms for solution of linear programs the algorithms are polynomial when the problem data satisfy certain conditions

differential equation for su 3 extension of bessel function with two indices for the coefficient function of the character expansion of the single link action in the su 3 lattice gauge theory the linear differential equation of the sixth order which is specified by two integer indices and of the su 3 representation of the young tableau is derived the asymptotic behavior of the function is derived from the recursion relations it will be shown that the coefficient function is regarded as the su 3 extension of the modified bessel function

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 new method to predict the evolution of the power spectral density for a finite amplitude sound wave a method to predict the effect of nonlinearity on the power spectral density of a plane wave traveling in a thermoviscous fluid is presented as opposed to time domain methods the method presented here is based directly on the power spectral density of the signal not the signal itself the burgers equation is employed for the mathematical description of the combined effects of nonlinearity and dissipation the burgers equation is transformed into an infinite set of linear equations that describe the evolution of the joint moments of the signal a method for solving this system of equations is presented only a finite number of equations is appropriately selected and solved by numerical means for the method to be applied all appropriate joint moments must be known at the source if the source condition has gaussian characteristics it is a gaussian noise signal or a gaussian stationary and ergodic stochastic process then all the joint moments can be computed from the power spectral density of the signal at the source numerical results from the presented method are shown to be in good agreement with known analytical solutions in the preshock region for two benchmark cases i sinusoidal source signal and ii a gaussian stochastic process as the source condition

inertial range and the finite reynolds number effect of turbulence the kolmogorov law which is the unique exact relationship of inertial range statistics is applied to investigate the finite reynolds number effect in particular to study how the width of the inertial range of finite reynolds number turbulence changes with the taylor microscale reynolds number mathrm r ensuremath lambda it is found that there is no inertial range when mathrm r ensuremath lambda ensuremath leqslant 2000 and within tolerance of 1 error mathrm r ensuremath lambda should be higher than 10 4 in order to have an inertial range wider than one decade the so called inertial range found in experiments and simulations is just a scaling range and is not the same as kolmogorov s inertial range the finite reynolds number effect cannot be neglected within such a scaling range and should be considered in comparing experiments or simulations with theories of the inertial range statistics

on the theory of statistical and isotropic turbulence the statistical theory of turbulence initiated by taylor 1935 and v karman howarth 1938 has recently been developed so far that a satisfactory explanation of the spectral distribution of energy among the turbulent eddies can be given in fact kolmogoroff 1941 a b and independently onsager 1945 and v weizsaecker 1948 have introduced a similarity hypothesis which allows a determination of the spectrum for eddies with large reynolds numbers and the author heisenberg 1948 has extended these calculations to include those frequency components which have small reynolds numbers since the distribution of energy among the largest eddies must be a geometrical and not a statistical problem one may say that the statistical part of the spectrum is now well understood recently batchelor townsend 1947 1948 a b have studied the decay of turbulent motion caused by a mesh grating in a wind tunnel and the following discussions will apply the statistical theory to this problem

energy transfer and dissipation in forced isotropic turbulence a model for the reynolds number dependence of the dimensionless dissipation rate ce was derived from the dimensionless karman howarth equation resulting in ce ce c rl o 1 rl where rl is the integral scale reynolds number the coefficients c and ce arise from asymptotic expansions of the dimensionless second and third order structure functions this theoretical work was supplemented by direct numerical simulations dns of forced isotropic turbulence for integral scale reynolds numbers up to rl 5875 r 435 which were used to establish that the decay of dimensionless dissipation with increasing reynolds number took the form of a power law rln with exponent value n 1 000 0 009 and that this decay of ce was actually due to the increase in the taylor surrogate u3 l the model equation was fitted to data from the dns which resulted in the value c 18 9 1 3 and in an asymptotic value for ce in the infinite reynolds number limit of ce 0 468 0 006

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

strong underrelaxation in kaczmarz s method for inconsistent systems we investigate the behavior of kaczmarz s method with relaxation for inconsistent systems we show that when the relaxation parameter goes to zero the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system this point minimizes the sum of the squares of the euclidean distances to the hyperplanes of the system if the starting point is chosen properly then the limits approach the minimum norm weighted least squares solution the proof is given for a block kaczmarz method

recursion relation of character expansion coefficient in su 3 lattice gauge theory in the su 3 lattice gauge theory the recursion relations of coefficients of the character expansion are derived through the schwinger dyson equation for the ensemble average containing multilink variables the variables form two kinds of recursion relations which correspond to two independent integers and of the su 3 representation of the young tableau the property of the recursion relations in comparison with those of the u 1 and su 2 groups is discussed

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

iterative algorithms for large partitioned linear systems with applications to image reconstruction abstract we present a unifying framework for a wide class of iterative methods in numerical linear algebra in particular the class of algorithms contains kaczmarz s and richardson s methods for the regularized weighted least squares problem with weighted norm the convergence theory for this class of algorithms yields as corollaries the usual convergence conditions for kaczmarz s and richardson s methods the algorithms in the class may be characterized as being group iterative and incorporate relaxation matrices as opposed to a single relaxation parameter we show that some well known iterative methods of image reconstruction fall into the class of algorithms under consideration and are thus covered by the convergence theory we also describe a novel application to truly three dimensional image reconstruction

extended self similarity in turbulent flows we report on the existence of a hitherto undetected form of self similarity which we call extended self similarity ess ess holds at high as well as at low reynolds number and it is characterized by the same scaling exponents of the velocity differences of fully developed turbulence

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

a strongly polynomial algorithm for linear systems having a binary solution this paper proposes a strongly polynomial algorithm which either finds a solution of a linear system ax b 0 x 1 or correctly decides that the system has no 0 1 solutions the algorithm can be used as the basis for the construction of a polynomial algorithm for linear programming

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

recursion relation in su n integral in quantum chromodynamics on the basis of the schwinger dyson variational method the recursion relations of the character expansion coefficients of the su n gauge action are derived three recursion relations that are associated with the casimir invariants of the second third and fourth orders are shown to be enough to determine the character expansion coefficient of every representation recursively in su n

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

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

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

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

on relaxation methods for systems of linear inequalities abstract in their classical papers agmon and motzkin and schoenberg introduced a relaxation method to find a feasible solution for a system of linear inequalities so far the method was believed to require infinitely many iterations on some problem instances since it could depending on the dimension of the set of feasible soltions converge asymptotically to a feasible solution if one exists hence it could not be used to determine infeasibility using two lemma s basic to khachian s polynomially bounded algorithm we can show that the relaxation method is finite in all cases and thus can handle infeasible systems as well in spite of more refined stopping criteria the worst case behaviour of the relaxation method is not polynomially bounded as examplified by a class of problems constructed here

magnetohydrodynamic turbulent flows existence results abstract in this paper we provide a detailed investigation of the problem of existence of solutions to the stochastic magnetohydrodynamic equations which describe the motion of turbulent particles of fluids in a magnetic field the results are proved through a delicate combination of galerkin approximation scheme with some deep compactness results of analytic and probabilistic nature

u 2 extension of modified bessel function with two indices a new set of functions that are given by the coefficients of the character expansion of the single link action in the u 2 lattice gauge theory is studied the set relies on the introduction of the character of the mixed term of the normal and adjoint u 2 variables from the schwinger dyson variational method the recursion relations among the functions are derived from the relations the asymptotic behaviors of the function at the inverse coupling 0 and are derived through the combination of the recursion relations and the differentiation the linear differential equation of the fourth order is derived the properties of the solution are discussed

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

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 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

kolmogorov constants for the second order structure function and the energy spectrum we examine the behavior of the kolmogorov constants c 2 c k and c k1 which are respectively the prefactors of the second order longitudinal structure function and the three dimensional and one dimensional longitudinal energy spectrum in the inertial range we show that their ratios c 2 c k1 and c k c k1 exhibit clear dependence on the microscale reynolds number r implying that they cannot all be independent of r in particular it is found that c k1 c 2 0 25 1 95r 0 68 the study further reveals that the widely used relation c 2 4 02c k1 holds only asymptotically when r 10 5 it is also found that c 2 has much stronger r dependence than either c k or c k1 if the latter indeed has a systematic dependence on r we further show that the varying dependence on r of these three numbers can be attributed to the difference of the inertial range in real and wave number space with the inertial range in real space known to be much shorter than that in wave number space

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

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

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

scaling exponents of the second order structure function of turbulence the kolmogorov equation which is an exact relationship between the second order structure function and the third order structure function is applied to study the relative scaling of against according to the extended self similarity ess method it is found that the relative ess scaling exponent is greater than the real or theoretical inertial range scaling exponent of in the case of normal scaling as well as anomalous scaling when and when therefore the experimental and numerical results favours the kolmogorov 2 3 law rather than anomalous scaling previously the results were interpreted as clear evidence of anomalous scaling based upon the assumption that

baxter t q equation for shape invariant potentials the finite gap potentials case the darboux transformation applied recurrently on a schrodinger operator generates what is called a dressing chain or from a different point of view a set of supersymmetric shape invariant potentials the finite gap potential theory is a special case of the chain for the finite gap case the equations of the chain can be expressed as a time evolution of a hamiltonian system we apply sklyanin s method of separation of variables to the chain we show that the classical equation of the separation of variables is the baxter t q relation after quantization