approximate solutions to the turbine balancing problem abstract the turbine balancing problem tbp is an np hard combinatorial optimization problem arising in the manufacturing and maintenance of turbine engines exact solution methods for solving the tbp are not appropriate since the problem has to be solved in real time and the input data is itself inaccurate in this paper the tbp is formulated as a quadratic assignment problem qap and we propose a heuristic algorithm for solving the resulting problem computational results on a set of instances provided by pratt whitney p w and from the literature indicate that the proposed algorithm outperforms the current methods used for solving the tbp and has the best overall performance with respect to other heuristic algorithms in the literature

regularized integral equation methods for elastic scattering problems in three dimensions abstract this paper presents novel methodologies for the numerical simulation of scattering of elastic waves by both closed and open surfaces in three dimensional space the proposed approach utilizes new integral formulations as well as an extension to the elastic context of the efficient high order singular integration methods 13 introduced recently for the acoustic case in order to obtain formulations leading to iterative solvers gmres which converge in small numbers of iterations we investigate theoretically and computationally the character of the spectra of various operators associated with the elastic wave calderon relation including some of their possible compositions and combinations in particular by relying on the fact that the eigenvalues of the composite operator ns are bounded away from zero and infinity new uniquely solvable low gmres iteration integral formulation for the closed surface case are presented the introduction of corresponding low gmres iteration equations for the open surface equations additionally requires for both spectral quality as well as accuracy and efficiency use of weighted versions of the classical integral operators to match the singularity of the unknown density at edges several numerical examples demonstrate the accuracy and efficiency of the proposed methodology

integer programming models for the multidimensional assignment problem with star costs abstract we consider a variant of the multidimensional assignment problem map with decomposable costs in which the resulting optimal assignment is described as a set of disjoint stars this problem arises in the context of multi sensor multi target tracking problems where a set of measurements obtained from a collection of sensors must be associated to a set of different targets to solve this problem we study two different formulations first we introduce a continuous nonlinear program and its linearization along with additional valid inequalities that improve the lower bounds second we state the standard map formulation as a set partitioning problem and solve it via branch and price these approaches were put to test by solving instances ranging from tripartite to 20 partite graphs of 4 to 30 nodes per partition computational results show that our approaches are a viable option to solve this problem a comparative study is presented

boundary integral equation methods for the elastic and thermoelastic waves in three dimensions in this paper we consider the boundary integral equation bie methods for solving the exterior neumann boundary value problems of elastic and thermoelastic waves in three dimensions based on the fredholm integral equations of the first kind the innovative contribution of this work lies in the proposal of the new regularized formulations for the hyper singular boundary integral operators bio associated with the time harmonic elastic and thermoelastic wave equations with the help of the new regularized formulations we only need to compute the integrals with weak singularities at most in the corresponding variational forms of the boundary integral equations the accuracy of the regularized formulations is demonstrated through numerical examples using the galerkin boundary element method bem

polygonal figures and their form spaces ii angle permutations in equilateral polygons abstract in a closed nonregular equilateral polygonal curve angles cannot be permuted arbitrarily without opening the curve here we investigate the possible angle permutations that leave a polygonal curve closed to this end we introduce angle sequences i e symbol sequences describing the number and arrangement of like angle values in classes of forms we find that there exist universal angle sequences giving closed polygons for arbitrary angle values provided that the angle sum is kept constant at one of certain permissible values the universal angle sequences are periodic they consist of at least two identical series of symbols within a series the symbols can be permuted freely on condition that in all other series the very same permutation is applied all these correlated permutations when applied to the angles of a corresponding polygonal curve leave this curve closed polygons belonging to a given angle sequence form class constitute a submanifold of the manifold m i of equilateral polygons within the form space these submanifolds structure m i in a hierarchical manner so that the form class concept gives a valuable complementary classification scheme for the forms of m i furthermore we find that polygons with constant angle composition the term composition is used here in its chemical sense show a wide variety of possible shapes from quasi circular to wormlike and even self intersecting ones corresponding to different degrees of segregation of like angles within the polygonal curve

on deriving a locally corrected nystrom scheme from a quadrature sampled moment method a novel high order method of moment procedure with quadrature point based discretization is presented the scheme is equivalent to a moment method employing smooth basis and testing functions applying a fixed point numerical quadrature approximation for the outer integral mapping the current to the quadrature points then leads to a formulation that is equivalent to that derived via the locally corrected nystrom method the convergence properties of the current density and rcs for smooth and singular geometries in two and three dimensions are also studied

an accurate boundary element method for the exterior elastic scattering problem in two dimensions this paper is concerned with a galerkin boundary element method solving the two dimensional exterior elastic wave scattering problem the original problem is first reduced to the so called burtonmiller 1 boundary integral formulation and essential mathematical features of its variational form are discussed in numerical implementations a newly derived and analytically accurate regularization formula 2 is employed for the numerical evaluation of hyper singular boundary integral operator a new computational approach is employed based on the series expansions of hankel functions for the computation of weakly singular boundary integral operators during the reduction of corresponding galerkin equations into a discrete linear system the effectiveness of proposed numerical methods is demonstrated using several numerical examples a new and accurate regularization formula which replaces the weak form of hypersingular boundary integral operator by a coupling of several weakly singular boundary integrals is given based on the series expansions of hankel functions several special integrals all weakly singular boundary integrals are evaluated exactly

frequency response of elastic bodies of revolution by the boundary element method a boundary element method bem is developed to analyze the elastodynamic frequency response of bodies of revolution the method computes both surface and internal variables for bilaterally symmetric geometries and boundary conditions discretization may be limited to one half of the generator of the body quadratic elements are employed for accuracy prior to numerical integration by gaussian quadrature singularities in the integral equation are removed analytically to validate the model results from numerical examples of a solid sphere a hollow sphere and a thick disk are compared to analytical and other bem solutions

a global solution to sparse correspondence problems we propose a new methodology for reliably solving the correspondence problem between sparse sets of points of two or more images this is a key step inmost problems of computer vision and so far no general method exists to solve it our methodology is able to handle most of the commonly used assumptions in a unique formulation independent of the domain of application and type of features it performs correspondence and outlier rejection in a single step and achieves global optimality with feasible computation feature selection and correspondence are first formulated as an integer optimization problem this is a blunt formulation which considers the whole combinatorial space of possible point selections and correspondences to find its global optimal solution we build a concave objective function and relax the search domain into its convex hull the special structure of this extended problem assures its equivalence to the original one but it can be optimally solved by efficient algorithms that avoid combinatorial search this methodology can use any criterion provided it can be translated into cost functions with continuous second derivatives

deformations isosymmetric manifolds and higher dimensional form space symmetries for point ensembles polygonal forms under o 2 symmetry i two and three points abstract the analysis of polygonal forms and their form transitions using normal deformations 1 has been extended to a global analysis of form spaces for polygonal forms specified by n points in a plane coincidences of the points are explicitly allowed and the origin of the form space is taken to correspond to a coincidence of all n points this choice is natural since for every form there exists a totally symmetric deformation leading to exactly this configuration the form space f for n points in a plane is a 2n dimensional euclidean space whose points represent all possible n vertex polygons simple and self intersecting ones by fixing the centres of gravity of the forms in the origin of the x y plane the dimension of the form space reduces to 2n 2 to give a reduced form space f within the form spaces isosymmetric manifolds geometric loci of all forms having the same symmetry g are determined these manifolds inscribed in a form space f define the corresponding symmetry space s the symmetry g s of the reduced symmetry space for three points has been determined in part the four dimensional symmetry group 21 04 has been found to be a subgroup of the full noncrystallographic symmetry of s for some cases the relation g f g s has been exemplified

a grasp for the biquadratic assignment problem the biquadratic assignment problem biqap is a generalization of the quadratic assignment problem qap it is a nonlinear integer programming problem where the objective function is a fourth degree multivariable polynomial and the feasible domain is the assignment polytope biqap problems appear in vlsi synthesis due to the difficulty of this problem only heuristic solution approaches have been proposed in this paper we propose a new heuristic for the biqap a greedy randomized adaptive search procedure grasp computational results on instances described in the literature indicate that this procedure consistently finds better solutions than previously described algorithms

on new classes of nonnegative symmetric tensors in this paper we introduce three new classes of nonnegative forms or equivalently symmetric tensors and their extensions the newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors order six or above for the special case of quartic forms they collapse into the set of convex quartic homogeneous polynomial functions we discuss the properties and applications of the new classes of nonnegative symmetric tensors in the context of polynomial and tensor optimization numerical experiments for solving certain polynomial optimization models based on the new classes of nonnegative symmetric tensors are presented

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

isogeometric indirect boundary element method for solving the 3d acoustic problems abstract this paper provides an isogeometric indirect boundary element method igibem based on nurbs non uniform rational b splines and pht splines polynomial splines over hierarchical t meshes for analyzing the three dimensional 3d acoustic problems in the classical procedure the geometries are discretized by lagrange polynomials elements which leads to both substantial geometrical error and time consuming meshing steps however these deficiencies can be eliminated by the isogeometric analysis iga directly incorporating the geometry description generated from the cad computer aided design software into cae computer aided engineering analysis unlike the dbem direct boundary element method the ibem indirect boundary element method allows different types of boundary conditions on the two sides of a surface moreover the hypersingular integrals in ibem can be transformed to a weakly singular form in addition the pht based ibem is used to investigate the influence of local refinement on the accuracy of solutions finally the non uniqueness problem is solved which is a fatal defect in the acoustic bem for the exterior problem four different methods to handle the non uniqueness problem are discussed and compared the results obtained by the proposed method were compared with analytical solutions and the results computed by lagrange based ibem several benchmark examples demonstrate 1 the present method i e igibem has super accuracy over conventional ibem for the acoustic problems 2 local refinement has a significant influence on the convergence rate of the solutions and the numerical accuracy is relevant to the distance to the boundary where the local refinement acted 3 as for the non uniqueness problem the imposition of specific interior boundary conditions not only obtains the best calculation result over the entire range of frequencies but also has a simple integral formulation

heuristics for biquadratic assignment problems and their computational comparison abstract the biquadratic assignment problem biqap is a generalization of the quadratic assignment problem qap as for any hard optimization problem also for biqap a reasonable effort to cope with the problem is trying to derive heuristics which solve it suboptimally and which possibly yield a good trade off between the solution quality and the time and memory requirements in this paper we describe several heuristics for biqaps in particular pair exchange algorithms improvement methods and variants of simulated annealing and taboo search we implement these heuristics as c codes and analyze their performances

quadrature over a pyramid or cube of integrands with a singularity at a vertex a simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex

absorbing boundary condition for the elastic wave equation velocity stress formulation this paper describes an absorbing boundary condition for finite difference modeling of elastic wave propagation in two and three dimensions the boundary condition is particularly effective for obliquely incident waves typically quite troublesome for absorbing boundaries analytical predictions of the boundary reflection coefficients of a few percent or less for angles of incidence up to 89 are verified in example finite difference applications the algorithm is appropriate for use in a velocity stress finite difference vs fd formulation it is computationally simpler than a similar absorbing boundary given previously for the standard displacement formulation a second algorithm is presented which may be advantageous when the boundary of interest is exposed to strong evanescent waves both algorithms require that the adjacent elastic medium be locally homogeneous

nystrom method for elastic wave scattering by three dimensional obstacles nystrom method is developed to solve for boundary integral equations bie s for elastic wave scattering by three dimensional obstacles to generate the matrix equation from a bie nystrom method applies a quadrature rule to the integrations of smooth integrands over a discretized element directly and chooses the values of the unknown function at quadrature points as the system s unknowns to be solved this leads to a simple procedure to form the off diagonal entries of matrix by simply evaluating the integrands without numerical integrations for the diagonal or near diagonal entries corresponding to the integrals over a singular or near singular element where the kernels are singular or near singular we develop a systematic singularity treatment technique known as the local correction scheme based on the linear approximation of elements the scheme differs from the singularity regularization or subtraction technique used in the boundary element method bem it applies the series expansion of scalar green s function to the kernels and derives analytical solutions for the strongly singular integrals under the cauchy principal value like cpv like sense since the approach avoids the need for reformulating the bie for singularity removal in bem and solves for the somigliana s equation directly it is easy to implement and efficient in calculation numerical examples are used to demonstrate its robustness

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

an improved annealing scheme for the qap abstract recently there has been some interest in the use of simulated annealing to obtain good solutions to a number of combinatorial problems this paper reports on the use of this method applied to the quadratic assignment problem i e the assignment of inter communicating objects to locations to minimize the total cost of communication between them the result is a much improved annealing scheme for this problem which performs well on a range of examples finding improved solutions for several of the largest problems available in the literature and requiring only modest amounts of computational effort

multivariate nonnegative quadratic mappings in this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain with nonnegativity defined by a prespecified conic order in particular we consider the set cone of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone and study when it can be represented by linear matrix inequalities we also discuss the applications of the results in robust optimization especially the robust quadratic matrix inequalities and the robust linear programming models in the latter application the implementational errors of the solution are taken into account and the problem is formulated as a semidefinite program

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

the asymptotic behaviour of quadratic sum assignment problems a statistical mechanics approach abstract in this paper we apply the statistical mechanics formalism to study quadratic sum assignment problems qsap this formalism allows us to exhibit in the limit of large problems the asymptotic behaviour of the optimal value of the cost function the conclusions of the study are confirmed by the results of metropolis computer simulations

moments tensors hilbert s identity and k wise uncorrelated random variables in this paper we introduce a notion to be called k wise uncorrelated random variables which is similar but not identical to the so called k wise independent random variables in the literature we show how to construct k wise uncorrelated random variables by a simple procedure the constructed random variables can be applied e g to express the quartic polynomial xtqx 2 where q is an n n positive semidefinite matrix by a sum of fourth powered polynomial terms known as hilbert s identity by virtue of the proposed construction the number of required terms is no more than 2n4 n this implies that it is possible to find a 2n4 n point distribution whose fourth moments tensor is exactly the symmetrization of q q moreover we prove that the number of required fourth powered polynomial terms to express xtqx 2 is at least n n 1 2 the result is applied to prove that computing the matrix 2 4 norm is np hard extensions of the results to complex random variables are discussed as well

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

isosymmetric manifolds in form spaces and the normal deformations of polygonal forms abstract form variations are described in an appropriately constructed form space f typically an r n where every point of f represents a different form regarding the symmetries of the forms f can be divided into disjunct isosymmetric manifolds i e points lines surfaces and volumes whose points correspond to forms with equal symmetries these manifolds are derived from a symmetry analysis of possible deformations of the forms this analysis is comparable to the construction of symmetry coordinates in a normal coordinate analysis of molecules and results in normal modes of deformation normal deformations of these forms from the symmetry species of a normal deformation the symmetry of the resulting form can be inferred transformation of the form space coordinates into normal coordinates the differentials of which are the normal deformations facilitates the description of the high dimensional form spaces and can be made the basis of an easy symmetry diagnosis of forms furthermore the problem of an ascent in symmetry by deformation is discussed

a thermodynamically motivated simulation procedure for combinatorial optimization problems abstract exchange algorithms are an important class of heuristics for hard combinatorial optimization problems as e g salesman problems or quadratic assignment problems in kirkpatrick s and cerny s exchange algorithms for the travelling salesman problem and placement problem they propose to perform an exchange not only if the objective function value decreases by this exchange but also in certain cases if the objective function value increases an exchange increasing the objective function value is performed stochastically depending on the size of the increment computational tests with quadratic assignment problems revealed an excellent behaviour in such an approach suboptimal solutions differing 1 2 from the best known solutions are obtained by a simple program in short time by starting this program several times with different starting values all known minimal objective function values were reached thus this approach is well suited also for smaller computers and leads in short time to acceptable solutions

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

polygonal figures and their form spaces i the structure of isodiastemic manifolds manifolds of equilateral polygons abstract the geometric structure of isodiastemic manifolds i e manifolds on which the equilaterality of polygons is preserved within a form space f spanned by n 2 consecutive vertex angles i internal coordinates of plane n gons is reported for n 5 and n 6 the curved isodiastemic manifolds are almost everywhere locally n 3 dimensional exceptions are singular points linear forms for even n where the local dimension is n 2 the isodiastemic manifolds for n 5 and n 6 are subdivided into five n 3 dimensional submanifolds of nondegenerate forms forms without coincident points each comprising forms with the same angle sum called a form family the submanifolds are bounded by n 4 dimensional dividing manifolds of degenerate forms which are structured hierarchically according to the types of degeneracy of their forms for n 6 the boundary polyhedra for two submanifolds are described in detail the manifold of spearhead shaped forms is equivalent to an octahedron whereas the hexagon manifold is equivalent to a special icosihexahedron 1

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

hypersingular boundary integral equations for radiation and scattering of elastic waves in three dimensions abstract a weakly singular form of the hypersingular boundary integral equation bie traction equation for 3 d elastic wave problems is developed in this paper all integrals involved are at most weakly singular and except for a stronger smoothness requirement on boundary elements regular quadrature and collocation procedures used for conventional bies are sufficient for the discretization of the original hypersingular bie this weakly singular form of the hypersingular bie is applied to the composite bie formulation which uses a linear combination of the conventional bie and the hypersingular bie to remove the fictitious eigenfrequencies existing in the conventional bie formulation for elastic wave problems numerical examples employing different types of boundary elements clearly demonstrate the effectiveness and efficiency of the developed formulation

deformations isosymmetric manifolds and higher dimensional form space symmetries for point ensembles polygonal forms under o 2 symmetry iii motifs in geometric figures and constraints in form spaces a polygonal form f specified by a set of n points in en here a plane can be represented by a point in an n n dimensional form space f a motif m within such a form has been defined as a subset m f on which certain constraints are imposed different types of constraints are discussed in any case every independent constraint reduces the dimension of the form space by one so that the resulting forms correspond to points in a constrained form space fc f the subspaces fc are investigated for some forms containing rigid and as a novel aspect deformable motifs and they are correlated with the full form spaces f described in previous papers 1 3 furthermore constraints which do not lead to motifs are treated generally constraints of distances define circles in certain planes of f so that angular coordinates have to be used as a special case cyclic polygons have been found to lie on hyperspheres in f on the other hand if the distances between points are constrained to such an extent that only two degrees of freedom the global rotation and one degree of shape variation are left then the resulting forms have been found to lie on hyperellipsoids in f these constitute special cases of isodiastemic manifolds consisting of forms with fixed point distances

a boundary integral equation method for radiation and scattering of elastic waves in three dimensions etude du rayonnement et de la diffusion d ondes elastiques par des obstacles de forme arbitraire

exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem we consider semidefinite programming relaxations of the quadratic assignment problem and show how to exploit group symmetry in the problem data thus we are able to compute the best known lower bounds for several instances of quadratic assignment problems from the problem library burkard et al in j global optim 10 291 403 1997

random assignment problems abstract analysis of random instances of optimization problems provides valuable insights into the behavior and properties of problem s solutions feasible region and optimal values especially in large scale cases a class of problems that have been studied extensively in the literature using the methods of probabilistic analysis is represented by the assignment problems and many important problems in operations research and computer science can be formulated as assignment problems this paper presents an overview of the recent results and developments in the area of probabilistic assignment problems including the linear and multidimensional assignment problems quadratic assignment problem etc

balancing hydraulic turbine runners a quadratic assignment problem abstract hydraulic turbine runners are used in electricity generation these consist of a cylinder around which are welded at regular spacings a number of blades whose weights differ slightly it is desired to locate the blades around the cylinder in order to minimize the distance between the center of mass of the blades and the geometric center of the cylinder the problem can be formulated as a quadratic assignment problem it is solved by adapting an interchange algorithm devised by or for the travelling salesman problem computational results are reported

multilevel fast multipole algorithm for elastic wave scattering by large three dimensional objects multilevel fast multipole algorithm mlfma is developed for solving elastic wave scattering by large three dimensional 3d objects since the governing set of boundary integral equations bie for the problem includes both compressional and shear waves with different wave numbers in one medium the double tree structure for each medium is used in the mlfma implementation when both the object and surrounding media are elastic four wave numbers in total and thus four fma trees are involved we employ nystrom method to discretize the bie and generate the corresponding matrix equation the mlfma is used to accelerate the solution process by reducing the complexity of matrix vector product from o n 2 to o nlogn in iterative solvers the multiple tree structure differs from the single tree frame in electromagnetics em and acoustics and greatly complicates the mlfma implementation due to the different definitions for well separated groups in different fma trees our nystrom method has made use of the cancellation of leading terms in the series expansion of integral kernels to handle hyper singularities in near terms this feature is kept in the mlfma by seeking the common near patches in different fma trees and treating the involved near terms synergistically due to the high cost of the multiple tree structure our numerical examples show that we can only solve the elastic wave scattering problems with 0 3 0 4 millions of unknowns on our dell precision 690 workstation using one core

approximation algorithms for homogeneous polynomial optimization with quadratic constraints in this paper we consider approximation algorithms for optimizing a generic multi variate homogeneous polynomial function subject to homogeneous quadratic constraints such optimization models have wide applications e g in signal processing magnetic resonance imaging mri data training approximation theory and portfolio selection since polynomial functions are non convex the problems under consideration are all np hard in general in this paper we shall focus on polynomial time approximation algorithms in particular we first study optimization of a multi linear tensor function over the cartesian product of spheres we shall propose approximation algorithms for such problem and derive worst case performance ratios which are shown to be dependent only on the dimensions of the model the methods are then extended to optimize a generic multi variate homogeneous polynomial function with spherical constraint likewise approximation algorithms are proposed with provable approximation performance ratios furthermore the constraint set is relaxed to be an intersection of co centered ellipsoids namely we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin and propose polynomial time approximation algorithms with provable worst case performance ratios numerical results are reported illustrating the effectiveness of the approximation algorithms studied