reducing the bandwidth of a sparse matrix with tabu search the bandwidth of a matrix ij a a is defined as the maximum absolute difference between i and j for which 0 ij a the problem of reducing the bandwidth of a matrix consists of finding a permutation of the rows and columns that keeps the nonzero elements in a band that is as close as possible to the main diagonal of the matrix this np complete problem can also be formulated as a labeling of vertices on a graph where edges are the nonzero elements of the corresponding symmetrical matrix many bandwidth reduction algorithms have been developed since the 1960s and applied to structural engineering fluid dynamics and network analysis for the most part these procedures do not incorporate metaheuristic elements which is one of the main goals of our current development another equally important goal is to design and test a special type of candidate list strategy and a move mechanism to be embedded in a tabu search procedure for the bandwidth reduction problem this candidate list strategy accelerates the selection of a move in the neighborhood of the current solution in any given iteration our extensive experimentation shows that the proposed procedure outperforms the best known algorithms in terms of solution quality consuming a reasonable computational effort

on periodic boundary conditions and ergodicity in computational homogenization of heterogeneous materials with random microstructure abstract due to high computational costs associated with stochastic computational homogenization a highly complex random material microstructure is often replaced by simplified parametric ergodic and sometimes periodic models this replacement is often criticized in the literature due to unclear error resulting from the periodicity and ergodicity assumptions in the current contribution we perform a validation of both assumptions through various numerical examples to this end we compare large scale non simplified and non ergodic models with simplified ergodic and periodic solutions in addition we analyze the hill mandel condition for stochastic homogenization problems and demonstrate that for a stochastic problem there are more than three classical types of boundary conditions as an example we propose two novel stochastic periodic boundary conditions which possess a clear physical meaning the effect of these novel periodic boundary conditions is also analyzed by comparing with non ergodic simulation results

computing preconditioners via subspace projection we give an alternative method for computing the matrix argmina pf where f is the frobenius norm and p is taken from the set of square matrices simultaneously diagonalized by a given orthogonal trigonometric transform specific examples are given

on the probable performance of heuristics for bandwidth minimization most research in algorithm design relies on worst case analysis for performance comparisons unfortunately worst case analysis does not always provide an adequate measure of an algorithm s effectiveness this is particularly true in the case of heuristic algorithms for hard combinatorial problems in such cases analysis of the probable performance can yield more meaningful results and can provide insight leading to better algorithms the problem of minimizing the bandwidth of a sparse symmetric matrix by performing simultaneous row and column permutations is an example of a problem for which there are well known heuristics whose practical success has lacked a convincing analytical explanation a class of heuristics introduced by cuthill and mckee in 1969 and referred to here as the level algorithms are the basis for bandwidth minimization routines that have been widely used for over a decade at the same time it is easy to construct examples showing that the level algorithms can produce solutions t

a new algorithm for solving toeplitz systems of equations abstract we present some recurrences that are the basis for an algorithm to invert an n n toeplitz system of linear equations with computational complexity o n log 2 n the recurrences used are closely related to the zohar trench and bareiss algorithms but do not have any obvious connection with other asymptotically fast algorithms for the inversion of toeplitz systems

a branch and bound algorithm for the matrix bandwidth minimization in this article we first review previous exact approaches as well as theoretical contributions for the problem of reducing the bandwidth of a matrix this problem consists of finding a permutation of the rows and columns of a given matrix which keeps the non zero elements in a band that is as close as possible to the main diagonal this np complete problem can also be formulated as a labeling of vertices on a graph where edges are the non zero elements of the corresponding symmetrical matrix we propose a new branch and bound algorithm and new expressions for known lower bounds for this problem empirical results with a collection of previously reported instances indicate that the proposed algorithm compares favourably to previous methods

principles of scatter search scatter search is an evolutionary method that has been successfully applied to hard optimization problems the fundamental concepts and principles of the method were first proposed in the 1970s based on formulations dating back to the 1960s for combining decision rules and problem constraints in contrast to other evolutionary methods like genetic algorithms scatter search is founded on the premise that systematic designs and methods for creating new solutions afford significant benefits beyond those derived from recourse to randomization it uses strategies for search diversification and intensification that have proved effective in a variety of optimization problems this paper provides the main principles and ideas of scatter search and its generalized form path relinking we first describe a basic design to give the reader the tools to create relatively simple implementations more advanced designs derive from the fact that scatter search and path relinking are also intimately related to the tabu search ts metaheuristic and gain additional advantage by making use of ts adaptive memory and associated memory exploiting mechanisms capable of being tailored to particular contexts these and other advanced processes described in the paper facilitate the creation of sophisticated implementations for hard problems that often arise in practical settings due to their flexibility and proven effectiveness scatter search and path relinking can be successfully adapted to tackle optimization problems spanning a wide range of applications and a diverse collection of structures as shown in the papers of this volume

virtual element methods for plate bending problems abstract we discuss the application of virtual elements to linear plate bending problems in the kirchhoff love formulation as we shall see in the virtual element environment the treatment of the c 1 continuity condition is much easier than for traditional finite elements the main difference consists in the fact that traditional finite elements for every element k and for every given set of degrees of freedom require the use of a space of polynomials or piecewise polynomials for composite elements for which the given set of degrees of freedom is unisolvent for virtual elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials whose degree determines the accuracy as we shall see the non polynomial part of our local spaces does not need to be known in detail and therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries

some aspects of circulant preconditioners this paper studies circulant approximations of hermitian toeplitz matrices that are solutions of certain minimization problems relative to the frobenius norm the lub 1 or the lub infty norms the examinations supplement the family of the so called optimal and superoptimal preconditioners that have been proposed for the preconditioned conjugate gradient method for solving toeplitz systems t n x b for the new frobenius norm approximations it is shown that they can be computed in o n log n arithmetic operations and that the eigenvalues of the preconditioned linear equations are asymptotically clustered around 1 if t n is a leading principal submatrix of an infinite toeplitz matrix connected with a positive function of the wiener class

on the virtual element method for three dimensional linear elasticity problems on arbitrary polyhedral meshes abstract we explore the recently proposed virtual element method vem for the numerical solution of boundary value problems on arbitrary polyhedral meshes more specifically we focus on the linear elasticity equations in three dimensions and elaborate upon the key concepts underlying the first order vem while the point of departure is a conforming galerkin framework the distinguishing feature of vem is that it does not require an explicit computation of the trial and test spaces thereby circumventing a barrier to standard finite element discretizations on arbitrary grids at the heart of the method is a particular kinematic decomposition of element deformation states which in turn leads to a corresponding decomposition of strain energy by capturing the energy of linear deformations exactly one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions the decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation as we show computing these projection maps and subsequently the local stiffness matrices in practice reduces to the computation of purely geometric quantities in addition to discussing aspects of implementation of the method we present several numerical studies in order to verify convergence of the vem and evaluate its performance for various types of meshes

circulant preconditioners for toeplitz matrices with piecewise continuous generating functions the authors consider the solution of n by n toeplitz systems t sub n x b by preconditioned conjugate gradient methods the preconditioner c sub n is the t chan circulant preconditioner which is defined to be the circulant matrix that minimizes parallel b sub n t sub n parallel sub f over all circulant matrices b sub n for toeplitz matrices generated by positive 2 pi periodic continuous functions they have shown earlier that the spectrum of the preconditioned system c sup minus 1 sub n t sub n is clustered around 1 and hence the convergence rate of the preconditioned system is superlinear however in this paper they show that if instead the generating function is only piecewise continuous then for all epsilon sufficiently small there are o log n eigenvalues of c sup minus 1 sub n t sub n that lie outside the interval 1 epsilon 1 epsilon in particular the spectrum of c sup minus 1 sub n t sub n cannot be clustered around 1 numerical examples are given to verify that the convergence rate of the method is no longer superlinear in general 20 refs

an effective version of dilworth s theorem 0 introduction loosely speaking a subset a of the natural numbers is recursive iff there exists an algorithm i e a finite computer program which upon input of a natural number n outputs 1 if n e a and 0 otherwise similarly a partial ordering a a is recursive iff there is an algorithm which upon input of an ordered pair of natural numbers a b outputs 1 if a b and 0 otherwise for a more careful definition of recursive relations see r one of the attractions of linite combinatorics over infinite combinatorics is its explicit constructions one never has to consider whether a finite object really exists this paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain we shall consider a recursive combinatorics whose domain is the recursive structures since finite structures are trivially recursive this domain does indeed extend that of finite combinatorics moreover each structure in this domain is explicitly exhibited by some finite computer program questions from the graph theory of this combinatorics have been studied by bean b bi kierstead k and schmerl s si generally their results relate the chromatic number of a recursive graph with certain properties to its recursive chromatic number the following example illustrates these ideas dilworth s theorem d asserts that any partial ordering of finite width n can be covered by n chains if the partial ordering is finite then one can actually exhibit these chains by trial and error if by no other method the following easy argument demonstrates how dilworth s theorem for countably infinite partial orderings follows from dilworth s theorem for finite partial orderings let p pi i e n we show by induction that for all i e n there exist chains co c 1 such that i ifj i and k n then ck c k ii pi e co l u u cn 1 iii if q is a finite subset of p then co c 1 can be extended to chains that cover q

algebraic cubature on polygonal elements with a circular edge abstract we compute low cardinality algebraic cubature formulas on convex or concave polygonal elements with a circular edge by subdivision into circular quadrangles blending formulas via subperiodic trigonometric gaussian quadrature and final compression via caratheodory tchakaloff subsampling of discrete measures we also discuss applications to the vem virtual element method in computational mechanics problems

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

generalized gaussian quadrature rules on arbitrary polygons in this paper we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons these quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required we have used this algorithm for the construction of symmetric and non symmetric quadrature rules over convex and concave polygons while in the case of symmetric quadratures our results are comparable to available rules the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions to demonstrate the efficiency of the new quadrature rules we have tested them for the integration of rational polygonal shape functions over a regular hexagon for a relative error of 10 8 in the computation of stiffness matrix entries one needs at least 198 evaluation points when the region is partitioned whereas 85 points suffice with our quadrature rule copyright 2009 john wiley sons ltd

gauss green cubature and moment computation over arbitrary geometries we have implemented in matlab a gauss like cubature formula over arbitrary bivariate domains with a piecewise regular boundary which is tracked by splines of maximum degree p spline curvilinear polygons the formula is exact for polynomials of degree at most 2n 1 using n cmn 2 nodes 1 c p m being the total number of points given on the boundary it does not need any decomposition of the domain but relies directly on univariate gauss legendre quadrature via green s integral formula several numerical tests are presented including computation of standard as well as orthogonal moments over a nonstandard planar region

polynomial interpolation and cubature over polygons we have implemented a matlab code to compute discrete extremal sets of fekete and leja type on convex or concave polygons together with the corresponding interpolatory cubature formulas the method works by qr and lu factorizations of rectangular vandermonde matrices on weakly admissible meshes wams of polygons constructed by polygon quadrangulation

quadrature schemes for arbitrary convex concave volumes and integration of weak form in enriched partition of unity methods abstract quadrature schemes are constructed based on moment fitting equations to integrate polynomials over arbitrary convex concave volumes that arise among others in enriched partition of unity finite element methods epum the building block of the scheme involves the divergence theorem of multivariable calculus which is used to integrate the base functions an efficient and robust point distribution method is proposed and the quadrature weights at the corresponding points are obtained by solving a least squares problem the method is applied initially to integrate given polynomial functions over complex volumes and further to simulate simple three dimensional fluid dynamic problems which involve very complex volumes when solved with epum accuracy of the present quadrature construction scheme is demonstrated by comparing the results with the available exact numerical solutions and efficiency of the method is proved by comparing the computational time with that of the widely used tessellation method

toeplitz preconditioners for hermitian toeplitz systems abstract we propose a new type of preconditioners for hermitian positive definite toeplitz systems a n x b where a n are assumed to be generated by functions f that are positive and 2 periodic our approach is to precondition a n by the toeplitz matrix a n generated by 1 f we prove that the resulting preconditioned matrix a n a n will have clustered spectrum when a n cannot be formed efficiently we use quadrature rules and convolution products to construct nearby approximations to a n we show that the resulting approximations are toeplitz matrices which can be written as sums of circulant matrices as a side result we prove that any toeplitz matrix can be written as a sum of circulant matrices we then show that our toeplitz preconditioners t n are generalizations of circulant preconditioners and the way they are constructed is similar to the approach used in the additive schwarz method for elliptic problems we finally prove that the preconditioned systems t n a n will have clustered spectra around 1

which circulant preconditioner is better the eigenvalue clustering of matrices s n 1 a n and c n 1 a n is experimentally studied where a n s n and c n respectively are toeplitz matrices strang and optimal circulant preconditioners generated by the fourier expansion of a function f x some illustrations are given to show how the clustering depends on the smoothness of f x and which preconditioner is preferable an original technique for experimental exploration of the clustering rate is presented this technique is based on the bisection idea and on the toeplitz decomposition of a three matrix product cac where a is a toeplitz matrix and c is a circulant in particular it is proved that the toeplitz displacement rank of cac is not greater than 4 provided that c and a are symmetric

grasp and path relinking for the matrix bandwidth minimization in this article we develop a greedy randomized adaptive search procedure grasp for the problem of reducing the bandwidth of a matrix this problem consists of finding a permutation of the rows and columns of a given matrix which keeps the nonzero elements in a band that is as close as possible to the main diagonal the proposed method may be coupled with a path relinking strategy to search for improved outcomes empirical results indicate that the proposed grasp implementation compares favourably to classical heuristics grasp with path relinking is also found to be competitive with a recently published tabu search algorithm that is considered one of the best currently available for bandwidth minimization

effective longitudinal shear moduli of periodic fibre reinforced composites with radially graded fibres this paper presents a closed form expression for the homogenized longitudinal shear moduli of a linear elastic composite material reinforced by long parallel radially graded circular fibres with a periodic arrangement an imperfect linear elastic fibre matrix interface is allowed the asymptotic homogenization method is adopted and the relevant cell problem is addressed periodicity is enforced by resorting to the theory of weierstrass elliptic functions the equilibrium equation in the fibre domain is solved in closed form by applying the theory of hypergeometric functions for new wide classes of grading profiles defined in terms of special functions the effectiveness of the present analytical procedure is proved by convergence analysis and comparison with finite element solutions a parametric analysis investigating the influence of microstructural and material features on the effective moduli is presented the feasibility of mitigating the shear stress concentration in the composite by tuning the fibre grading profile is shown

symmetric gaussian elimination for cauchy type matrices with application to positive definite toeplitz matrices the problem of solving linear equations with a toeplitz matrix t appears in many applications often t is positive definite but ill conditioned with many small eigenvalues in this case fast and superfast algorithms may show a very poor behavior or even break down in recent papers the transformation of a toeplitz matrix into a cauchy type matrix is proposed the resulting new linear equations can be solved in o n 2 operations using standard pivoting strategies which leads to very stable fast methods also for ill conditioned systems the basic tool is the formulation of gaussian elimination for matrices with low displacement rank in this paper we will transform a hermitian toeplitz matrix t into a cauchy type matrix b ftf rm h by applying the fourier transform we will prove some useful properties of b and formulate a symmetric gaussian elimination algorithm for positive definite b using the symmetry and persymmetry of t we can reduce the total costs of this algorithm compared with unsymmetric gaussian elimination for complex hermitian t the complexity of the new algorithm is then nearly the same as for the schur algorithm furthermore it is possible to include some strategies for ill conditioned positive definite matrices that are well known in optimization numerical examples show that this new algorithm is fast and reliable

the online graph bandwidth problem abstract the online graph bandwidth problem is defined and we present an online algorithm that always outputs a 2k 1 n 1 2k bandwidth function for any n vertex graph with bandwidth k a lower bound of k k 1 n 2 is shown for any such algorithm two other protocols for online problems are given and we prove lower bounds for the bandwidth problem under both of these alternative protocols

transformation techniques for toeplitz and toeplitz plus hankel matrices part i transformations transformations of the form a c1ac2 are investigated that transform toeplitz and toeplitz plus hankel matrices into generalized cauchy matrices c1 and c2 are matrices related to the discrete fourier transformation or to various real trigonometric transformations combining these results with pivoting techniques in part ii algorithms for toeplitz and toeplitz plus hankel systems will be presented that are more stable than classical algorithms

numerical integration over arbitrary polygonal domains based on schwarz christoffel conformal mapping this paper presents a new numerical integration technique oil arbitrary polygonal domains the polygonal domain is mapped conformally to the unit disk using schwarz christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used this method eliminates the need for a two level isoparametric mapping usually required moreover the positivity of the jacobian is guaranteed numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results

the spectrum of a family of circulant preconditioned toeplitz systems the solutions of symmetric positive definite toeplitz systems ax b are studied by the preconditioned conjugate gradient method the preconditioner is the circulant matrix c that minimizes the frobenius norm c a f t chan an optimal circulant preconditioner for toeplitz systems ucla department of mathematics cam report 87 06 june 1987 the convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of c 1 a for toeplitz matrix a with entries which are fourier coefficients of a positive function in the wiener class this paper establishes the invertibility of c finds the asymptotic behaviour of the eigenvalues of the preconditioned matrix c 1 a as the dimension increases and proves that they are clustered around 1

fast superoptimal preconditioning of multiindex toeplitz matrices abstract in this article three well known methods of circulant preconditioning of finite multiindex toeplitz linear systems that is linear systems indexed by integers i 1 i d with 0 i s n s s 1 d are studied in detail a general algorithm for the construction of the so called superoptimal preconditioner is also given and it is shown that this procedure requires o n 1 n d log n 1 n d floating point operations

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity

a decomposition of toeplitz matrices and optimal circulant preconditioning abstract a factorization of the given toeplitz matrix t obtained in this work is used to derive an optimal circulant preconditioner c in the sense of minimizing the frobenius norm of i ct by use of the fast fourier transform the computation of the new preconditioner can be performed in o n log n operations