superconvergent patch recovery with equilibrium and conjoint interpolant enhancements the superconvergent patch derivative recovery method of zienkiewicz and zhu is enhanced by adding the squares of the residuals of the equilibrium equation and natural boundary conditions in addition a new conjoint polynomial for interpolating the local patch stresses over the element which significantly improves the local projection scheme is presented results show that in the 4 node quadrilateral the equilibrium and boundary condition residuals usually improve accuracy but not the rate of convergence whereas in the 9 node quadrilateral results are mixed the conjoint polynomial always improves the accuracy of the derivative field within the element as compared to the standard nodal interpolation particularly in 4 node quadrilaterals

gradient systems in view of information geometry abstract dualistic properties of a gradient flow on a manifold m associated with a dualistic structure g is studied from an information geometrical viewpoint some useful applications are also investigated

recovery procedures in error estimation and adaptivity part ii adaptivity in nonlinear problems of elasto plasticity behaviour the first part of this paper 1 discusses in some detail the use of recovery procedures for stress strain or gradients for efficient error estimator and the methods of arriving at adaptive h refinement in this paper we extend such procedure to deal with elasto plasticity and discuss in detail some special features of such computation

structure of metastable states in the hopfield model an upper bound for the number of metastable states in the hopfield model is calculated as a function of the hamming fraction from an input pattern for all finite values of alpha the ratio of number of patterns to nodes the hamming fraction from the input pattern to the nearest metastable state is infinite when alpha 0 113 the bound also implies that there is a gap between a set of states close to the input pattern and another set centred around the hamming fraction 0 5 from it

a graduated assignment algorithm for graph matching a graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise by combining graduated nonconvexity two way assignment constraints and sparsity large improvements in accuracy and speed are achieved its low order computational complexity o lm where l and m are the number of links in the two graphs and robustness in the presence of noise offer advantages over traditional combinatorial approaches the algorithm not restricted to any special class of graph is applied to subgraph isomorphism weighted graph matching and attributed relational graph matching to illustrate the performance of the algorithm attributed relational graphs derived from objects are matched then results from twenty five thousand experiments conducted on 100 mode random graphs of varying types graphs with only zero one links weighted graphs and graphs with node attributes and multiple link types are reported no comparable results have been reported by any other graph matching algorithm before in the research literature twenty five hundred control experiments are conducted using a relaxation labeling algorithm and large improvements in accuracy are demonstrated

extrema of sums of heterogeneous quadratic forms abstract we analyze the following problem arising in various situations in multivariate statistical analysis we are given k symmetric positive definite n n matrices a1 a2 ak k n and we would like to maximize the function ki 1 xtiaixi under the constraint that x1 x2 xk rn form an orthonormal system some theoretical results as well as an algorithm are presented

patch recovery based on superconvergent derivatives and equilibrium in this paper a postprocessing technique is developed for determining first order derivatives fluxes stresses at nodal points based on derivatives in superconvergent points it is an extension of the superconvergent patch recovery technique presented by zienkiewicz and zhu in contrast to that technique all flux or stress components are interpolated at the same time coupled by equilibrium equations at the superconvergent points the equilibrium equations and use of one order higher degree of interpolation polynomials of stress give a dramatic decrease in error of recovered derivatives even at boundaries

what do multistep methods approximate standard analysis of multistep methods for ode s assumes the application of an initialization routine that generates the starting points here ak step method is considered directly as a mappingr kn r n it is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution

rigid body constrained noisy point pattern matching noisy pattern matching problems arise in many areas e g computational vision robotics guidance and control stereophotogrammetry astronomy genetics and high energy physics least squares pattern matching over the euclidean space e sup n for unordered sets of cardinalities p and q is commonly formulated as a combinatorial optimization problem having complexity p p 1 spl middot spl middot spl middot p q 1 q spl les p since p and q may be 10 sup 3 or larger in typical applications less than satisfactory suboptimal methods are usually employed a hybrid approach is described for solving the pattern matching problem under rigid motion constraints which often apply the method reduces the complexity to l sub 21 spl middot n sup 4 l sub 12 spl middot p sup 3 where l sub 12 and l sub 21 are the number of iterations required by steepest ascent and singular value decomposition svd based procedures respectively

a note on multistep methods and attracting sets of dynamical systems we consider a dynamical system described by an autonomous ode with an asymptotically stable attractor a compact set of orbitrary shape for which the stability can be characterized by a lyapunov function using recent results of eirola and nevanlinna 1 we establish a uniform estimate for the change in value of this lyapunov function on discrete trajectories of a consistent strictly stable multistep method approximating the dynamical system this estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in kloeden and lorenz 3 4 for 1 step methods

continuous methods for extreme and interior eigenvalue problems in this paper continuous methods are introduced to compute both the extreme and interior eigenvalues and their corresponding eigenvectors for real symmetric matrices the main idea is to convert the extreme and interior eigenvalue problems into some optimization problems then a continuous method which includes both a merit function and an ordinary differential equation ode is introduced for each resulting optimization problem the convergence of each ode solution is proved for any starting point the limit of each ode solution for any starting point is fully studied both the extreme and the interior eigenvalues and their corresponding eigenvectors can be easily obtained under a very mild condition promising numerical results are also presented

a continuous approximation to the generalized schur decomposition abstract we consider the problem of approximating the generalized schur decomposition of a matrix pencil a b by a family of differentiable orthogonal transformations it is shown that when b is nonsingular this approach is feasible and can be fully expressed as an autonomous differential system when b is singular we show that the location of zero diagonal entries of b affects the feasibility of such an approach and hence we conclude that at least the qz algorithm cannot be extended continuously

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

vector fields global flows and the calculus of pulses the point of this paper is to show a remarkable property of functional expansions in the form of iterated integrals in capturing the dynamics of digital computation when realized in the form of pulse spike driven controlled smooth dynamical systems the key features of the structure thus captured by the formalism are a the combinatorics of spike trains in various input channels and their effect on the discrete time dynamics of the realized automaton and b the relative timing precedence of the spike trains in various input channels we do this by providing globally valid functional expansions for a certain kind of hybrid dynamical system when subject to inputs which resemble pulse trains we give these functional expansions in the form of a product of a nonlinear holonomy factor and a local fliess functional expansion we also obtain a computational scheme to determine the response of finite automata to input sequences which is meant as a substitute for the naive iteration of the next state maps we obtain this by embedding finite automata in suitable controlled dynamical systems on the special orthogonal group incidentally we show that our realization of automata can also be modified to realize hybrid machines in which a finite automaton is interacting with an external system some of whose analog variables are converted into interspike time intervals which in turn are simultaneously tokenized and acted upon by the modified realization of the finite automaton the latter is achieved intrinsically by exploiting the dynamics of the realization

content addressability and learning in neural networks the content addressability of patterns stored in ising spin neural network models with symmetric interactions is studied numerical results from simulations on the icl distributed array processor dap involving systems with up to 2048 neurons are presented behaviour consistent with finite size scaling characteristic of a first order phase transition is shown to be exhibited by the basins of attraction of the stored patterns both in the case of the hopfield model and for systems using a local iterative learning algorithm designed to optimise the basins of attraction estimates are obtained for the critical minimum overlaps which an input pattern must have with a stored pattern in order to successfully retrieve it

isospectral flows and abstract matrix factorizations a general framework for constructing isospectral flows in the space operatorname gl n of n by n matrices is proposed depending upon how operatorname gl n is split this framework gives rise to different types of abstract matrix factorizations when sampled at integer times these flows naturally define special iterative processes and each flow is associated with the sequence generated by the corresponding abstract factorization the proposed theory unifies as special cases the well known matrix decomposition techniques used in numerical linear algebra and is likely to offer a broader approach to the general matrix factorization problem

a lie bracket decomposition and its application to flows on symmetric matrices abstract the paper studies the problem of finding a canonical form for differential equations on symmetric matrices that applies in particular to isospectral flows and spectrum increasing flows such a canonical form is obtained using the result for any lie algebra u and elements w z u such that z is ad semisimple there exist unique a b u with w z a b such that a z c for some c u and z b 0 taking u gl n r the ad semisimplicity condition holds for symmetric matrices z and differential equations then have a canonical form z t z t a t b t where a t z c t and z b t 0 a flow on symmetric matrices is isospectral if and only if b t 0 and spectrum increasing if and only if b t is positive definite applying this canonical form twice to the finite nonperiodic toda flow we obtain a double bracket form for this differential equation that differs from one recently found by bloch

finite size effects in separable recurrent neural networks we perform a systematic analytical study of finite size effects in separable recurrent neural network models with sequential dynamics away from saturation we find two types of finite size effects thermal fluctuations and disorder induced frozen corrections to the mean field laws the finite size effects are described by equations that correspond to a time dependent ornstein uhlenbeck process we show how the theory can be used to understand and quantify various finite size phenomena in recurrent neural networks with and without detailed balance

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

dynamical methods for polar decomposition and inversion of matrices we show how one may obtain polar decomposition as well as inversion of fixed and time varying matrices using a class of nonlinear continuous time dynamical systems first we construct a dynamic system that causes an initial approximation of the inverse of a time varying matrix to flow exponentially toward the true time varying inverse using a time parameterized homotopy from the identity matrix to a fixed matrix with unknown inverse and applying our result on the inversion of time varying matrices we show how any positive definite fixed matrix may be dynamically inverted by a prescribed time without an initial guess at the inverse we then construct a dynamical system that solves for the polar decomposition factors of a time varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition as a byproduct this method gives another method of inverting time varying matrices finally using homotopy again we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time

convergence time and finite size effects in neural networks the time needed for convergence to a stable state starting from an initial state having macroscopic overlap with a stored state is studied numerically for networks up to n 105 neurons although such systems are much larger than those traditionally used in studying neural networks they are still too small to extract reliable information about the asymptotic behaviour of the convergence time

qr type factorization the yang baxter equation and an eigenvalue problem of control theory abstract it is shown that every solution to the yang baxter equation corresponds with the o r type algorithm the structure of solutions of the yang baxter equation is described and examples relevant to the algebraic riccati equation are considered

on the local convergence of an iterative approach for inverse singular value problems the purpose of this paper is to provide the convergence theory for the iterative approach given by m t chu numerical methods for inverse singular value problems siam j numer anal 29 1992 pp 885 903 in the context of solving inverse singular value problems we provide a detailed convergence analysis and show that the ultimate rate of convergence is quadratic in the root sense numerical results which confirm our theory are presented it is still an open issue to prove that the method is q quadratic convergent as claimed by m t chu

exponential basis functions in solution of incompressible fluid problems with moving free surfaces in this paper a new simple meshless method is presented for the solution of incompressible inviscid fluid flow problems with moving boundaries a lagrangian formulation established on pressure as a potential equation is employed in this method the approximate solution is expressed by a linear combination of exponential basis functions ebfs with complex valued exponents satisfying the governing equation constant coefficients of the solution series are evaluated through point collocation on the domain boundaries via a complex discrete transformation technique the numerical solution is performed in a time marching approach using an implicit algorithm in each time step the governing equation is solved at the beginning and the end of the step with the aid of an intermediate geometry the use of ebfs helps to find boundary velocities with high accuracy leading to a precise geometry updating the developed lagrangian meshless algorithm is applied to variety of linear and nonlinear benchmark problems non linear sloshing fluids in rigid rectangular two dimensional basins are particularly addressed

partial retrieval of cad models based on the gradient flows in lie group based on the gradient flows in lie group a partial retrieval approach for cad models is presented in this paper first a representation of the face attributed relational graph arg for a cad model is created from its b rep model and thus partial retrieval is converted to a subgraph matching problem then an optimization method is adopted to solve the matching problem where the optimization variable is the vertex mapping and the objective function is the measurement of compatibility between the mapped vertices and between the mapped edges different from most previously proposed methods a homogeneous transformation matrix is introduced to represent the vertex mapping in subgraph matching whose translational sub matrix gives the vertex selection in the larger graph and whose orthogonal sub matrix presents the vertex permutation for the same sized mapping from the selected vertices to the smaller graph s vertices finally a gradient flow method is developed to search for optimal matching matrix in special euclidean group se n here a penalty approach is used to handle the constraints on the elements of the matching matrix which leads its orthogonal part to be a permutation matrix and its translational part to have different integer elements experimental results show that it is a promising method to support the partial retrieval of cad models

simulating highly diluted neural networks the asymmetric diluted hopfield model is studied through numerical simulations and reliable results are obtained even when the network sizes are not exponentially large the mean final overlap the mean convergence time and a quantity that gives information on the relative importance of spurious states are measured as a function of the load capacity alpha

homotopy algorithm for symmetric eigenvalue problems the homotopy method can be used to solve eigenvalue eigenvector problems the purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results in practice it is rerely of any interest to compute all the eigenvalues the homotopy method having the order preserving property can provide any specific eigenvalue without calculating any other eigenvalues besides this advantage we note that the homotopy algorithm is to a large degree a parallel algorithm numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices

a differential equation approach to the singular value decomposition of bidiagonal matrices we consider the problem of approximating the singular value decomposition of a bidiagonal matrix by a one parameter family of differentiable matrix flows it is shown that this approach can be fully expressed as an autonomous homogeneous and cubic dynamical system asymptotic behavior is justified by the established theory of the toda lattice

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

exponential basis functions in solution of static and time harmonic elastic problems in a meshless style in this paper exponential basis functions ebfs are used in a boundary collocation style to solve engineering problems whose governing partial differential equations pdes are of constant coefficient type complex valued exponents are considered for the ebfs two dimensional elasto static and time harmonic elasto dynamic problems are chosen in this paper the solution procedure begins with first finding a set of appropriate ebfs and then considering the solution as a summation of such ebfs with unknown coefficients the unknown coefficients are determined by the satisfaction of the boundary conditions through a collocation method with the aid of a consistent and complex discrete transformation technique the basis and various forms of the transformation have been addressed and discussed we shall propose several strategies for selection of ebfs with the aid of the basis explained for the transformation while using the transformation the number of ebfs should not necessarily be equal to or less than the number of boundary information data a library of ebfs has also been presented for further use the effect of body forces is included in the solution via construction of particular solution by the use of the discrete transformation and another series of ebfs a number of sample problems are solved to demonstrate the capabilities of the method it has been shown that the time harmonic problems with high wave number can be solved without much effort the method categorized in meshless methods can be applied to many other problems in engineering mechanics and general physics since ebfs can easily be found for almost all problems with constant coefficient pdes

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

scaled toda like flows abstract this paper discusses the class of isospectral flows x x a x where denotes the hadamard product and is the lie bracket the presence of a allows arbitrary and independent scaling for each element in the matrix x the time 1 mapping of the scaled toda like flow still enjoys a qr like iteration the scaled structure includes the classical toda flow brockett s double bracket flow and other interesting flows as special cases convergence proof is thus unified and simplified the effect of scaling on a variety of applications is demonstrated by examples

an eigendecomposition approach to weighted graph matching problems an approximate solution to the weighted graph matching problem is discussed for both undirected and directed graphs the weighted graph matching problem is that of finding the optimum matching between two weighted graphs which are graphs with weights at each arc the proposed method uses an analytic instead of a combinatorial or iterative approach to the optimum matching problem using the eigendecompositions of the adjacency matrices in the case of the undirected graph matching problem or hermitian matrices derived from the adjacency matrices in the case of the directed graph matching problem a matching close to the optimum can be found efficiently when the graphs are sufficiently close to each other simulation results are given to evaluate the performance of the proposed method

the superconvergent patch recovery spr and adaptive finite element refinement abstract the error estimation processes which have been developed by the authors depend on the accuracy with which stresses gradients can be recovered the recovery methods available currently such as the l 2 projection and averaging are inaccurate and an improved process is needed this paper presents a very much improved recovery process yielding superconvergent values throughout the domain details of this method named the superconvergent patch recovery are presented with test results showing its efficiency examples of adaptive h based refinement are given showing high local and overall efficiency indices for error estimation

algebraic aspects of brockett type equations abstract it is well known that soliton equations can be written in lax form where the bracket of two linear differential n kdv or pseudo differential kp operators appears in this work we introduce a new hierarchy each equation of which is defined by a double bracket of two pseudo differential operators these double bracket equations arose originally in the study by brockett of the steepest descent equations corresponding to certain least squares matching and sorting problems we deal with some algebraic properties of these equations in particular we show that as in the classical case they are related to the presence of an infinite sequence of first integrals

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

matrix factorizations for symplectic qr like methods abstract symplectic qr like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the qr process a fundamental problem for the development of such methods is the choice of a suitable type of decomposition a sr corresponding to the qr decomposition where s is symplectic or unitary symplectic decompositions of this type are studied with regard to their application in a qr like process

model indexing and object recognition using 3d viewpoint invariance object recognition and location of 3d objects from range data are described based on invariant representations of characteristic views of single objects depth data is processed to remove outliers then smoothed and segmented into surface patches fitted by polynomial equations mathematical invariants are derived from groups of these surface patches and used for indexing a model database stored in the form of a kd tree the pose of the known object may then be determined from the euclidean centre and orientation of the higher order surfaces formed from the feature groups the approach is complete scalable and extendable the robustness of the method is evaluated in terms of the stability and discriminatory power of the representation and the accuracy of the determined pose

a posteriori estimation and adaptive control of the pollution error in the h version of the finite element method abstract we studied the pollution error in the h version of the finite element method and its effect on the quality of the local error indicators resp the quality of the derivatives recovered by local postprocessing in the interior of the mesh here we show that it is possible to construct a posteriori estimates of the pollution error in a patch of elements by employing the local error indicators over the entire mesh we also give an adaptive algorithm for the local control of the pollution error in a patch of elements of interest

iterative point matching for registration of free form curves and surfaces a heuristic method has been developed for registering two sets of 3 d curves obtained by using an edge based stereo system or two dense 3 d maps obtained by using a correlation based stereo system geometric matching in general is a difficult unsolved problem in computer vision fortunately in many practical applications some a priori knowledge exists which considerably simplifies the problem in visual navigation for example the motion between successive positions is usually approximately known from this initial estimate our algorithm computes observer motion with very good precision which is required for environment modeling e g building a digital elevation map objects are represented by a set of 3 d points which are considered as the samples of a surface no constraint is imposed on the form of the objects the proposed algorithm is based on iteratively matching points in one set to the closest points in the other a statistical method based on the distance distribution is used to deal with outliers occlusion appearance and disappearance which allows us to do subset subset matching a least squares technique is used to estimate 3 d motion from the point correspondences which reduces the average distance between points in the two sets both synthetic and real data have been used to test the algorithm and the results show that it is efficient and robust and yields an accurate motion estimate

a fictitious domain method using equilibrated basis functions for harmonic and bi harmonic problems in physics abstract in this paper we present a new meshless method to solve well known problems in physics and engineering with either constant or variable material properties in 2d space harmonic and bi harmonic problems are considered in this paper the method constructs a set of basis functions called equilibrated basis functions eqbfs through a weighted residual integration over a fictitious domain embedding the main one the bases can satisfy the governing partial differential equations approximately chebyshev polynomials of the first kind are employed to construct the eqbfs and exponential functions are used as the weights in the integrals the parameters of the solution are arranged so that all the integrals can be decomposed into much simpler 1d ones over a normalized intervals this reduces the computational efforts significantly either the eqbfs or the results of the integration process may be stored for further use the validity of the results is examined through an extended patch test a set of physical sample problems with variable non variable material properties as the potential flow over a cylinder steady state heat conduction problems in an anisotropic inhomogeneous functionally graded material potential and stokes flow through a channel of finite width obstructed by periodic cylinders and the bending of thin elastic plates having constant or variable thickness are solved to demonstrate the capabilities of the method as a preliminary study we show that the method may effectively be used in a domain decomposition approach

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

a quadratically convergent algorithm for inverse eigenvalue problems with multiple eigenvalues abstract in 2017 for inverse symmetric eigenvalue problems a new quadratically convergent algorithm has been derived from simple matrix equations although this algorithm has some nice features compared with the other quadratically convergent methods it is not applied to multiple eigenvalues in this paper we improve this algorithm with the aid of an optimization problem for the eigenvectors associated with multiple eigenvalues the proposed algorithm is adapted to an arbitrary set of given eigenvalues the main contribution is our convergence theorem formulated in a different manner from previous work for the existing quadratically convergent methods our theorem ensures the quadratic convergence in a neighborhood of the solutions that satisfy a mild condition

validation of a posteriori error estimators by numerical approach abstract a numerical methodology which determines the quality or robustness of a posteriori error estimators is described the methodology accounts precisely for the factors which affect the quality of error estimators for finite element solutions of linear elliptic problems namely the local geometry of the grid and the structure of the solution the methodology can be employed to check the robustness of any estimator for the complex grids which are used in engineering computations

enhanced superconvergent patch recovery incorporating equilibrium and boundary conditions patch recovery based on superconvergent derivatives and equilibrium spre an enhancement of the superconvergent patch recovery spr is studied for linear elasticity problems the paper also presents a further improvement for recovery of derivatives near boundaries spreb where either tractions or displacements are prescribed this is made by inclusion of weighted residual errors at boundary points in the patch recovery a pronounced improvement in the post processed gradients of the finite element solution is observed by this method

bayesian segmentation via asymptotic partition functions asymptotic approximations to the partition function of gaussian random fields are derived textures are characterized via gaussian random fields induced by stochastic difference equations determined by finitely supported stationary linear difference operators adjusted to be nonstationary at the boundaries it is shown that as the scale of the underlying shape increases the log normalizer converges to the integral of the log spectrum of the operator inducing the random field fitting the covariance of the fields amounts to fitting the parameters of the spectrum of the differential operator induced random field model matrix analysis techniques are proposed for handling textures with variable orientation examples of texture parameters estimated from training data via asymptotic maximum likelihood are shown isotropic models involving powers of the laplacian and directional models involving partial derivative mixtures are explored parameters are estimated for mitochondria and actin myocin complexes in electron micrographs and clutter in forward looking infrared images deformable template models are used to infer the shape of mitochondria in electron micrographs with the asymptotic approximation allowing easy recomputation of the partition function as inference proceeds

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

brief paper a connection between formation infeasibility and velocity alignment in kinematic multi agent systems in this paper a feedback control strategy that achieves convergence of a multi agent system to a desired formation configuration is proposed for both the cases of agents with single integrator and nonholonomic unicycle type kinematics when inter agent objectives that specify the desired formation cannot occur simultaneously in the state space the desired formation is infeasible it is shown that under certain assumptions formation infeasibility forces the agents velocity vectors to a common value at steady state this provides a connection between formation infeasibility and flocking behavior for the multi agent system we finally also obtain an analytic expression of the common velocity vector in the case of formation infeasibility

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

a fast algorithm for connectivity graph approximation using modified manhattan distance in dynamic networks connectivity graphs are widely used in different branches of engineering specially in ad hoc network analysis decentralized control of unmanned air vehicles uav and robot control in large network simulations the extensive cost and the huge amount of memory required for performing the calculations is a major problem therefore methods should be developed that are fast and require less memory to determine the connectivity graph to expedite such calculations in this paper we focus on approximate connectivity graphs for dynamic networks with moving agents we use the velocity information of each agent to develop a fast algorithm to approximate the connectivity graph also we used mean degree distribution instead of degree distribution to investigate congestion interference of network when approximate method is used we introduced a modification factor for radio range r in manhattan measure to increase the accuracy in a wide range of r the results for two random distributions of agents based on monte carlo simulation are compared to the present real connectivity method to show the superiority of our approach

an algorithm for weighted sub graph matching based on gradient flows an algorithm for matching the weighted sub graphs based on gradient flows is proposed in this paper first the smaller and larger graphs for matching are represented by means of their weighted adjacency matrices then an objective function is introduced to measure the differences between two weighted adjacency matrices because the permutation matrix for graph matching is usually relaxed to an orthogonal matrix with non negative elements an optimization based approach is adopted to navigate the solution toward an appropriate permutation matrix to accomplish this goal two gradient flows in the space of orthogonal matrices are defined in such a way that they minimize the objective functions one minimizes the objective function in the space of orthogonal matrices and the other minimizes the distance of an orthogonal matrix from the set of permutations in addition two cost functions are introduced to force satisfaction approximately of the constraint that a translation matrix must have integer entries of different values the experimental results show that the proposed sub graph matching is feasible

a dynamical systems approach to weighted graph matching graph matching is a fundamental problem that arises frequently in the areas of distributed control computer vision and facility allocation in this paper we consider the optimal graph matching problem for weighted graphs which is computationally challenging due the combinatorial nature of the set of permutations contrary to optimization based relaxations to this problem in this paper we develop a novel relaxation by constructing dynamical systems on the manifold of orthogonal matrices in particular since permutation matrices are orthogonal matrices with nonnegative elements we define two gradient flows in the space of orthogonal matrices the first minimizes the cost of weighted graph matching over orthogonal matrices whereas the second minimizes the distance of an orthogonal matrix from the finite set of all permutations the combination of the two dynamical systems converges to a permutation matrix which provides a suboptimal solution to the weighted graph matching problem finally our approach is shown to be promising by illustrating it on nontrivial problems

recognition by linear combinations of models an approach to visual object recognition in which a 3d object is represented by the linear combination of 2d images of the object is proposed it is shown that for objects with sharp edges as well as with smooth bounding contours the set of possible images of a given object is embedded in a linear space spanned by a small number of views for objects with sharp edges the linear combination representation is exact for objects with smooth boundaries it is an approximation that often holds over a wide range of viewing angles rigid transformations with or without scaling can be distinguished from more general linear transformations of the object by testing certain constraints placed on the coefficients of the linear combinations three alternative methods of determining the transformation that matches a model to a given image are proposed

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

a model study of the quality of a posteriori error estimators for linear elliptic problems error estimation in the interior of patchwise uniform grids of triangles abstract this paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems in this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles a completely numerical methodology for the analysis of the quality of estimators is presented results from the application of the methodology to the study of the quality of several well known error estimators are reported in subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals non uniform grids at boundaries grid interfaces and near singular points

fractional transformation group induced by qr factorization and linear prediction problems abstract a new fractional transformation group is found which acts transitively on the space of linear predictors for nonstationary processes by using the qr factorization of nonsingular matrices

self similar flows abstract large classes of self similar isospectral flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms in particular there exist families of flows associated with the qr lr and cholesky eigenvalue algorithms this paper uses lie theory to develop a general theory of self similar flows which includes the qr lr and cholesky flows as special cases also included are new families of flows associated with the sr and hr eigenvalue algorithms the basic theory produces analogues of unshifted single step eigenvalue algorithms but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple step eigenvalue algorithms

exponential basis functions in solution of problems with fully incompressible materials in this paper exponential basis functions ebfs satisfying the governing equations of elastic problems with incompressible materials are introduced due to similarity between elasticity problems and steady state fluid problems the bases found for the former problems are used for latter problems we discuss on using single field form known as displacement velocity based formulation and also on using a two field form known as u p formulation in the first formulation we find the pressure bases through performing a limit analysis using a fictitious bulk modulus while in the second formulation the bases are found directly by considering the pressure as a separate variable in the second formulation we directly apply the condition of incompressibility it is shown that both formulations yield identical bases meaning that the first one may be used in a standard approach however it is also shown that when the incompressibility condition is applied by a laplacian of pressure in the second formulation some additional spurious ebfs may be obtained having defined appropriate bases we follow the solution strategy recently introduced by the authors for other engineering problems some well known benchmark problems are solved to show the capabilities of the method

containment control in mobile networks in this paper the problem of driving a collection of mobile robots to a given target destination is studied in particular we are interested in achieving this transfer in an orderly manner so as to ensure that the agents remain in the convex polytope spanned by the leader agents while the remaining agents only employ local interaction rules to this aim we exploit the theory of partial difference equations and propose hybrid control schemes based on stop go rules for the leader agents non zenoness liveness and convergence of the resulting system are also analyzed

steepest descent algorithms for optimization under unitary matrix constraint in many engineering applications we deal with constrained optimization problems with respect to complex valued matrices this paper proposes a riemannian geometry approach for optimization of a real valued cost function t of complex valued matrix argument w under the constraint that w is an n times n unitary matrix we derive steepest descent sd algorithms on the lie group of unitary matrices u n the proposed algorithms move towards the optimum along the geodesics but other alternatives are also considered we also address the computational complexity and the numerical stability issues considering both the geodesic and the nongeodesic sd algorithms armijo step size 1 adaptation rule is used similarly to 2 but with reduced complexity the theoretical results are validated by computer simulations the proposed algorithms are applied to blind source separation in mimo systems by using the joint diagonalization approach 3 we show that the proposed algorithms outperform other widely used algorithms

numerical solution of a class of deficient polynomial systems most systems of polynomials which arise in applications have fewer than the expected number of solutions the amount of computation required to find all solutions of such a deficient system using current homotopy continuation methods is proportional to the expected number of solutions and roughly to the size of the system much time is wasted following paths which do not lead to solutions we suggest methods for solving some deficient polynomial systems for which the amount of computational effort is instead proportional to the number of solutions

on the least squares solution of inverse eigenvalue problems an inverse eigenvalue problem where a matrix is to be constructed from some or all of its eigenvalues may not have a real valued solution at all an approximate solution in the sense of least squares is sometimes desirable two types of least squares problems are formulated and explored in this paper in spite of their different appearance the two problems are shown to be equivalent thus one new numerical method modified from the conventional alternating projection method is proposed the method converges linearly and globally and can be used to generate good starting values for other faster but more expensive and locally convergent methods the idea can be applied to multiplicative inverse eigenvalue problems for the purpose of preconditioning numerical examples are presented

a continuous jacobi like approach to the simultaneous reduction of real matrices abstract the problem of simultaneous reduction of real matrices by either orthogonal similarity or orthogonal equivalence transformations is considered based on the jacobi idea of minimizing the sum of squares of the complementary part of the desired form to which matrices are reduced the projected gradient method is used in this paper it is shown that the projected gradient of the objective function can be formulated explicitly this gives rise to a system of ordinary differential equations that can be readily solved by numerical software the advantages of this approach are that the desired form to which matrices are reduced can be almost arbitrary and that if a desired form is not attainable then the limit point of the corresponding differential equation gives a way of measuring the distance from the best reduced matrices to the nearest matrices that have the desired form the general procedure for deriving these differential equations is discussed the framework can be generalized to the complex valued case some applications are given

the continuous time rayleigh quotient flow on the sphere a continuous time differential equation inspired by the rayleigh quotient iteration for a symmetric matrix a is studied the set of all continuous solutions termed the rayleigh quotient flow is shown to be a time scaled version of the newton flow for rayleigh quotient cost functional the scaling factor ensures that the rate of variation of the rayleigh quotient is constant and positive along solutions this interpretation leads to a precise phase portrait for the rayleigh quotient flow it is shown that a complete non degenerate solution of the rayleigh quotient flow visits each of the eigenvectors of a in ascending order

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

generalization of robustness test procedure for error estimators part ii test results for error estimators using spr and rep in this part of the paper we shall use the formulation given in the first part to assess the quality of recovery based error estimators using two recovery methods i e superconvergent patch recovery spr and recovery by equilibrium in patches rep the recovery methods have been shown to be asymptotically robust and superconvergent when applied to two dimensional problems in this study we shall examine the behaviour of the recovery methods on several three dimensional mesh patterns for patches located either inside or at boundaries this is performed by first finding an asymptotic finite element solution irrespective of boundary conditions at far ends of the domain and then applying the recovery methods the test procedure near kinked boundaries is explained in a step by step manner the results are given in a series of tables and figures for various cases of three dimensional mesh patterns it has been experienced that the full superconvergent property is generally lost due to presence of boundary layer solution and the definition of the recoveries near boundaries though the results of the robustness test is still within an acceptable range

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

pollution error in the h version of the finite element method and the local quality of the recovered derivatives abstract in this paper we address the quality of the solution derivatives which are recovered from finite element solutions by local averaging schemes as an example we consider the zienkiewicz zhu superconvergent patch recovery scheme the zz spr scheme and we study its accuracy in the interior of the mesh for finite element approximations of solutions of laplace s equation in polygonal domains we will demonstrate the following 1 in general the accuracy of the derivatives recovered by the zz spr or any other local averaging scheme may not be higher than the accuracy of the derivatives computed directly from the finite element solution 2 if the mesh is globally adaptive i e it is nearly equilibrated in the energy norm then we can practically always gain in accuracy by employing the recovered derivatives instead of the derivatives computed directly from the finite element solution 3 it is possible to guarantee that the recovered solution derivatives have higher accuracy than the derivatives computed directly from the finite element solution in any patch of elements of interest by employing a mesh which is adaptive only with respect to the patch of interest i e it is nearly equilibrated in a weighted energy norm 4 in practice we are often interested in obtaining highly accurate derivatives or heat fluxes stresses etc only in a few critical regions which are identified by a preliminary analysis a grid which is adaptive only with respect to the critical regions of interest may be much more economical for this purpose because it may achieve the desired accuracy by employing substantially fewer degrees of freedom than a globally adaptive grid which achieves comparable accuracy in the critical regions

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

recovery procedures in error estimation and adaptivity part i adaptivity in linear problems abstract the procedures of error estimation using stress or gradient recovery were introduced by zienkiewicz and zhu in 1987 3 and with the improvement of recovery procedures introduction of superconvergent patch recovery spr since that date the authors have succeeded in making these error estimators the most robust of those currently available very recently introduced methods of recovery such as rep recovery by equilibrium in patches allow general application in this paper the general idea of using recovery based error estimation in adaptive procedure of linear problems is explained through which the latest developments in recovery techniques is described the fundamental basis of a patch test introduced by babuska et al 11 14 is explained and applied to the recovery based error estimators using both spr and rep

quadratic assignment problems abstract this paper surveys quadratic assignment problems qap at first several applications of this problem class are described and mathematical formulations of qaps are given then some exact solution methods and good heuristics are outlined their computational behaviour is illustrated by numerical results further recent results on the asymptotic probabilistic behaviour of qaps are outlined

a model study of the quality ofa posteriori error estimators for finite element solutions of linear elliptic problems with particular reference to the behavior near the boundary

connectivity graphs as models of local interactions in this paper we study graphs that arise from certain sensory and communication limitations on the local interactions in multi agent systems in particular we show that the set of graphs that can represent formations corresponds to a proper subset of all graphs and we denote such graphs as connectivity graphs these graphs have a special structure that allows them to be composed from a small number of atomic generators using a certain kind of graph amalgamation this structure moreover allows us to give connectivity graphs a topological characterization in terms of their simplicial complexes finally we outline some applications of this topological characterization to the construction of decentralized algorithms