on domain symmetry and its use in homogenization abstract the present paper focuses on solving partial differential equations in domains exhibiting symmetries and periodic boundary conditions for the purpose of homogenization we show in a systematic manner how the symmetry can be exploited to significantly reduce the complexity of the problem and the computational burden this is especially relevant in inverse problems when one needs to solve the partial differential equation the primal problem many times in an optimization algorithm the main motivation of our study is inverse homogenization used to design architected composite materials with novel properties which are being fabricated at ever increasing rates thanks to recent advances in additive manufacturing for example one may optimize the morphology of a two phase composite unit cell to achieve isotropic homogenized properties with maximal bulk modulus and minimal poisson ratio typically the isotropy is enforced by applying constraints to the optimization problem however in two dimensions one can alternatively optimize the morphology of an equilateral triangle and then rotate and reflect the triangle to form a space filling d 3 symmetric hexagonal unit cell that necessarily exhibits isotropic homogenized properties one can further use this d 3 symmetry to reduce the computational expense by performing the unit strain periodic boundary condition simulations on the single triangle symmetry sector rather than the six fold larger hexagon in this paper we use group representation theory to derive the necessary periodic boundary conditions on the symmetry sectors of unit cells the developments are done in a general setting and specialized to the two dimensional dihedral symmetries of the abelian d 2 i e orthotropic square unit cell and nonabelian d 3 i e trigonal hexagon unit cell we then demonstrate how this theory can be applied by evaluating the homogenized properties of a two phase planar composite over the triangle symmetry sector of a d 3 symmetric hexagonal unit cell

singular behaviour of the lattice green function for the d dimensional hypercubic lattice the analytic properties of the d dimensional hypercubic lattice green function are investigated where w u iv is a complex variable in the u v plane in particular the detailed behaviour of g d w in the immediate neighbourhood of the branch point singularities w d is determined these results are used to derive an asymptotic expansion in powers of 1 n for the number of random walks on the hypercubic lattice which return to their starting point not necessarily for the first time after a walk of 2n steps finally it is shown that this asymptotic expansion enables one to calculate extremely accurate values for the generalized d dimensional watson integral where s d 2 and s 1 2

group theoretic exploitations of symmetry in computational solid and structural mechanics the use of group theory in simplifying the study of problems involving symmetry is a well established approach in various branches of physics and chemistry and major applications in these areas date back more than 70 years within the engineering disciplines the search for more systematic and more efficient strategies for exploiting symmetry in the computational problems of solid and structural mechanics has led to the development of group theoretic methods over the past 40 years this paper reviews the advances made in the application of group theory in areas such as bifurcation analysis vibration analysis and finite element analysis and summarizes the various implementation procedures currently available illustrative examples of typical solution procedures are drawn from recent work of the author it is shown how the group theoretic approach through the characteristic vector space decomposition enables considerable simplifications and reductions in computational effort to be achieved in many cases group theoretic considerations also allow valuable insights on the behaviour or properties of a system to be gained before any actual calculations are carried out copyright 2009 john wiley sons ltd

cost approximation a unified framework of descent algorithms for nonlinear programs this paper describes and analyzes the cost approximation algorithm this class of iterative descent algorithms for nonlinear programs and variational inequalities places a large number of algorithms within a common framework and provides a means for analyzing relationships among seemingly unrelated methods a common property of the methods included in the framework is that their subproblems may be characterized by monotone mappings which replace an additive part of the original cost mapping in an iterative manner alternately a step is taken in the direction obtained in order to reduce the value of a merit function for the original problem the generality of the framework is illustrated through examples and the convergence characteristics of the algorithm are analyzed for applications to nondifferentiable optimization the convergence results are applied to some example methods demonstrating the strength of the analysis compared to existing results

asymptotic expansions of lattice green s functions we derive asymptotic expansions for the green functions associated with coercive difference equations on general lattices these expansions lead to rigorous methods for approximating the lattice green functions by polyharmonic rational functions

on the identification property of a projected gradient method the authors study the convergence properties of a projected gradient algorithm for the general problem min f x x in omega where f r n to r is a mapping continuously differentiable on a closed convex set omega subseteq r n the algorithm which requires only one projection per iteration is a special version of the method of projection of the gradient by demyanov and rubinov approximate methods in optimization problems elsevier new york 1970 where the step choice is made according to a scheme similar to the one used by calamai and more math programming 39 1987 pp 93 116 the authors are mainly interested in analysing the identification property of the algorithm for the case where the set omega is a polyhedron that is the ability to identify in a finite number of steps the face in which the final solution lies the convergence results that are shown are very similar to those shown in 6 for the standard projected gradient method

application of mahler measure theory to the face centred cubic lattice green function at the origin and its associated logarithmic integral the mathematical properties of the face centred cubic lattice green function and the associated logarithmic integral are investigated where c cos and w w1 iw2 is a complex variable in the w plane in particular the theory of mahler measures is used to obtain a closed form expression for s w in terms of 5f4 generalized hypergeometric functions the method of analytic continuation is then applied to this result in order to prove that next the relation ds dw g w where w 3 is used to derive the simple formula where w lies in a restricted region of the cut w plane the limit function is also evaluated in the intervals w1 1 0 and w1 0 3 it is shown that gr w1 and gi w1 can be expressed in terms of 2f1 z w1 hypergeometric functions where the independent variable z w1 is a real valued rational function of w1 finally new formulae are derived for the number of random walks rn on the face centred cubic lattice which return to their starting point not necessarily for the first time after n nearest neighbour steps

on the cubic lattice green functions wheatstone physics laboratory king s college university of london strand london wc2r 2ls u k it is proved that k k 4 eta frac 1 2 1 eta frac 1 2 k k where eta is a complex variable which lies in a certain region scr r 2 of the eta plane and k k pm are complete elliptic integrals of the first kind with moduli k pm which are given by k pm 2 equiv k pm 2 eta frac 1 2 pm frac 1 4 eta 4 eta frac 1 2 frac 1 4 2 eta 1 eta frac 1 2 this basic result is then used to express the face centred cubic and simple cubic lattice green functions at the origin in terms of the square of a complete elliptic integral of the first kind several new identities involving the heun function f a b alpha beta gamma delta eta are also derived next it is shown that the three cubic lattice green functions all have parametric representations which involve the green function for the two dimensional honeycomb lattice finally the results are applied to a variety of problems in lattice statistics in particular a new simplified formula for the generating function of staircase polygons on a four dimensional hypercubic lattice is derived

on the cubic modular transformation and the cubic lattice green functions the elliptic modular transformation of order 3 is used to prove that the lattice green functions at the origin for the face centred cubic and simple cubic lattices can both be expressed in terms of the square of a complete elliptic integral of the first kind

exploiting partial or complete geometrical symmetry in 3d symmetric galerkin indirect bem formulations procedures based on group representation theory allowing the exploitation of geometrical symmetry in symmetric galerkin bem formulations are investigated in particular this investigation is based on the weaker assumption of partial geometrical symmetry where the boundary has two disconnected components one of which is symmetric e g this can be very useful for defect identification problems the main development is expounded in the context of 3d neumann elastostatic problems considered as model problems and then extended to sgbie formulations for dirichlet and or scalar problems both abelian and non abelian finite symmetry groups are considered the effectiveness of the present approach is demonstrated through numerical examples where both partial and complete symmetry are considered in connection with both abelian and non abelian symmetry groups

recursive evaluation of space time lattice green s functions up to a multiplicative constant the lattice green s function lgf as defined in condensed matter physics and lattice statistical mechanics is equivalent to the z domain counterpart of the finite difference time domain green s function gf on a lattice expansion of a well known integral representation for the lgf on a dimensional hyper cubic lattice in powers of z 1 and application of the chu vandermonde identity results in 1 nested finite sum representations for discrete space time gfs due to severe numerical cancellations these nested finite sums are of little practical use for 2 the finite sum may be evaluated in closed form in terms of a generalized hypergeometric function for special lattice points that representation simplifies considerably while on the other hand the finite difference stencil may be used to derive single lattice point second order recurrence schemes for generating 2d discrete space time gf time sequences on the fly for arbitrary symbolic lattice points zeilberger s algorithm produces a third order recurrence operator with polynomial coefficients of the sixth degree the corresponding recurrence scheme constitutes the most efficient numerical method for the majority of lattice points in spite of the fact that for explicit numeric lattice points the associated third order recurrence operator is not the minimum recurrence operator as regards the asymptotic bounds for the possible solutions to the recurrence scheme perron s theorem precludes factorial or exponential growth along horizontal lattices directions rapid initial growth does occur but poses no problems in augmented dynamic range fixed precision arithmetic by analysing long distance wave propagation along a horizontal lattice direction we have concluded that the chirp up oscillations of the discrete space time gf are the root cause of grid dispersion anisotropy with each factor of ten increase in the lattice distance one would have to roughly double the pulse width of the source signature to keep pulse distortion at bay the gf time sequences can also be used for an efficient computation of discrete space frequency lgfs especially if one employs aitken s 2 process for the acceleration of the convergence of the consecutive partial sums

algorithms for the continuous nonlinear resource allocation problem new implementations and numerical studies patriksson 2008 provided a then up to date survey on the continuous separable differentiable and convex resource allocation problem with a single resource constraint since the publication of that paper the interest in the problem has grown several new applications have arisen where the problem at hand constitutes a subproblem and several new algorithms have been developed for its efficient solution this paper therefore serves three purposes first it provides an up to date extension of the survey of the literature of the field complementing the survey in patriksson 2008 with more then 20 books and articles second it contributes improvements of some of these algorithms in particular with an improvement of the pegging that is variable fixing process in the relaxation algorithm and an improved means to evaluate subsolutions third it numerically evaluates several relaxation primal and breakpoint dual algorithms incorporating a variety of pegging strategies as well as a quasi newton method our conclusion is that our modification of the relaxation algorithm performs the best at least for problem sizes up to 30 million variables the practical time complexity for the breakpoint and relaxation algorithms is linear

on the regularizing behavior of the sda and sdc gradient methods in the solution of linear ill posed problems we analyze the regularization properties of two recently proposed gradient methods sda and sdc applied to discrete linear inverse problems by studying their filter factors we show that the tendency of these methods to eliminate first the eigencomponents of the gradient corresponding to large singular values allows to reconstruct the most significant part of the solution thus yielding a useful filtering effect this behavior is confirmed by numerical experiments performed on some image restoration problems furthermore the experiments show that for severely ill conditioned problems and high noise levels the sda and sdc methods can be competitive with the conjugate gradient cg method since they are slightly slower than cg but exhibit a better semiconvergence behavior

global and asymptotic convergence rate estimates for a class of projected gradient processes projected gradient processes of the goldstein levitin polyak type are considered for constrained minimization problems min omega f with omega a convex set in a hilbert space x and f x to mathbb r 1 a differentiable functional global and local convergence theorems are established for a large class of these processes including those generated with implicit step length rules proposed by bertsekas and goldstein in this analysis traditional uniform strong positivity conditions on the hessian nabla 2 f are replaced by weaker pseudoconvexity conditions and growth conditions on f when f has a unique minimizes in omega convergence rates are shown to depend on how rapidly the function gamma sigma inf r f x f xi mid x in omega x xi geqq sigma grows with increasing sigma 0 if gamma sigma geqq b sigma nu for some b 0 the processes f n in question converge to inf f like o n nu nu 2 linear

competitive facility location model with concave demand abstract we consider a spatial interaction model for locating a set of new facilities that compete for customer demand with each other as well as with some pre existing facilities to capture the market expansion and the market cannibalization effects customer demand is assumed to be a concave non decreasing function of the total utility derived by each customer from the service offered by the facilities the problem is formulated as a non linear knapsack problem for which we develop a novel solution approach based on constructing an efficient piecewise linear approximation scheme for the objective function this allows us to develop exact and optimal solution approaches capable of dealing with relatively large scale instances of the model we also develop a fast heuristic algorithm for which a tight worst case error bound is established

on the laurent series for the epstein zeta function the epstein zeta function d m s n z d n m n t s 2 where is a real symmetric and invertible d d matrix and is a d dimensional row vector with integer coordinates is considered the prime on the sum indicates that the term should be excluded it is known that has a laurent series expansion about the singular point s d which can be written in the form d m s 1 a m d s d in this paper we shall show that the coefficient can be accurately calculated using rapidly convergent series which involve the meijer g function exact formulae are also derived for when m u n 1 0 0 n with the results for are then used to establish several mathematical identities involving summations of generalized stieltjes constants next the laurent series for where is the d d unit matrix is briefly discussed for the cases and 8 finally a new application of the results in lattice statistics is described

on the number of spanning trees for the 3d simple cubic lattice the number of spanning trees on a large lattice is evaluated exactly for the 3d simple cubic lattice graph similarities to the evaluation of the lattice green function are pointed out

singular behaviour of the cubic lattice green functions and associated integrals the behaviour of the three cubic lattice green functions where is determined in the neighbourhood of the singular point w 1 the results are used to investigate the singular behaviour of the associated integrals finally a new method is developed which enables one to calculate the numerical values of the constants lj 1 j 1 2 3 with extremely high precision

convex quadratic programming with one constraint and bounded variables in this paper we propose an iterative algorithm for solving a convex quadratic program with one equality constraint and bounded variables at each iteration a separable convex quadratic program with the same constraint set is solved two variants are analyzed one that uses an exact line search and the other a unit step size preliminary testing suggests that this approach is efficient for problems with diagonally dominant matrices

on the continuous quadratic knapsack problem we introduce a new algorithm for the continuous bounded quadratic knapsack problem this algorithm is motivated by the geometry of the problem is based on the iterative solution of a series of simple projection problems and is easy to understand and implement in practice the method compares favorably to other well known algorithms some of which have superior worst case complexity on problem sizes up ton 4000

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

projected barzilai borwein methods for large scale box constrained quadratic programming this paper studies projected barzilai borwein pbb methods for large scale box constrained quadratic programming recent work on this method has modified the pbb method by incorporating the grippo lampariello lucidi gll nonmonotone line search so as to enable global convergence to be proved we show by many numerical experiments that the performance of the pbb method deteriorates if the gll line search is used we have therefore considered the question of whether the unmodified method is globally convergent which we show not to be the case by exhibiting a counter example in which the method cycles a new projected gradient method pabb is then considered that alternately uses the two barzilai borwein steplengths we also give an example in which this method may cycle although its practical performance is seen to be superior to the pbb method with the aim of both ensuring global convergence and preserving the good numerical performance of the unmodified methods we examine other recent work on nonmonotone line searches and propose a new adaptive variant with some attractive features further numerical experiments show that the pabb method with the adaptive line search is the best bb like method in the positive definite case and it compares reasonably well against the gpcg algorithm of more and toraldo in the indefinite case the pbb method with the adaptive line search is shown on some examples to find local minima with better solution values and hence may be preferred for this reason

evaluation of the watson integral and associated logarithmic integral for the d dimensional hypercubic lattice the watson integral for the d dimensional hypercubic lattice and the associated logarithmic integral are investigated in particular a new method is developed which enables one to calculate the numerical values of ld d 1 2 and wd d 3 4 with extremely high precision the asymptotic behaviour of ld and wd as d is also determined finally some generalizations of the results are briefly discussed

on the exact evaluation of the face centred cubic lattice green function the mathematical properties of the lattice green function open image in new window are investigated where w w 1 iw 2 lies in a complex plane which is cut from w 1 to w 3 and l 1 l 2 l 3 is a set of integers with l 1 l 2 l 3 equal to an even integer in particular it is proved that g 2n 0 0 w where n 0 1 2 is a solution of a fourth order linear differential equation of the fuchsian type with four regular singular points at w 1 0 3 and it is also shown that g 2n 0 0 w satisfies a five term recurrence relation with respect to the integer variable n the limiting function g 2n 0 0 w 1 equiv lim epsilon rightarrow0 g 2n 0 0 w 1 mathrm i epsilon g mathrm r 2n 0 0 w 1 mathrm i g mathrm i 2n 0 0 w 1 nonumber where w 1 1 3 is evaluated exactly in terms of 2 f 1 hypergeometric functions and the special cases g 2n 0 0 0 g 2n 0 0 1 and g 2n 0 0 3 are analysed using singular value theory more generally it is demonstrated that g l 1 l 2 l 3 w can be written in the form open image in new window where open image in new window are rational functions of the variable k k and e k are complete elliptic integrals of the first and second kind respectively with k 2 equiv k 2 w 1 over2 2 over w biggl 1 1 over w biggr 3 over2 1 over2 biggl 1 1 over w biggr biggl 1 1 over w biggr 3 over2 biggl 1 3 over w biggr 1 over2 nonumber and the parameter is defined as xi equiv xi w biggl 1 sqrt 1 3 over w biggr 1 biggl 1 sqrt 1 1 over w biggr nonumber this result is valid for all values of w which lie in the cut plane the asymptotic behaviour of g 2n 0 0 w 1 and g 2n 0 0 w 1 as n is also determined in the final section of the paper a new 2 f 1 product form for the anisotropic face centred cubic lattice green function is given