direct calculation of buckling strength of imperfect structures abstract this paper is concerned with the direct calculation of buckling strength of an imperfect structure the main assumptions are that for the perfect structure there is a symmetric or a linear fundamental path and the corresponding first critical point is a simple symmetry breaking or a simple bifurcation point an extended system of limit points is proposed for which the newly introduced scaling parameters are regular solution and thus standard methods can be used to compute them using the extended system one can directly obtain the exact buckling loads without tracing the postbuckling paths an efficient implementation of newton s method solving the extended system is presented and numerical examples are given

a product decomposition bound for bezout numbers most polynomial systems that arise in practice are not completely general but have special structures a common form is that each equation must be a sum of products where each factor has an identifiable generic type a theorem is proven for such systems which offers a method for obtaining a tighter upper bound on the number of nonsingular solutions than is generally available at the same time this theorem provides an approach for solving such systems via polynomial continuation which results in less computational work to illustrate the practical usefulness of these ideas we show that a significant design of mechanisms problem can be solved with an order of magnitude less work than the published solution

tracing the equilibrium path beyond compound critical points a unified presentation of some popular continuation procedures used in the non linear finite element analysis of structural mechanics is introduced an extension of the elliptical constraint equation proposed by crisfield is given it is shown that in the proposed procedure real roots can always be obtained in solving the iterative change of the load parameter updated weighting factors are introduced in the constraint equation in order to get better convergence characteristics in the case when localized deformations occur for bifurcation points a modification of rheinboldt s branching procedure is presented post critical response after limit and bifurcation points is determined in some numerical examples

direct computation of lower bound dynamic buckling loads of imperfection sensitive systems the computation of lower bound dynamic buckling loads of imperfection sensitive systems is examined under step load of ine nite duration an extended system based on the energy criterion for establishing the lower bound dynamic buckling loads without solving the highly nonlinear initial value problems is proposed the newly introduced scaling parameters are nonsingular solutions to the extended system thus standard methods can be used to compute them using the extended system one can directly obtain the lower bound dynamic buckling loads without tracing the postbuckling equilibrium paths an efe cient implementation of newton s method for solving the extended system is presented and numerical examples are given

postbuckling behavior and imperfection sensitivity of elastic structures by the lyapunov schmidt koiter approach beginning with the work of koiter in 1945 valuable insights into the postbuckling behavior of structures have been gained by lyapunov schmidt decomposition of the displacements followed by an asymptotic expansion about the bifurcation point here this methodology is generalized to include nonlinear prebuckling behavior as well as multiple not necessarily coincident buckling modes the expansion of the reduced equilibrium equations is performed about a reference point which need not coincide with any of the bifurcation points and applies no matter whether the modes are coincident closely spaced or well separated from a variety of possible decompositions of the admissible space of displacements two are incorporated into a finite element program theoretical considerations and numerical examples in which asymptotic results are compared to exact results indicate that one of the decompositions has some important advantages over the other examples include a shallow arch and a beam on elastic foundation problem exhibiting symmetry breaking modal interaction

maximum likelihood estimation using square root information filters the maximum likelihood parameter estimation algorithm is known to provide optimal estimates for linear time invariant dynamic systems however the algorithm is computationally expensive and requires evaluations of the gradient of a log likelihood function and the fisher information matrix by using the square root information filter a numerically reliable algorithm to compute the required gradient and the fisher information matrix is developed the algorithm is a significant improvement over the methods based on the conventional kalman filter the square root information filter relies on the use of orthogonal transformations that are well known for numerical reliability this algorithm can be extended to real time system identification and adaptive control

extended chandrasekhar recursions we extend the discrete time chandrasekhar recursions for least squares estimation in constant parameter state space models to a class of structured time variant state space models special cases of which often arise in adaptive filtering it can be shown that the much studied exponentially weighted recursive least squares filtering problem can be reformulated as an estimation problem for a state space model having this special time variant structure other applications arise in the multichannel and multidimensional adaptive filtering context

symmetric homotopy construction abstract homotopy continuation methods can be applied to compute all finite solutions to a given polynomial system computations will be performed more efficiently if the symmetric structure of the system can be exploited this paper presents the construction of a symmetric homotopy using this homotopy only the paths according to the generating solutions have to be traced during continuation

minimizing multi homogeneous bezout numbers by a local search method consider the multi homogeneous homotopy continuation method for solving a system of polynomial equations for any partition of variables the multi homogeneous bezout number bounds the number of isolated solution curves one has to follow in the method this paper presents a local search method for finding a partition of variables with minimal multi homogeneous bezout number as with any other local search method it may give a local minimum rather than the minimum over all possible homogenizations numerical examples show the efficiency of this local search method

computing the homology of basic semialgebraic sets in weak exponential time we describe and analyze an algorithm for computing the homology betti numbers and torsion coefficients of basic semialgebraic sets that works in weak exponential time that is of a set of exponentially small measure in the space of data the cost of the algorithm is exponential in the size of the data all algorithms previously proposed for this problem have a complexity that is doubly exponential and this is so for almost all data

on stability and the worst imperfection shape in solids with nearly simultaneous eigenmodes abstract of interest here is the influence of imperfections on the stability of elastic systems discrete or continuous with nearly simultaneous modes the majority of such structures show an extreme sensitivity of their first instability load to the shape of these small amplitude but unavoidable imperfections the determination of the worst such possible shape is thus a very important issue from the design standpoint following a general analysis of the stability of arbitrary elastic systems with nearly simultaneous bifurcation eigenmodes in the presence of imperfections conditions are given for the determination of the worst imperfection shape which minimizes the first local load maximum for the case of coincident eigenmodes the answer to the worst shape problem is considerably simplified for it is determined from the equilibrium branch of the perfect structure on which the load drops more rapidly

a transformation to avoid solutions at infinity for polynomial systems in finding all solutions to polynomial systems the existence of solutions at infinity makes the problem more difficult particularly when a continuation method is being used as the solution technique systems with solutions at infinity do arise in applications for example in computer graphics and geometric modeling in this paper a simple transformation of the system is given which eliminates solutions at infinity

a numerically efficient implementation of the expectation maximization algorithm for state space models empirical time series are subject to observational noise naive approaches that estimate parameters in stochastic models for such time series are likely to fail due to the error in variables challenge state space models ssm explicitly include observational noise applying the expectation maximization em algorithm together with the kalman filter constitute a robust iterative procedure to estimate model parameters in the ssm as well as an approach to denoise the signal the em algorithm provides maximum likelihood parameter estimates at convergence the drawback of this approach is its high computational demand here we present an optimized implementation and demonstrate its superior performance to naive algorithms or implementations

a perturbation method for the determination of the buckling strength of imperfection sensitive structures a perturbation method for determining the buckling strength of imperfection sensitive structures is presented the method is based on an extended system which was first used by seydel in computation of limit points of nonlinear systems the effects of various possible sources of structural imperfections on perfect structures with simple buckling loads are investigated by the method the derivations and results of the present exposition are equally applicable to discrete and continuous systems in the finite element implementation of the new formulation the perturbation expansions are determined by solving numerically several linear problems with a single positive definite sub stiffness matrix the method is illustrated with its application to the analyses of imperfection sensitivity of a simply supported column on a linear elastic foundation subjected to axial compression

singular perturbations for sensitivity analysis in symmetric bifurcation buckling a direct procedure for the evaluation of imperfection sensitivity in bifurcation problems is presented the problems arise in the context of the general theory of elastic stability koiter s theory for discrete structural systems in which the total potential energy is employed together with a stability criterion based on energy derivatives the imperfection sensitivity of critical states such as bifurcations and trifurcations is usually represented as a plot of the critical load versus the amplitude of the imperfection considered however such plots have a singularity at the point with 0 so that a regular perturbation expansion of the solution is not possible in this work we describe a direct procedure to obtain the sensitivity of the critical load eigenvalue of the bifurcation problem and the sensitivity of the critical direction eigenvector of the bifurcation problem using singular perturbation analysis the perturbation expansions are constructed as a power series in terms of the imperfection amplitude in which the exponents and the coe cients are the unknowns of the problem the solution of the exponents is obtained by means of trial and error using a least degenerate criterion or by geometrical considerations to compute the coe cients a detailed formulation is presented which employs the conditions of equilibrium and stability at the critical state and their contracted forms the formulation is applied to symmetric bifurcations and the coe cients are solved up to third order terms in the expansion the algorithm is illustrated by means of a simple example a beam on an elastic foundation under axial load for which the coe cients are computed and the imperfection sensitivity is plotted copyright 2001 john wiley sons ltd

computing the optimal partition of variables in multi homogeneous homotopy methods the multi homogenous homotopy continuation method is one of the most efficient approaches in finding all isolated solutions of polynomial systems a different partition of variables leads to a different homotopy system the homotopy using the optimal partition of variables reduces the computational cost in curve following to the minimum however finding the optimal variable partition is likely an np hard problem an approximate algorithm is introduced in this paper to avoid exhaustive search in finding the approximate optimal variable partition the global convergence of this algorithm is proved with markov chain theory numerical comparisons with algorithms existed show the efficiency of the new method

theory of globally convergent probability one homotopies for nonlinear programming for many years globally convergent probability one homotopy methods have been remarkably successful on difficult realistic engineering optimization problems most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective convergence theory has been derived for a few particular problems and considerable fixed point theory exists but generally convergence theory for the homotopy maps used in practice for nonlinear constrained optimization has been lacking this paper derives some probability one homotopy convergence theorems for unconstrained and inequality constrained optimization for linear and nonlinear inequality constraints and with and without convexity some insight is provided into why the homotopies used in engineering practice are so successful and why this success is more than dumb luck by presenting the theory as variations on a prototype probability one homotopy convergence theorem the essence of such convergence theory is elucidated

a hybrid polynomial system solving method for mixed trigonometric polynomial systems mixed trigonometric polynomial systems arise in many fields of science and engineering commonly this class of systems is transformed into polynomial systems by variable substituting and adding some quadratic equations and then solved by some polynomial system solving method in this paper by exploiting the special structure of the additional quadratic equations an efficient hybrid method for solving polynomial systems coming from mixed trigonometric polynomial systems is presented it combines the homotopy method in which the homotopy is a combination of coefficient parameter homotopy and the random product homotopy with decomposition variable substitution and reduction techniques numerical tests are given to show its effectiveness and it is applied to solve a practical problem

completeness of solutions of bethe s equations we consider the bethe equations for the isotropic spin 1 2 heisenberg quantum spin chain with periodic boundary conditions we formulate a conjecture for the number of solutions with pairwise distinct roots of these equations in terms of numbers of so called singular or exceptional solutions using homotopy continuation methods we find all such solutions of the bethe equations for chains of length up to 14 the numbers of these solutions are in perfect agreement with the conjecture we also discuss an indirect method of finding solutions of the bethe equations by solving the baxter t q equation we briefly comment on implications for thermodynamical computations based on the string hypothesis

postbuckling analysis and imperfection sensitivity of general shells by the finite element method abstract buckling and imperfection sensitivity are the primary considerations in analysis and design of thin shell structures the objective here is to develop accurate and efficient capabilities to predict the postbuckling behavior of shells including imperfection sensitivity the approach used is based on the lyapunov schmidt koiter lsk decomposition and asymptotic expansion in conjunction with the finite element method this lsk formulation for shells is derived and implemented in a finite element code the method is applied to cylindrical and spherical shells cases of linear and nonlinear prebuckling behavior coincident as well as non coincident buckling modes and modal interactions are studied the results from the asymptotic analysis are compared to exact solutions obtained by numerically tracking the bifurcated equilibrium branches the accuracy of the lsk asymptotic technique its range of validity and its limitations are illustrated

a homotopy for solving general polynomial systems that respects m homogeneous structures methods for finding all solutions to a polynomial system f z 0 of n equations in n unknowns using a generic homotopy have been presented by several researchers over the last ten years in this paper we describe a new method for defining a homotophy to find all solutions to a polynomial system a major feature of this new approach is that the homotopy need not to be generic or random the practical consequences are that the homotopy may be chosen to more closely resemble the system being solved with a consequent potential for reducing the arc length of paths and exploiting special structures in the original system in particular if the system is naturally m homogeneous then the number of homotopy paths that need to be tracked is reduced since many systems that arise in applications have such a structure this represents a significant advance over previous homotopies we also provide a proof that the homotopy parameter is strictly increasing on homotopy paths as a function of arc length a new result which has implications for the choice of numerical path tracking algorithms

heuristic methods for computing the minimal multi homogeneous bezout number the multi homogeneous bezout number of a polynomial system is the number of paths that one has to follow in computing all its isolated solutions with continuation method each partition of variables corresponds to a multi homogeneous bezout number it is a challenging problem to find a partition with minimal multi homogeneous bezout number two heuristic numerical methods for computing the minimal multi homogeneous bezout number are presented in this paper some analysis of computational complexity are given numerical examples show the efficiency of these two methods

a nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets a roadmap for a semi algebraic set s is a curve which has a non empty and connected intersection with all connected components of s hence this kind of object introduced by canny can be used to answer connectivity queries with applications for instance to motion planning but has also become of central importance in effective real algebraic geometry since it is used in higher level algorithms in this article we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets its output size and running time are polynomial in nd nlog d where d is the maximum of the degrees of the input polynomials d is the dimension of the set under consideration and n is the number of variables more precisely the running time of the algorithm is essentially subquadratic in the output size even under our assumptions it is the first roadmap algorithm with output size and running time polynomial in nd nlog d

on the solutions to polynomial systems obtained by homotopy methods a natural class of homotopy methods for solving polynomial systems is considered it is shown that at least one solution from each connected component of the solution set is obtained this generalizes the results of previous papers which concentrated on isolated solutions i e connected components with one single point the number of solution paths ending in a connected component is independent of the particular homotopy in use and defines in a natural way the multiplicity of the connected component a few numerical experiments illustrate the obtained results

likelihood gradient evaluation using square root covariance filters using the array form of numerically stable square root implementation methods for kalman filtering formulas we construct a new square root algorithm for the log likelihood gradient score evaluation this avoids the use of the conventional kalman filter with its inherent numerical instabilities and improves the robustness of computations against roundoff errors the new algorithm is developed in terms of covariance quantities and based on the ldquocondensed formrdquo of the array square root filter

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

the gbq algorithm for constructing start systems of homotopies for polynomial systems homotopy methods have become a standard tool for the computation of all solutions of a polynomial system this paper concerns the solution of deficient polynomial systems which appear to be typical in many engineering applications the gbq algorithm presented consists of two parts the computation of a generalized bezout number gb and the construction of a multi homogeneous start system q the approach generalizes m homogenization into multihomogenization it can also be regarded as a generalization towards the random product homotopy however without making assumptions on the coefficients of the polynomials in the system as is illustrated in the examples symmetric polynomial systems also can be solved more efficiently

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

numerical algebraic geometry for optimal control applications a new technique for solving polynomial nonlinear constrained optimal control problems is presented the problem is reformulated into a parametric optimization problem which in turn is solved in a two step procedure first in a precomputation step the equation part of the corresponding first order optimality conditions is solved for a generic value of the parameter relying on the underlying algebraic geometry this first solution makes it possible to solve efficiently and in real time the corresponding optimal control problem at the measured parameter value for each subsequent time step this approach has a probability one guarantee of finding the global optimal solution at each step controller synthesis for two applications from the area of power electronics featuring a dc ac converter and a dc dc converter are discussed to motivate the proposed approach

worst shapes of imperfections for space trusses with multiple global and local buckling modes abstract the initial postbuckling behavior and imperfection sensitivity of truss type structures in which the joints do not transmit moments is determined by means of the lyapunov schmidt koiter approach such structures possess local buckling modes involving the buckling of individual members and global modes involving axial deformations of all members but no member buckling of particular interest is the case in which through optimization or otherwise a number of local and global modes are coincident the worst shape of imperfection for this case is determined from the bifurcated equilibrium branch on which the load drops most rapidly it is shown that this critical bifurcated branch initially involves the buckling of one member only the general theory is illustrated by a number of examples involving two and three dimensional lattice columns the leading order asymptotic results for these examples are compared to exact results obtained by tracking the appropriate equilibrium branch numerically

solving systems of polynomial equations by bounded and real homotopy the homotopy method for solving systems of polynomial equations proposed by chow mallet paret and yorke is modified in two ways the first modification allows to keep the homotopy solution curves bounded the second one to work with real polynomials when solving a system of real equations for the first method numerical results are presented

algebraic methods for image processing and computer vision many important problems in image processing and computer vision can be formulated as the solution of a system of simultaneous polynomial equations crucial issues include the uniqueness of solution and the number of solutions if not unique and how to find numerically all the solutions the goal of this paper is to introduce to engineers and scientists some mathematical tools from algebraic geometry which are very useful in resolving these issues three dimensional motion structure estimation is used as the context however these tools should also be helpful in other areas including surface intersection in computer aided design and inverse position problems in kinematics robotics the tools to be described are bezout numbers grobner bases homotopy methods and a powerful theorem which states that under rather general conditions one can draw general conclusions on the number of solutions of a polynomial system from a single numerical example

a robust elliptic grid generator a variational principle is proposed that results in a robust elliptic grid generator having many of the strengths of original winslow or homogeneous thompson thames mastin method httm the new grid generator places grid lines more uniformly over the domain than does httm without loss of orthogonality numerically generated examples are given to demonstrate these effects grid quality measures are introduced to quantify differences between discrete grids both the httm and the new grid generator can generate folded grids on certain pathological regions but overall they are very robust grid weighting for solution adaptive calculations is briefly considered generalization of the new method to surface and volume grid generation is straightforward

post bifurcation and stability of a finitely strained hexagonal honeycomb subjected to equi biaxial in plane loading abstract the buckling and crushing mechanics of cellular honeycomb materials is an important engineering problem motivated by the pioneering experimental and numerical studies of papka and kyriakides 1994 1999a b we review the literature on finitely strained honeycombs subjected to in plane loading and identify two open questions i how does the mechanical response of the honeycomb depend on the applied loading device and ii what can the bloch wave representation of all bounded perturbations contribute to our understanding of the stability of post bifurcated equilibrium configurations to address these issues we model the honeycomb as a two dimensional infinite perfect periodic medium we use analytical group theory methods as opposed to the more common but less robust imperfection method to study the honeycomb s bifurcation behavior under three different far field loadings that produce initially the same equi biaxial contractive dilatation using an fem discretization of the honeycomb walls struts we solve the equilibrium equations to find the principal and bifurcated equilibrium paths for each of the three loading cases we evaluate the structure s stability using two criteria rank one convexity of the homogenized continuum long wavelength perturbations and bloch wave stability bounded perturbations of arbitrary wavelength we find that the post bifurcation behavior is extremely sensitive to the applied loading device in spite of a common principal solution we confirm that the flower mode is always unstable as previously reported however our first ever bloch wave stability analysis of the post bifurcated equilibrium paths shows that the flower mode is stable for all sufficiently short wavelength perturbations this new result provides a realistic explanation for why this mode has been observed in the finite size specimen experiments of papka and kyriakides 1999a

worst shapes of imperfections for space trusses with many simultaneously buckling members abstract optimally designed truss type structures whose joints do not transmit moments often have a large number of coincident buckling modes each mode corresponds to the buckling of an individual member due to the interaction between various simultaneous modes such trusses can be sensitive to imperfections this problem is analysed using the lyapunov schmidt koiter decomposition and asymptotic expansion technique the shape of the imperfection that maximizes the load drop is determined from the postbifurcated equilibrium branch of the perfect structure on which the load drops most rapidly it is shown that this branch is obtained by minimizing a homogeneous quadratic form subject to linear inequality constraints the general theory is illustrated by several examples involving two and three dimensional trusses

robust maximum likelihood estimation of multivariable dynamic systems this paper examines the problem of estimating linear time invariant state space system models in particular it addresses the parametrization and numerical robustness concerns that arise in the multivariable case these difficulties are well recognised in the literature resulting for example in extensive study of subspace based techniques as well as recent interest in data driven local co ordinate approaches to gradient search solutions the paper here proposes a different strategy that employs the expectation maximisation em technique the consequence is an algorithm that is iterative with associated likelihood values that are locally convergent to stationary points of the gaussian likelihood function furthermore theoretical and empirical evidence presented here establishes additional attractive properties such as numerical robustness avoidance of difficult parametrization choices the ability to naturally and easily estimate non zero initial conditions and moderate computational cost moreover since the methods here are maximum likelihood based they have associated known and asymptotically optimal statistical properties

round off error propagation in four generally applicable recursive least squares estimation schemes abstract the numerical robustness of four generally applicable recursive least squares estimation schemes is analysed by means of a theoretical round off propagation study this study highlights a number of practical interesting insights into the widely used recursive least squares schemes these insights have been confirmed in an experimental verification study

a homotopy method for eigenvalue assignment using decentralized state feedback this paper addresses the following problem given an interconnected system m composed of n subsystems of the form a i b i k i i 1 n a i b i a controllable pair and where the off diagonal blocks of m lie in the image of the appropriate b i then is it possible to arbitrarily assign the characteristic polynomial of m by a suitable selection of the characteristic polynomials of a i b i k i moreover is it possible to compute the appropriate characteristic polynomials of the a i b i k i or equivalently construct the k i needed to do so the first question is answered by constructing a mapping f r n rightarrow r n which maps a prescribed set of n of the feedback gains elements of k i i 1 n to the n coefficients of the characteristic polynomial of m the question then becomes given a p in r n does f x p have a solution the answer is found by constructing a homotopy h r n x o 1 rightarrow r n where h x 1 f x and h x 0 is some trivial function degree theory is then applied to guarantee that there exists an x t such that h x t t p for all t in 0 1 the parameterized sard s theorem is then utilized to prove that with probability 1 x t is a smooth curve and hence can be followed numerically from x 0 to x 1 by the solution of a differential equation davidenko s method

signal and image segmentation using pairwise markov chains the aim of this paper is to apply the recent pairwise markov chain model which generalizes the hidden markov chain one to the unsupervised restoration of hidden data the main novelty is an original parameter estimation method that is valid in a general setting where the form of the possibly correlated noise is not known several experimental results are presented in both gaussian and generalized mixture contexts they show the advantages of the pairwise markov chain model with respect to the classical hidden markov chain one for supervised and unsupervised restorations

a new start system for solving deficient polynomial systems using continuation we are interested in all finite solutions of polynomial systems the start system normally used in the homotopy causes divergent continuation paths when applied to a deficient system our aim is to avoid this wasted computation by constructing a new start system the start system presented in this paper respects the m homogeneous structure of the system we want to solve

an efficient start system for multi homogeneous polynomial continuation when solving a system of polynomial equations using homotopy continuation the number of solution paths can often be significantly reduced by casting the equations in multi homogeneous form this requires a multi homogeneous start system having a full set of nonsingular solutions that can be easily computed we give a general form of such a start system that works for any multi homogeneous structure and that can be used efficiently in continuation

on the solvability of polynomial systems arising in control we consider a compactification of a general system of polynomial equations in weighted projective space and give some sufficient conditions about the solvability of a polynomial system over c and r we also prove that for generic linear subspaces the inverse eigenvalue problem with perturbations in the linear subspace always has n solutions

space time discontinuous galerkin method for nonlinear water waves a space time discontinuous galerkin dg finite element method for nonlinear water waves in an inviscid and incompressible fluid is presented the space time dg method results in a conservative numerical discretization on time dependent deforming meshes which follow the free surface evolution the algorithm is higher order accurate both in space and time and closely related to an arbitrary lagrangian eulerian ale approach a detailed derivation of the numerical algorithm is given including an efficient procedure to solve the nonlinear algebraic equations resulting from the space time discretization numerical examples are shown on a series of model problems to demonstrate the accuracy and capabilities of the method

new square root algorithms for kalman filtering presents some new square root algorithms that allow more reliable computation of the state estimates using as far as possible quantities obtained via orthogonal operations new algorithms are given for covariance quantities and information quantities and a new combined algorithm is also presented

gradient based parameter estimation in pairwise linear gaussian system this technical note addresses the gradient based parameter estimation problem for pairwise linear gaussian systems the new adaptive filtering scheme is based on gradient based optimization methods for estimating the uncertain system parameters and a robust square root variant of the pairwise kalman filter used for estimating the unknown states of the pairwise linear gaussian systems hence in the adaptive filtering techniques the dynamic state and system parameters are estimated simultaneously the method of the filter sensitivities computation required in gradient evaluation is derived in terms of square root factors of covariance matrices

the continuation method for rational and deficient systems deficient polynomial systems those with fewer than the full number of solutions cannot be solved efficiently with the usual continuation methods this paper describes a continuation technique which allows some deficient and rational systems of equations to be solved with little wasted computation several examples from the literature are discussed

kalman filter sensitivity evaluation with orthogonal and j orthogonal transformations this technical note addresses the array square root kalman filtering smoothing algorithms with the conventional orthogonal and j orthogonal transformations in the adaptive filtering context j orthogonal matrices arise in computation of the square root of the covariance or smoothed covariance by solving an equation of the form cct ddt bbt the latter implies an application of the qr decomposition with j orthogonal transformations in each iteration step in this paper we extend functionality of array square root kalman filtering schemes and develop an elegant and simple method for computation of the derivatives of the filter variables to unknown system parameters required in a variety of applications for instance our result can be implemented for an efficient sensitivity analysis and in gradient search optimization algorithms for the maximum likelihood estimation of unknown system parameters it also replaces the standard approach based on direct differentiation of the conventional kalman filtering equations with their inherent numerical instability and hence improves the robustness of computations against roundoff errors

a globally convergent method for finding zeros of smooth functions computing a zero of a smooth function is an old and extensively researched problem in numerical computation while a large body of results and algorithms has been reported on this problem in the literature to the extent we are aware the published literature does not contain a globally convergent algorithm for finding a zero of an arbitrary smooth function in this paper we present the first globally convergent algorithm for computing a zero if one exists of a general smooth function after presenting the algorithm and a proof of global convergence we also clarify the connection between our algorithm and some known results in topological degree theory

mathematical aspects of variational grid generation ii abstract variational grid generation techniques are now used to produce grids suitable for solving numerical partial differential equations in irregular geometries in this paper the existence and uniqueness of solutions of the volume and smoothness problems that are used in variational grid generation are studied an analysis of the euler lagrange el equations near the identity shows that the volume problem is difficult these variational problems use a reference grid to specify the properties of the desired grid replication of reference grid properties is analyzed examples are given that show the effectiveness of the reference grid concept

a computational model for organism growth based on surface mesh generation the generation of curvilinear coordinate meshes has been a powerful tool in computational fluid dynamics calculation in the computational modelling of the fluid flows around complex bodies such as an airfoil or a complete aircraft this same technique may be applied to many other computational models in this work the approach is used as part of a computational model to generate simple geometries associated with biological forms or organisms the model adopted was first proposed by cummings and simulates morphogenesis in terms of the geometrical changes occurring during the growth and development of simple organisms this model depends on the generation of a curvilinear coordinate mesh on the surface of an organism previous work has concentrated on the model and its use in generating axisymmetric shapes that are simple models of elementary organisms in this work we describe how the model may be extended to geometrical symmetry breaking this paper describes the methodology of this extension and demonstrates it in the simulation of tentacle growth the resulting computational technique makes it possible to link models of cell bio chemistry and surface deformation

finding all solutions to a system of polynomial equations given a polynomial equation of degree d over the complex domain the fundamen tal theorem of algebra tells us that there are d solutions assuming that the solutions are counted by multiplicity these solutions can be approximated by deforming a standard n th degree equation into the given equation and following the solutions through the deformation this is called the homotopy method the fundamental theorem of algebra can be proved by the same technique in this paper we extend these results and methods to a system of n polynomial equations in n complex variables we show that the number of solutions to such a system is the product of the degrees of the equations assuming that infinite solutions are included and solutions are counted by multiplicity the proof is based on a homotopy or deformation from a standard system of equations with the same degrees and known solutions this homotopy provides a computational method of approximating all solutions computational results demonstrating the feasibility of this method are also presented

parameter estimation in variational grid generation two approaches to numerical variational grid generation use a linear combination of three variational problems to control the properties of the grid previously the parameters in the linear combination were determined by experimentation good estimates for these parameters can be obtained from simple model problems

numerical aspects of different kalman filter implementations a theoretical analysis is made of the error propagation due to numerical roundoff for four different kalman filter implementations the conventional kalman filter the square root covariance filter the square root information filter and the chandrasekhar square root filter an experimental analysis is performed to validate the new insights gained by the theoretical analysis

achieving finite element mesh quality via optimization of the jacobian matrix norm and associated quantities part i a framework for surface mesh optimization structured mesh quality optimization methods are extended to optimization of unstructured triangular quadrilateral and mixed finite element meshes n ew interpretations of well known nodally bssed objective functions are made possible using matrices and matrix norms the matrix perspective also suggests several new objective functions particularly significant is the interpretation of the oddy metric and the smoothness objective functions in terms of the condition number of the metric tensor and jacobian matrix respectively objective functions are grouped according to dimensionality to form weighted combinations a simple unconstrained local optimum is computed using a modiiied n ewton iteration the optimization approach was implemented in the cubit mesh generation code and tested on several problems results were compared against several standard element based quaiity measures to demonstrate that good mesh quality can be achieved with nodally based objective functions

a polyhedral method for solving sparse polynomial systems a continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations we introduce mixed subdivisions of newton polytopes and we apply them to give a new proof and algorithm for bernstein s theorem on the expected number of roots this results in a numerical homotopy with the optimal number of paths to be followed in this homotopy there is one starting system for each cell of the mixed subdivision and the roots of these starting systems are obtained by an easy combinatorial construction

robust blind pairwise kalman algorithms using qr decompositions the pairwise kalman filter pkf w pieczynski and f desbouvries kalman filtering using pairwise gaussian models in proc int conf acoust speech signal process icassp hong kong apr 2003 is an extension of the classical kalman filter that keeps propagation equations explicit i e it does not require time consuming simulations the contribution of this note is twofold first new robust equations for filtering smoothing and unsupervised off lined parameters estimation based on qr decompositions are presented second since the model is over parametrized we give a simple condition to uniquely characterize a filter of interest when the dimension of observations is equal to the dimension of states unsupervised experiments based on simulated data confirm the nice behavior of the robust pkf even for a limited number of observations

surface grid generation in the tangent plane abstract recent work has shown that parametrically formulated surface grid generation algorithms are severely limited by solution bifurcations of the discretized equations such bifurcations become apparent if the given surface has points of high curvature the present study demonstrates that discrete equations derived from an alternative physical variables formulation do not result in similar bifurcations however the natural formulation using physical variables is not entirely successful on surface of high curvature due to excessive truncation error in the solution grid it is shown that both bifurcations and truncation error are overcome by formulating the grid generation equations in terms of projections of the physical variables onto the surface tangent plane

a homotopy for solving polynomial systems this paper presents a continuation method for finding all solutions to polynomial systems it features a simpler homotopy than has been previously published

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

a methodology for solving chemical equilibrium systems this paper describes a methodology for solving chemical equilibrium systems the methodology is especially appropriate when many systems with the same structure must be solved quickly as in finite difference models of fluid flow and combustion the approach features identifying a canonical form for such systems and exploiting this canonical form to effect a preliminary algebraic reduction then numerical scaling and iterative solution techniques are evoked both homotopy continuation and a variant of newton s method are effective solution methods this methodology is a significant advance over previous approaches to solving chemical equilibrium systems the problem of solving such systems has heretofore been regarded as challenging solution techniques have tended to emphasize case by case analyses and hit or miss numerical methods we show that the problem can be solved by a unified approach quickly and reliably

on gradient based search for multivariable system estimates this paper addresses the design of gradient based search algorithms for multivariable system estimation in particular the paper here considers so called ldquofull parametrizationrdquo approaches and establishes that the recently developed ldquodata driven local coordinaterdquo methods can be seen as a special case within a broader class of techniques that are designed to deal with rank deficient jacobians this informs the design of a new algorithm that via a strategy of dynamic jacobian rank determination is illustrated to offer enhanced performance

modified stiffness iteration to pinpoint multiple bifurcation points abstract a modified stiffness iteration to precisely compute a bifurcation point with multiple zero eigenvalues is presented the iteration combines the direct pinpointing iteration for simple bifurcation points with a procedure to modify the tangent stiffness matrix based on pertinent congruent transformation to reduce the multiple eigenvalues to single ones two different transformation matrices are employed to modify the stiffness matrix one is a non orthogonal transformation matrix amplifying the values of the entries in a certain row and a column of the stiffness matrix the other is an orthogonal one which is chosen with reference to the symmetry of the system under consideration the use of the non orthogonal transformation matrix is a key idea termed stiffness amplification method in this paper whereas the orthogonal matrix is based on block diagonalization which is becoming popular in group theoretic analysis of symmetric structures extensive numerical examples show the robustness and usefulness of the proposed iteration method for pinpointing multiple bifurcation points

robust partial learning in linear gaussian systems this technical note deals with unsupervised and off line learning of parameters involved in linear gaussian systems i e the estimation of the transition and the noise covariances matrices of a state space system from a finite series of observations only in practice these systems are the result of a physical problem for which there is a partial knowledge either on the sensors from which the observations are issued or on the state of the studied system we therefore propose in this work an expectation maximization learning type algorithm that takes into account constraints on parameters such as the fact that two identical sensors have the same noise characteristics and so estimation procedure should exploit this knowledge the algorithms are designed for the pairwise linear gaussian system that takes into account supplementary cross dependences between observations and hidden states w r t the conventional linear system while still allowing optimal filtering by means of a kalman like filter the algorithm is made robust through qr decompositions and the propagation of a square root of the covariance matrices instead of the matrices themselves it is assessed through a series of experiments that compare the algorithms which incorporate or not partial knowledge for short as well as for long signals

a hybrid algorithm for multi homogeneous bezout number the multi homogenous homotopy continuation method can solve all isolated solutions of polynomial systems different variable partition yields different multi homogenous bezout number which gives the upper bound of the number of isolated solutions however the computation of the multi homogenous bezout number is hard in this paper the permanent formulation of the multi homogenous bezout number is considered the intensive and systemic computations are made for the method of row expansion with memory the precise and the approximate permanent methods each of these methods has its own advantage hence a hybrid algorithm is naturally presented this method works for n about 30 contrasting with 15 before where n is the number of the variables of the polynomial system

post bifurcation equilibria in the plane strain test of a hyperelastic rectangular block abstract of interest here is the bifurcated equilibrium solution of a homogeneous hyperelastic rectangular block under finite plane strain tension or compression a general asymptotic analysis of the bifurcated equilibrium path about the principal solution s lowest critical load is presented using lagrangian kinematics the analysis is valid for any compressible hyperelastic material with axes of orthotropy aligned with the block s axes of symmetry in the reference stress free configuration the general theory is subsequently applied to blocks of different constitutive laws results are presented in the form of bifurcated equilibrium branch s curvature at the critical load as function of the block s aspect ratio since the sign of this curvature determines the branch s stability for small aspect ratios there is agreement with existing structural models while for relatively higher aspect ratios some rather counter intuitive stability results appear which strongly depend on the constitutive law

errata computing all solutions to polynomial systems using homotopy continuation in a previous paper we described a new method for defining homotopies for finding all solutions to polynomial systems a major feature of this new approach is that the start system for the homotopy need not be a random or generic system also homotopy paths are strictly increasing in the homotopy parameter in this paper we establish some principles of implementation and report on the performance of programs that use the new homotopies a feature of our implementation is that we eliminate divergent paths entirely we include performance statistics for homotopies derived from more traditional approaches for comparison generally the new approach is faster and more reliable

state sensitivity evaluation within ud based array covariance filters this technical note addresses the ud factorization based kalman filtering kf algorithms using this important class of numerically stable kf schemes we extend its functionality and develop an elegant and simple method for computation of sensitivities of the system state to unknown parameters required in a variety of applications for instance it can be used for efficient calculations in sensitivity analysis and in gradient search optimization algorithms for the maximum likelihood estimation the new theory presented in this technical note is a solution to the problem formulated by bierman in which has been open since 1990s as in the cited paper our method avoids the standard approach based on the conventional kf and its derivatives with respect to unknown system parameters with its inherent numerical instabilities and hence improves the robustness of computations against roundoff errors

analysis of post buckling branches at multiple symmetric bifurcations a method to analyse and solve symmetric bifurcations by establishing the bifurcation equations using an asymptotic expansion method is presented the bifurcation equations are obtained using a decomposition of the spaces by means of the theory of lyapunov schmidt to solve the bifurcation equations an asymptotic expansion method along the lines of koiter is applied the expansion is presented in a form suited for implementation in a finite element context the present paper is focused on the treatment of symmetric multiple bifurcations where new forms of the bifurcation equations are established the accuracy of the method is verified with three examples copyright 2001 john wiley sons ltd