new block quadrature rules for the approximation of matrix functions abstract golub and meurant have shown how to use the symmetric block lanczos algorithm to compute block gauss quadrature rules for the approximation of certain matrix functions we describe new block quadrature rules that can be computed by the symmetric or nonsymmetric block lanczos algorithms and yield higher accuracy than standard block gauss rules after the same number of steps of the symmetric or nonsymmetric block lanczos algorithms the new rules are block generalizations of the generalized averaged gauss rules introduced by spalevic applications to network analysis are presented

rank revealing decomposition of symmetric indefinite matrices via block anti triangular factorization abstract we present an algorithm for computing a symmetric rank revealing decomposition of a symmetric n n matrix a as defined in the work of hansen yalamov 9 we factorize the original matrix into a product a q m q t with q orthogonal and m symmetric and in block form with one of the blocks containing the dominant information of a such as its largest eigenvalues moreover the matrix m is constructed in a form that is easy to update when adding to a a symmetric rank one matrix or when appending a row and symmetrically a column to a the cost of such an updating is o n 2 floating point operations the proposed algorithm is based on the block anti triangular form of the original matrix m as introduced by the authors in 11 via successive orthogonal similarity transformations this form is then updated to a new form a q m q t whereby the first k rows and columns of m have elements bounded by a given threshold and the remaining bottom right part of m is maintained in block anti triangular form the updating transformations are all orthogonal guaranteeing the backward stability of the algorithm and the algorithm is very economical when the near rank deficiency is detected in some of the anti diagonal elements of the block anti triangular form numerical results are also given showing the reliability of the proposed algorithm

sparse matrix approximations for multigrid methods abstract we discuss the application of sparse matrix approximations for two grid and v cycle multigrid methods sparse approximate inverses can be used as smoothers further the galerkin coarse matrix can be sparsified by sparse approximation techniques also the projection can be defined by combining sparse approximation with side conditions related to high frequency components numerical results are given which demonstrate the efficiency and accuracy of the proposed strategies

design sensitivity analysis for rate independent elastoplasticity abstract a new incremental direct differentiation method for design sensitivity analysis of structures with rate independent elastoplastic behavior is presented we formulate analytical sensitivity expressions that are consistent with numerical algorithms for elastoplasticity that use implicit methods to integrate the constitutive equations and return mappings to enforce the consistency conditions the sensitivity expressions can be evaluated with only a modest increase in computational expense beyond the cost of simulation combined with the inherent advantages of implicit integration strategies this represents a significant improvement over previous sensitivity formulations for history dependent materials first order sensitivity expressions involving the complete set of design variables including shape design variables are derived for a generic response functional the reduced form of the consistent tangent stiffness matrix obtained at the end of each time or load step in the finite element procedure is used to update the response sensitivities for that time step no iterations are needed in the sensitivity computations a numerical example demonstrates the accuracy and efficiency of the new sensitivity analysis method for an elastoplastic analysis problem explicit sensitivities from the new method are confirmed by finite difference estimates

a new look at the john nirenberg and john stromberg theorems for bmo abstract we develop some techniques for studying various versions of the function space bmo special cases of one of our results give alternative proofs of the celebrated john nirenberg inequality and of related inequalities due to john and to wik our approach enables us to pose a simply formulated geometric question for which an affirmative answer would lead to a version of the john nirenberg inequality with dimension free constants

an approach to optimal design of structures with non linear response a commonly adopted procedure to solve a structural optimum design problem is to improve the design iteratively by solving a sequence of simpler problems where each of these is obtained as an approximation of the original problem considering structures with linear response this procedure works very well and can be implemented quite straightforwardly however considering structures with non linear response the situation is more sophisticated since structural stability properties have to be taken into account the paper presents an approach to optimal design of statically loaded structures with non linear response it is proposed to replace the original problem with a sequence of similar problems from which corresponding approximate problems can easily be obtained these similar problems additionally contain an estimation of the critical load constraint while other constraints are imposed at some properly chosen load level which is lower or at most equal to the critical load level by this approach the structural stability requirements are taken into account in an effective manner the efficiency is illustrated with several examples involving kinematic as well as material non linear effects

essential spectral equivalence via multiple step preconditioning and applications to ill conditioned toeplitz matrices essential spectral equivalence via multiple step preconditioning and applications to ill conditioned toeplitz matrices

on the construction of general cubature formula by flat extensions we describe a new method to compute general cubature formulae the problem is initially transformed into the computation of truncated hankel operators with flat extensions we then analyze the algebraic properties associated to flat extensions and show how to recover the cubature points and weights from the truncated hankel operator we next present an algorithm to test the flat extension property and to additionally compute the decomposition to generate cubature formulae with a minimal number of points we propose a new relaxation hierarchy of convex optimization problems minimizing the nuclear norm of the hankel operators for a suitably high order of convex relaxation the minimizer of the optimization problem corresponds to a cubature formula furthermore cubature formulae with a minimal number of points are associated to faces of the convex sets we illustrate our method on some examples and for each we obtain a new minimal cubature formula

optimal preconditioning for image deblurring with anti reflective boundary conditions abstract inspired by the theoretical results on optimal preconditioning stated by ng chan and tang in the framework of reflective boundary conditions bcs in this paper we present analogous results for anti reflective bcs here a key technical difficulty is represented by the non orthogonal character of the anti reflective transform and indeed the proof proposed by ng chan and tang does not work nevertheless in both cases the optimal preconditioner is the blurring matrix associated to the symmetrized point spread function psf the geometrical idea on which our proof is based is very simple and general so it may be useful in the future to prove theoretical results for new proposed bcs numerical tests show that the optimal preconditioning strategy is effective when using both preconditioned conjugate gradient methods and recently introduced nonstationary preconditioned iterations

quasiseparable hessenberg reduction of real diagonal plus low rank matrices and applications abstract we present a novel algorithm to perform the hessenberg reduction of an n n matrix a of the form a d u v where d is diagonal with real entries and u and v are n k matrices with k n the algorithm has a cost of o n 2 k arithmetic operations and is based on the quasiseparable matrix technology applications are shown to solving polynomial eigenvalue problems and some numerical experiments are reported in order to analyze the stability of the approach

structural design sensitivity analysis with general boundary conditions dynamic problem this paper presents and analyses methods for design sensitivity analysis of static response for structural systems when general boundary conditions are used during the analysis phase the design sensitivity analysis methods where the boundary conditions are used to reduce the equilibrium equation have been presented earlier in such procedures the reduced stiffness matrix is non singular this fact is used to show differentiability of the displacement vector when reduction procedures are not used the global structural stiffness matrix may be singular in such a case differentiability of the displacement vector may be questionable however it is shown that the displacement vector is still differentiable several forms of the structural design sensitivity equations are presented although these equations are derived with lagrange multiplier treatment of general boundary conditions it is shown that they are applicable with any treatment of general boundary conditions for example they can also be used with co ordinate reduction schemes therefore design sensitivity analysis can be carried out with any existing finite element analysis computer code moreover the procedures presented can be used to calculate sensitivity of a force of constraint can that is desired the formulations are kept quite general such that design dependent boundary conditions can be routinely treated an example is used to illustrate the procedures

sparse block factorization of saddle point matrices abstract the factorization method presented in this paper takes advantage of the special structures and properties of saddle point matrices a variant of gaussian elimination equivalent to the cholesky s factorization is suggested and implemented for factorizing the saddle point matrices block wise with small blocks of orders 1 and 2 the gaussian elimination applied to these small blocks on block level also induces a block 3 3 structured factorization of which the blocks have special properties we compare the new block factorization with the schilders factorization in terms of sparsity and computational complexity the factorization can be used as a direct method and also anticipate for preconditioning techniques

collocation quadrature methods and fast summation for cauchy singular integral equations with fixed singularities abstract basing on recent results on the stability of collocation methods applied to cauchy singular integral equations with additional fixed singularities we give necessary and sufficient conditions for the stability of collocation quadrature methods for such equations these methods have the advantage that the respective system of equations has a very simple structure and allows to apply fast summation methods which results in a fast algorithm with o n log n complexity we present numerical results of the application of the proposed collocation quadrature methods to the notched half plane problem of two dimensional elasticity theory

reduced basis technique for calculating sensitivity coefficients of nonlinear structural response an efficient reduced basis technique is presented for calculating the sensitivity of nonlinear structural response to variations in the design variables the structure is discretized by using two field mixed finite element models the vector of structural response and its sensitivity coefficients derivatives with respect to design variables are each expressed as a linear combination of a small number of basis or global approximation vectors the bubnov galerkin technique is then used to approximate each of the finite element equations governing the response and the sensitivity coefficients by a small number of algebraic equations in the amplitudes of these vectors the path derivatives derivatives of the response vector with respect to path parameters e g load parameters are used as basis vectors for approximating the response a combination of the path derivatives and their derivatives with respect to the design variables is used for approximating the sensitivity coefficients the potential of the proposed technique is discussed and its effectiveness is demonstrated by means of numerical examples of laminated composite plates subjected to mechanical and thermal loads

generalized modularity matrices various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems in this paper we put in evidence certain common traits of various modularity matrices and shed light on their spectral properties that are at the basis of various theoretical results and practical spectral type algorithms for community detection

minimum weight design of non linear elastic structures with multimodal buckling constraints it well known that multimodal instability is an event particularly relevant in structural optimization here in the context of non linear stability theory an exact method is developed for minimum weight design of elastic structures with multimodal buckling constraints given an initial design the method generates a sequence of improved designs by determining a sequence of critical equilibrium points related to decreasing values of the structural weight multimodal buckling constraints are imposed without repeatedly solving an eigenvalue problem and the difficulties related to the non differentiability in the common sense of state variables in multimodal critical states are overcome by means of the lagrange multiplier method further constraints impose that only the first critical equilibrium states local maxima or bifurcation points on the initial equilibrium path of the actual designs are taken into account copyright 2002 john wiley sons ltd

some matrix nearness problems suggested by tikhonov regularization abstract the numerical solution of linear discrete ill posed problems typically requires regularization i e replacement of the available ill conditioned problem by a nearby better conditioned one the most popular regularization methods for problems of small to moderate size are tikhonov regularization and truncated singular value decomposition tsvd by considering matrix nearness problems related to tikhonov regularization several novel regularization methods are derived these methods share properties with both tikhonov regularization and tsvd and can give approximate solutions of higher quality than either one of these methods

minimum weight design of truss structures with geometric nonlinear behavior the paper presents an optimization method based on optimality criterion for minimum weight design of structures with geometric nonlinear behavior the nonlinear critical load is determined by finding the load level at which the hessian of the potential energy ceases to be positive definite a recurrence relation based on the criterion that at optimum the nonlinear strain energy density be equal in all the members is used to develop an algorithm sample problems are given to illustrate the application of the method to truss type structures with a large number of design variables number of algorithms based on the optimality criterion approach have been developed to design a minimum weight structure with specified constraints on nodal displacements element stresses system stability etc 1 2 the optimization algorithm consisted of analyzing the structure by the finite element method and using a recurrence relation derived from the appropriate optimality criterion to modify the design variables the optimality criterion was derived by differentiating the lagrangian with respect to the design variables the displacements were assumed to be small and in the finite element analysis linear equilibrium equations were solved to determine the response of the structure to the ap plied loads in the case of system stability the constraints were defined see refs 3 6 by the associated linear eigenvalue problem this definition of system stability for some structures may not be valid because of the nonlinear behavior of the structure which may be due to the geometry of the structure or the presence of geometric imperfections the correct procedure for these structures would be to analyze the structure by using nonlinear equilibrium equations this will be particularly true for large space structures lss in this paper an optimization method based on the optimality criterion approach is presented for structures with geometric nonlinear behavior in the case of a structure optimized with constraints on linear stability the optimum structure can have more than one critical buckling mode this design tends to become im perfection sensitive and a small deviation in the geometry of the structure can reduce its load carrying capacity sub stantially the imperfection sensitivity of the optimized structure can be reduced by designing the structure so that buckling loads associated with the critical buckling modes are not equal see ref 7 a real structure has geometric im perfections and these structures tend to have nonlinear

polynomial computations for blind image deconvolution this paper considers the problem of blind image deconvolution bid when the blur arises from a spatially invariant point spread function psf h which implies that a blurred image g is formed by the convolution of h and the exact form f of g since the multiplication of two bivariate polynomials is performed by convolving their coefficient matrices the equivalence of the formation of a blurred image and the product of two bivariate polynomials implies that bid can be performed by considering f g and h to be bivariate polynomials on which polynomial operations are performed these operations allow the psf to be computed which is then deconvolved from the blurred image g thereby obtaining a deblurred image that is a good approximation of the exact image f computational results show that the deblurred image obtained using polynomial computations is better than the deblurred image obtained using other methods for blind image deconvolution

lu factorization for matrices in quasiseparable form via orthogonal transformations abstract the paper presents a new algorithm to compute the lu factorization of a matrix represented in a quasiseparable or semiseparable form i e using generators it obtains the quasiseparable representations of the factors l and u of an n n block matrix via o n arithmetic operations on the block entries the algorithm uses recursions based exclusively on unitary transformations which provide numerical stability even in singular cases the method of the paper is based on the theory developed in 1 and provides an alternative to the approach proposed in 7 for strongly regular matrices the algorithm presented here works also for some matrices with possibly singular principle submatrices the results of numerical tests show that also for strongly regular matrices the new algorithm is comparable with the previous methods

structural optimization of guyed trusses here is a strong trend in the design of skeletal structures to wherever practical make increasing use of tensioned cables one of the best examples of the convergence of structural efficiency to a best design can probably be found when studying the evolution of the configurations of power transmission towers from the early days of cantilever truss structures these have now evolved to the much lighter cross rope type in which two separate vertical cable stayed lattice struts support the power conductors through suspension from a cross rope apart from the savings in material significant savings in transportation cost can also be obtained especially since erection often has to occur in remote areas although the cable stayed structures are easy to analyze the determination of a best or least weight design can be complex due to the variety of design variables available these are typically configuration geometry member sizing and prestraining the optimization of a configuration is complex and is best accomplished manually perhaps with the aid of layout optimization methods on the contrary geometric and prestrain optimization and sizing can be treated by direct sensitivity analysis of a chosen topology this paper describes the sizing and prestrain optimization of guyed trusses for the nonlinear analysis the total lagrangian method is used in combination with the pure newton method and adaptive arc length procedures because of the lack of stiffness of a cable structure in its undeformed state soft spring elements are used to stabilize the structure temporarily during the load stepping phase the choice of prestrain instead of prestress variables is justified by both the simplicity of inclusion of prestrain in the sensitivity analysis and by the independence of the prestrain quantity with respect to the sequence of pretensioning of the structure the prestraining of a member can also simply be seen as the lack of fit of a member in a fixed geometry and can in practice be accomplished by specifying member lengths to create a predetermined lack of fit or by specifying turns of a turnbuckle included in the member for the optimization process two optimization methods are used namely the sequential quadratic programming sqp method idesign plba authored by arora1 and the successive approximation method sam algorithm of snyman and stander2 and stander et al 3 using two methods gives optima in which greater confidence can be placed additionally some idea of the degree of difficulty of the newly formulated optimization problems is gained five examples are given to demonstrate the analysis and optimization methodology the optimization of example structures subjected to second order effects show that unexpected results can be obtained a number of papers 4 8 related to the general topic of the sensitivity analysis and optimization of structures with nonlinear geometric

adjusted least squares fitting of algebraic hypersurfaces we consider the problem of fitting a set of points in euclidean space by an algebraic hypersurface we assume that points on a true hypersurface described by a polynomial equation are corrupted by zero mean independent gaussian noise and we estimate the coefficients of the true polynomial equation the adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator the adjusted least squares estimator is based on constructing a quasi hankel matrix which is a bias corrected matrix of moments for the case of unknown noise variance the estimator is defined as a solution of a polynomial eigenvalue problem in this paper we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for an arbitrary set of monomials in the polynomial equation

design sensitivity analysis of elastoplastic structures an iterative and incremental algorithm is presented for the design sensitivity analysis dsa of elastoplastic structures geometrical non linearity is included in the formulation and the structure may be subjected to a cyclic loading the key concept is discussed using an elastoplastic truss it is shown that the design sensitivity coefficients dscs are path dependent and discontinuous at the time at which yielding takes place in a member yield time in the proposed algorithm the yield time is considered to be a function of the design variables in this way the discontinuity in the dscs at the material transition points can be overcome in a simple and routine way incremental response analysis and dsa are carried out simultaneously using the newton raphson type iterative procedure since the proposed algorithm is completely consistent with the analysis procedure it can be implemented into an existing code for structural analysis application to distributed parameter structures with kinematic and or isotropic hardening is discussed using the von mises yield condition and the elastic predictor radial return method dsa is carried out for a ten bar truss with a piecewise linear constitutive relation it is shown that the dscs found using the proposed method agree quite well with those calculated by the central difference method finally an alternate method without any iteration for dsa is proposed and the results from the two methods are compared

a direct application of higher order parametric programming techniques to structural optimization computational methods based on a sequence of parametric programming problems are presented for solving constrained optimization problems cop without any parameter an auxiliary parametric programming problem appp is formulated in order to solve cop the procedure is started with an arbitrary initial solution which is the trivial solution of appp corresponding to the initial value of the parameter then the optimal solution for the final value of the parameter which is the optimal solution of cop is estimated by taylor s expansion with respect to the parameter where higher order terms are incorporated it is shown that the incorporation of the higher order terms indeed leads to a faster convergence of the solution as an extension of the method a general algorithm is presented for optimum design problems with state variable constraints which are implicit functions of the design variables logarithmic penalty functions are incorporated and the weight coefficients for the penalty terms are updated continuously the derivatives of the state variables with respect to the parameter and their sensitivity coefficients are expressed explicitly in terms of those of the design variables finally a method of simultaneous analysis and optimization is developed for trusses with geometrical non linearity

design sensitivity analysis and optimization of nonlinear structuralresponse using incremental procedure this paper describes procedures for design sensitivity analysis of nonlinear structural response using the incremental solution scheme the total and updated lagrangian approaches are discussed sensitivity analyses for the stress strain and displacement constraints are developed in the optimization process the buckling load constraint is also imposed numerical aspects of design sensitivity analysis are discussed three numerical examples are solved to illustrate the design sensitivity analysis and optimization process the numerical procedures are stable and work well they can be applied to more complex structures to optimize them

on a class of matrix pencils and l ifications equivalent to a given matrix polynomial a new class of linearizations and l ifications for m m matrix polynomials p x of degree n is proposed the l ifications in this class have the form a x d x e im w x where d is a block diagonal matrix polynomial with blocks bi x of size m w is an m qm matrix polynomial and e 1 1 t cq for a suitable integer q the blocks bi x can be chosen a priori subjected to some restrictions under additional assumptions on the blocks bi x the matrix polynomial a x is a strong l ification i e the reversed polynomial of a x defined by a x xdeg a x a x 1 is an l ification of p x the eigenvectors of the matrix polynomials p x and a x are related by means of explicit formulas some practical examples of l ifications are provided a strategy for choosing bi x in such a way that a x is a well conditioned linearization of p x is proposed some numerical experiments that validate the theoretical results are reported

shape optimization in elasticity and elasto viscoplasticity by the boundary element method abstract this paper is concerned with the optimal shape design of solids undergoing small strain small rotation elasto viscoplastic deformation shape sensitivities for this class of problems are determined by using a direct differentiation approach dda to the governing boundary element method bem equations of the problem the standard bem and the sensitivity equations are discretized and solved numerically shape optimization is carried out by coupling the standard and sensitivity analyses with an optimizer the optimization algorithm chosen here uses sequential quadratic programming to obtain the desired optimal shape of a body in an iterative manner numerical solutions for optimal shapes of cutouts in plates two dimensional plane stress are presented the difference between optimal shapes of solids undergoing purely elastic and elasto viscoplastic deformation is shown clearly in these examples

shaping optimal design of elastic planar frames with frequency constraints an approach to shaping optimal design of planar frames subject to eigenfrequency constraints displacement constraints and stress constraints simultaneously is described the eigenfrequency constraints may be imposed at the deformed due to the static load cone guration of the structure for the shape representation of the structure a design element technique is used a frame structure is treated as assembled from several design elements each of them being dee ned as a bezier curve the design variables may ine uence the position and the shape of each design element plus the cross sectional properties of beam elements the optimal design problem is dee ned in a general form and its solution by employing gradient based methods of mathematical programming is discussed briee y the structural response analysis is based on a nonlinear model that can handle virtually arbitrary large structural displacements two numerical examples illustrate the theory nomenclature ai area of the cross section asi shear area of the cross section b vector of design variables ci centroid curve e elastic modulus of the material f vector of internal forces fi eigenfrequencies g shear modulus of the material hi constrained quantitites ii moment of inertia of the cross section k stiffness matrix m mass matrix pi lagrange interpolation polynomial p external distributed force per unit length qi position vector of a control point r vector of external forces ri position vector u vector of structural displacements v vector of structural response variables w external distributed force per unit length shear strain extensional strain i eigenvalues angle of inclination at the undeformed cone guration i eigenvector a angle of rotation

regularization matrices determined by matrix nearness problems abstract this paper is concerned with the solution of large scale linear discrete ill posed problems with error contaminated data tikhonov regularization is a popular approach to determine meaningful approximate solutions of such problems the choice of regularization matrix in tikhonov regularization may significantly affect the quality of the computed approximate solution this matrix should be chosen to promote the recovery of known important features of the desired solution such as smoothness and monotonicity we describe a novel approach to determine regularization matrices with desired properties by solving a matrix nearness problem the constructed regularization matrix is the closest matrix in the frobenius norm with a prescribed null space to a given matrix numerical examples illustrate the performance of the regularization matrices so obtained

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

variational method for design sensitivity analysis in nonlinear structural mechanics this paper describes a unified variational theory of design sensitivity analysis of linear and nonlinear structures for shape nonshape and material selection problems the concepts of reference volume and adjoint structure are used to develop the unified viewpoint a general formula for design sensitivity analysis is derived several analytical linear and nonlinear examples are used to interpret various terms of the formula and demonstrate its use such analytical examples give insight into the application of the general formula to more complex problems that must be treated numerically adjoint state fields are also interpreted as sensitivities for the functional related to the primary structure

designing rational filter functions for solving eigenvalue problems by contour integration abstract solving nonlinear eigenvalue problems by contour integration requires an effective discretization for the corresponding contour integrals in this paper it is shown that good rational filter functions can be computed using nonlinear least squares optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis the conditions that such an effective filter function should satisfy are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from tensorlab numerical experiments illustrate the validity of this approach

estimation of the bilinear form y f a x for hermitian matrices abstract for a hermitian matrix a c p p given vectors x y c p and for suitable functions f the bilinear form y f a x is estimated by extending the extrapolation method proposed by c brezinski in 1999 families of one term and two term estimates e f c and e f n k n k z respectively are derived by extrapolation of the moments of the matrix a for the positive definite case bounds for the optimal value of which leads to an efficient one term estimate in only one matrix vector product are derived for f a a 1 a formula approximating this optimal value of is specified numerical results for several matrix functions and comparisons are provided to demonstrate the effectiveness of the extrapolation method

computing the exponential of large block triangular block toeplitz matrices encountered in fluid queues the erlangian approximation of markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block triangular block toeplitz structure to this end we propose some algorithms which exploit the toeplitz structure and the properties of generators such algorithms allow us to compute the exponential of very large matrices which would otherwise be untreatable with standard methods we also prove interesting decay properties of the exponential of a generator having a block triangular block toeplitz structure