pseudo skeleton approximations with better accuracy estimates abstract we propose a priori accuracy estimates for low rank matrix approximations that use just a small number of the rows and columns this number is greater than the approximation rank unlike the existing methods of pseudo skeleton approximation but the estimates are more accurate than previously known ones this paper generalizes the results of 12 13

symmetric prismatic tensegrity structures part ii symmetry adapted formulations this paper presents analytical formulations for the symmetry adapted equilibrium force density and geometrical stiffness matrices for prismatic tensegrity structures with dihedral symmetries an analytical expression for the infinitesimal mechanisms is also derived

rigidity and energy suppose one holds two sticks in the form of a cross in one hand and places a rubber band in tension around the four ends when it is released it comes to rest in the shape of a convex quadrilateral in a plane it always returns to the same shape no matter how it is distorted as long as the ends do not slip and the rubber band does not break this is a very simple example of a rigid framework of the type we discuss here energy explains why a framework such as the one above is rigid the rubber band deforms in such a way as to minimize the total energy in the framework only in the final deformed shape will the framework have a minimum energy this idea of introducing energy functions is very useful it can be used to prove a key lemma that was used by cauchy 5 in 1813 to show that convex polyhedral surfaces are rigid it can explain why some and perhaps all of r buckminster fuller s tensegrity structures 10 stay up when applied to spider webs it shows why they can only take on certain geometric shapes it also can be used to prove conjecture 6 of branko gri inbaum and g c shephard in their lectures on lost mathematics 12 this conjecture says that if a framework in the shape of a convex polygon with rods sticks on the boundary and cables inside is rigid in the plane so is the framework obtained by reversing the roles of rods and cables it is interesting to compare some of these results with the opening a rm theorem of axel schur 18 see chern 6 also which is a very close smooth analogue to cauchy s lemma schur s theorem says that if a convex planar smooth arc the arm is opened that is it is moved to another position with the same length but with corresponding points having smaller curvature then the two ends are moved apart another amusing application is to show that a regular pentagon in 4 space a pentagon with all 5 sides equal and all 5 angles between the sides equal has its angles bounded between 36 and 108 a comment of o bottema in 4

on the asymptotical behavior of nonautonomous dynamical systems abstract the uniform asymptotical behavior of nonautonomous dynamical systems and their attractors is investigated in particular it is shown that parametrically inflated pullback attractors are uniformly forward attracting and also that under appropriate conditions the component sets of the pullback attractor a of a system form an almost periodic setvalued mapping t a t p when its driving system is almost periodic this together with the attraction properties of a demonstrates that almost periodic systems exhibit global almost periodic asymptotic behavior

algebraic tensegrity form finding abstract this paper concerns the form finding problem for general and symmetric tensegrity structures with shape constraints a number of different geometries are treated and several fundamental properties of tensegrity structures are identified that simplify the form finding problem the concept of a tensegrity invariance similarity transformation is defined and it is shown that tensegrity equilibrium is preserved under affine node position transformations this result provides the basis for a new tensegrity form finding tool the generality of the problem formulation makes it suitable for the automated generation of the equations and their derivatives state of the art numerical algorithms are applied to solve several example problems examples are given for tensegrity plates shell class symmetric tensegrity structures and structures generated by applying similarity transformation

kernel sections and uniform attractors of multi valued semiprocesses abstract we present the existence of kernel sections which are all compact invariant and pullback attracting of an infinite dimensional general multi valued process constructed by the set valued backward extension of multi valued semiprocesses moreover the structure of the uniform attractors of a family of multi valued semiprocesses and the uniform forward attraction of kernel sections of a family of general multi valued processes are investigated finally we explain our abstract results by considering the mixed wave systems with supercritical exponent and ordinary differential equations

on the sauer xu formula for the error in multivariate polynomial interpolation use of a new notion of multivariate divided difference leads to a short proof of a formula by sauer and xu for the error in interpolation by polynomials of total degree n in d variables at a correct point set

a monte carlo form finding method for large scale regular and irregular tensegrity structures we propose a monte carlo form finding method that employs a stochastic procedure to determine equilibrium configurations of a tensegrity structure this method does not involve complicated matrix operations or symmetry analysis works for arbitrary initial configurations and can handle large scale regular or irregular tensegrity structures with or without material geometrical constraints

effective algorithms with circulant block matrices abstract effective numerical algorithms for circulant block matrices a whose blocks are circulant are obtained the eigenvalues of such matrices are determined in terms of the eigenvalues of matrices of reduced dimension and systems of linear equations involving these matrices are solved efficiently using fast fourier transforms

global attractivity in monotone and subhomogeneous almost periodic systems abstract a global attractivity theorem is first proved for a class of skew product semiflows then this result is applied to monotone and subhomogeneous almost periodic reaction diffusion equations ordinary differential systems and delay differential equations for their global dynamics

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

on the dynamics of nonautonomous periodic general dynamical systems and differential inclusions abstract we establish the existence and stability results for periodic nonautonomous uniform forward attractors of periodic general dynamical systems set valued dynamical systems we also investigate the dynamical behavior of nonautonomous periodic differential inclusion x t f t x t on r m with only upper semi continuous right hand side by applying the abstract results firstly we show that if the system has a compact uniformly attracting set then it has a periodic nonautonomous uniform forward attractor a secondly we prove that a is robust with respect to both internal and external perturbations finally we apply the robustness result to discuss the effects of small time delays to asymptotic stability of the system

boundary element formulations for the numerical solution of two dimensional diffusion problems with variable coefficients this paper presents new formulations of the radial integration boundary integral equation ribie and the radial integration boundary integro differential equation ribide methods for the numerical solution of two dimensional diffusion problems with variable coefficients the methods use either a specially constructed parametrix levi function or the standard fundamental solution for the laplace equation to reduce the boundary value problem bvp to a boundary domain integral equation bdie or boundary domain integro differential equation bdide the radial integration method rim is then employed to convert the domain integrals arising in both bdie and bdide methods into equivalent boundary integrals the resulting formulations lead to pure boundary integral and integro differential equations with no domain integrals furthermore a subdomain decomposition technique sdbdie is proposed which leads to a sparse system of linear equations thus avoiding the need to calculate a large number of domain integrals numerical examples are presented for several simple problems for which exact solutions are available to demonstrate the efficiency of the proposed approaches

a theory of pseudoskeleton approximations abstract let an m n matrix a be approximated by a rank r matrix with an accuracy e we prove that it is possible to choose r columns and r rows of a forming a so called pseudoskeleton component which approximates a with o e r m n accuracy in the sense of the 2 norm on the way to this estimate we study the interconnection between the volume i e the determinant in the absolute value and the minimal singular value r of r r submatrices of an n r matrix with orthogonal columns we propose a lower bound better than one given by chandrasekaran and ipsen and by hong and pan for the maximum of r over all these submatrices and formulate a hypothesis on a tighter bound

on dynamical properties of general dynamical systems and differential inclusions abstract let f be a general dynamical system defined on a complete locally compact metric space x we give a slightly improved version of the lyapunov characterization of asymptotic stability in one of our previous works and provide a short self contained proof for the existence of arbitrarily small positively invariant neighborhoods of compact asymptotically stable sets in the present context based on these results we prove that the uniform attractors of f are connected and are stable with respect to perturbations under appropriate conditions we are also interested in the dynamical properties of differential inclusion x t f x t on r m first we show that if no solutions of the system blow up in finite future time then its reachable mapping f is a general dynamical system then we discuss some asymptotic stability properties of the system in particular we prove that if there exists a nonempty compact connected subset m of r m such that m attracts a neighborhood of itself then the system has a connected uniform attractor a we also prove that a is stable with respect to both internal and external perturbations more precisely we prove that when 0 is sufficiently small the perturbed system x t con f x t b 1 b 1 has a connected uniform attractor a moreover h a a 0 as 0 where h is the hausdorff distance in r m

reshaping of tensegrities using a geometrical variation approach abstract we propose a novel approach for the reshaping of tensegrities and pre stressed pin jointed frameworks both free standing and restrained frameworks are studied first a geometrical variation is introduced and considered as an initial geometrical change to a self equilibrated tensegrity a new nonlinear set of equilibrium equations is then established based on this variation and subsequently linearized using an expansion the linearization provides us with an underdetermined system of linear equations with a rectangular jacobian matrix which is derived in a very simple and compact matrix form the system is finally solved iteratively using a newton raphson method employing a moore penrose pseudo inverse operation to establish a new equilibrium state the method promises to generate very accurate results with a remarkable convergence rate the outstanding features and performance of the proposed method are demonstrated and studied comprehensively through three examples

dissipative quasi geostrophic motion under temporally almost periodic forcing the full nonlinear dissipative quasi geostrophic model is shown to have a unique temporally almost periodic solution when the wind forcing is temporally almost periodic under suitable constraints on the spatial square integral of the wind forcing and the parameter ekman number viscosity and the domain size the proof involves the pullback attractor for the associated nonautonomous dynamical system

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

a new approach to the analytical and numerical form finding of tensegrity structures we develop a new formulation for the form finding of tensegrity structures in which the primary variables are the cartesian components of element lengths both an analytical and a numerical implementation of the formulation are described each require a description of the connectivity of the tensegrity with the iterative numerical method also requiring a random starting vector of member force densities the analytical and numerical form finding of tensegrity structures is demonstrated through six examples and the results obtained are compared and contrasted with those available in the literature to verify the accuracy and viability of the suggested methods

symmetric prismatic tensegrity structures part i configuration and stability this paper presents a simple and efficient method to determine the self equilibrated configurations of prismatic tensegrity structures nodes and members of which have dihedral symmetry it is demonstrated that stability of this class of structures is not only directly related to the connectivity of members but is also sensitive to their geometry height radius ratio and is also dependent on the level of self stress and stiffness of members a catalogue of the structures with relatively small number of members is presented based on the stability investigations

communication optimal parallel and sequential qr and lu factorizations we present parallel and sequential dense qr factorization algorithms that are both optimal up to polylogarithmic factors in the amount of communication they perform and just as stable as householder qr we prove optimality by deriving new lower bounds for the number of multiplications done by non strassen like qr and using these in known communication lower bounds that are proportional to the number of multiplications we not only show that our qr algorithms attain these lower bounds up to polylogarithmic factors but that existing lapack and scalapack algorithms perform asymptotically more communication we derive analogous communication lower bounds for lu factorization and point out recent lu algorithms in the literature that attain at least some of these lower bounds the sequential and parallel qr algorithms for tall and skinny matrices lead to significant speedups in practice over some of the existing algorithms including lapack and scalapack for example up to 6 7 times over scalapack a performance model for the parallel algorithm for general rectangular matrices predicts significant speedups over scalapack

on bivariate hermite interpolation with minimal degree polynomials a newton type approach is used to deal with bivariate polynomial hermite interpolation problems when the data are distributed in the intersections of two families of straight lines as a generalization of regular grids the interpolation operator is degree reducing and the interpolation space is a minimal degree space integral remainder formulas are given for the lagrange case then extended to the hermite case and finally used to obtain error estimates

a direct approach to design of geometry and forces of tensegrity systems in the process of designing a tensegrity system some constraints are usually introduced for geometry and or forces to ensure uniqueness of the solution because the tensegrity systems are underdetermined in most cases in this paper a new approach is presented to enable designers to specify independent sets of axial forces and nodal coordinates consecutively under the equilibrium conditions and the given constraints to satisfy the distinctly different requirements of architects and structural engineers the proposed method can be used very efficiently for practical applications because only linear algebraic equations are to be solved and no equation of kinematics or material property is needed some numerical examples are given to show not only efficiency of the proposed method but also its ability of searching new configurations

topological entropy for local processes abstract the aim of this article is to formalize definition of chaos in terms of topological entropy for dynamics of processes described by nonautonomous differential equations we state a formal definition of topological entropy in this setting and provide tools for estimation of its value its upper or lower bounds in terms of poincare sections

why finite element discretizations can be factored by triangular hierarchical matrices although the asymptotic complexity of direct methods for the solution of large sparse finite element systems arising from second order elliptic partial differential operators is far from being optimal these methods are often preferred over modern iterative methods this is mainly due to their robustness in this article it is shown that an approximate lu decomposition exists which can be computed in the algebra of hierarchical matrices with almost linear complexity and with the same robustness as the classical lu decomposition low precision approximations may be used for preconditioning iterative solvers as a byproduct we prove that schur complements of stiffness matrices can be approximated with almost linear complexity

on the convergence of the aca algorithm for radiation and scattering problems the adaptive cross approximation aca algorithm when used to accelerate the numerical solution of integral equations for radiation and scattering problems sometimes suffers from inaccuracies these inaccuracies occur when the aca convergence criterion which is based on an approximation of the residual relative error is prematurely satisfied this paper identifies the two sources of this problem and proposes adaptations of the algorithm that remedy them

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

path planning for active tensegrity structures abstract this paper presents a path planning method for actuated tensegrity structures with quasi static motion the valid configurations for such structures lay on an equilibrium manifold which is implicitly defined by a set of kinematic and static constraints the exploration of this manifold is difficult with standard methods due to the lack of a global parameterization thus this paper proposes the use of techniques with roots in differential geometry to define an atlas i e a set of coordinated local parameterizations of the equilibrium manifold this atlas is exploited to define a rapidly exploring random tree which efficiently finds valid paths between configurations however these paths are typically long and jerky and therefore this paper also introduces a procedure to reduce their control effort a variety of test cases are presented to empirically evaluate the proposed method

low rank approximation of a sparse matrix based on lu factorization with column and row tournament pivoting in this paper we present an algorithm for computing a low rank approximation of a sparse matrix based on a truncated lu factorization with column and row permutations we present various approaches for determining the column and row permutations that show a trade off between speed versus deterministic probabilistic accuracy we show that if the permutations are chosen by using tournament pivoting based on qr factorization then the obtained truncated lu factorization with column row tournament pivoting lu crtp satisfies bounds on the singular values which have similarities with the ones obtained by a communication avoiding rank revealing qr factorization experiments on challenging matrices show that lu crtp provides a good low rank approximation of the input matrix and it is less expensive than the rank revealing qr factorization in terms of computational and memory usage costs while also minimizing the communication cost we also compare the computational complexity of our algorithm with randomized a

adaptive force density method for form finding problem of tensegrity structures a numerical method is presented for form finding of tensegrity structures eigenvalue analysis and spectral decomposition are carried out iteratively to find the feasible set of force densities that satisfies the requirement on rank deficiency of the equilibrium matrix with respect to the nodal coordinates the equilibrium matrix is shown to correspond to the geometrical stiffness matrix in the conventional finite element formulation a unique and non degenerate configuration of the structure can then be obtained by specifying an independent set of nodal coordinates a simple explanation is given for the required rank deficiency of the equilibrium matrix that leads to a non degenerate structure several numerical examples are presented to illustrate the robustness as well as the strong ability of searching new configurations of the proposed method

pullback attractors for a non autonomous incompressible non newtonian fluid this paper studies the pullback asymptotic behavior of solutions for a non autonomous incompressible non newtonian fluid in two dimensional 2d bounded domains we first prove the existence of pullback attractors av in space v has h2 regularity see notation in section 2 and ah in space h has l2 regularity for the cocycle corresponding to the solutions of the fluid then we verify the regularity of the pullback attractors by showing av ah which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data

application of the multiplicative regularized gauss newton algorithm for three dimensional microwave imaging we apply the so called multiplicative regularized gauss newton inversion algorithm for solving three dimensional electromagnetic microwave inverse problems this inversion algorithm automatically adjusts the regularization parameter and when combined with the total variation type regularization function it can provide inversion results with excellent edge preserving characteristics in addition in order to deal with an extensive memory requirement for the gauss newton method we employ an implicit jacobian calculation scheme by using this scheme we do not have to explicitly store the jacobian matrix hence we are able to significantly reduce the memory requirement of the gauss newton method albeit at an additional computational overhead furthermore in order to be able to handle a large scale computational problem both the forward and the inversion algorithms are parallelized using the mpi library where we obtain a nearly linear speedup factor we demonstrate efficiency and robustness of this algorithm by inverting synthetic data fresnel experimental data and biomedical experimental data

linear time cur approximation of bem matrices in this paper we propose linear time cur approximation algorithms for admissible matrices obtained from the hierarchical form of boundary element matrices we propose a new approach called geometric sampling to obtain indices of most significant rows and columns using information from the domains where the problem is posed our strategy is tailored to boundary element methods bem since it uses directly and explicitly the cluster tree containing information from the problem geometry our cur algorithm has precision comparable with low rank approximations created with the truncated qr factorization with column pivoting qrcp and the adaptive cross approximation aca with full pivoting which are quadratic cost methods when compared to the well known linear time algorithm aca with partial pivoting we show that our algorithm improves in general the convergence error and overcomes some cases where aca fails we provide a general relative error bound for cur approximations created with geometrical sampling finally we evaluate the performance of our algorithms on traditional bem problems defined over different geometries

the black box fast multipole method a new o n fast multipole formulation is proposed for non oscillatory kernels this algorithm is applicable to kernels k x y which are only known numerically that is their numerical value can be obtained for any x y this is quite different from many fast multipole methods which depend on analytical expansions of the far field behavior of k for x y large other black box or kernel independent fast multipole methods have been devised our approach has the advantage of requiring a small pre computation time even for very large systems and uses the minimal number of coefficients to represent the far field for a given l 2 tolerance error in the approximation this technique can be very useful for problems where the kernel is known analytically but is quite complicated or for kernels which are defined purely numerically

pullback attractors for the norm to weak continuous process and application to the nonautonomous reaction diffusion equations abstract in this paper we introduce the concept of norm to weak continuous process in a banach space and obtain the existence of pullback attractors for this kind of process then we give a new method for proving the existence of the pullback attractors as an application we obtain the existence of pullback attractors for nonautonomous reaction diffusion equation in h 0 1 with exponential growth of the external force

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

the h2 wavelet method abstract in the present paper we introduce the h 2 wavelet method for the fast solution of nonlocal operator equations on unstructured meshes on the given mesh we construct a wavelet basis which provides vanishing moments with respect to the traces of polynomials in the space with this basis at hand the system matrix in wavelet coordinates is compressed to o n log n relevant matrix coefficients where n denotes the number of boundary elements the compressed system matrix is computed with nearly linear complexity by using the h 2 matrix approach numerical results in three spatial dimensions validate that we succeeded in developing a fast wavelet galerkin scheme on unstructured triangular or quadrangular meshes

efficient algorithms for computing a strong rank revealing qr factorization given anm n matrixm withm n it is shown that there exists a permutation fi and an integer k such that the qr factorization myi q ak ckbk reveals the numerical rank of m the k k upper triangular matrix ak is well conditioned ilckll2 is small and bk is linearly dependent on ak with coefficients bounded by a low degree polynomial in n existing rank revealing qr rrqr algorithms are related to such factorizations and two algorithms are presented for computing them the new algorithms are nearly as efficient as qr with column pivoting for most problems and take o ran2 floating point operations in the worst case

comparison of fast boundary element methods on parametric surfaces we compare fast black box boundary element methods on parametric surfaces in r3 these are the adaptive cross approximation the multipole method based on interpolation and the wavelet galerkin scheme the surface representation by a piecewise smooth parameterization is in contrast to the common approximation of surfaces by panels nonetheless parametric surface representations are easily accessible from computer aided design cad and are recently topic of the studies in isogeometric analysis especially we can apply two dimensional interpolation in the multipole method a main feature of this approach is that the cluster bases and the respective moment matrices are independent of the geometry this results in a superior compression of the far field compared to other cluster methods

structure of the set of bounded solutions for a class of nonautonomous second order differential equations abstract some precise descriptions of the structure and behavior of bounded solutions for a class of nonautonomous second order differential equations are obtained the nonautonomous differential equations here are in the form x f t h x x where the driving system is defined in a compact base space h and the vector field f h x p is strictly increasing in state variable x on some interval a b the stability of the structure of bounded solutions with respect to parameter perturbations in f is also addressed

static and dynamic characterization of regular truncated icosahedral and dodecahedral tensegrity modules static and dynamic properties of a pair of dual spherical tensegrity modules invented by buckminster fuller are investigated they are regular truncated icosahedral and dodecahedral tensegrity modules the computation of the maxwell number and the use of calladine s relation reveal that regular truncated icosahedral and dodecahedral tensegrity modules possess 55 infinitesimal mechanism modes a reduced equilibrium matrix is presented for the initial shape finding to economically impose the existence of a pre stress mode both the initial shape and the corresponding pre stress mode are analytically obtained by using graphs of the icosahedral group and the reduced equilibrium matrix for both icosahedral and dodecahedral modules the maximum values of the cable tension is always less than the absolute value of bar compression in order to classify a large number of infinitesimal mechanism modes modal analyses are conducted infinitesimal mechanism modes have the stiffness due to pre stress and are associated with lowest natural frequencies their natural frequencies increase proportionally to the square root of the amplitude of pre stress it is found that there are only 15 distinct natural frequencies associated with the infinitesimal mechanism modes

form finding of tensegrity structures via genetic algorithm abstract we propose an efficient method for the form finding of tensegrity structures the force densities of each tensegrity are obtained by the minimisation of a particular objective function leading to a semi positive definite force density matrix a super stable tensegrity with a required rank deficiency a genetic algorithm is used as a global search technique for the minimisation the geometry of a tensegrity is subsequently formed based on those eigenvectors of the force density matrix corresponding to zero eigenvalues furthermore two other methods are introduced to convert the asymmetrical geometry obtained from the main algorithm into its symmetrical counterparts this transformation in geometry is performed by finding a suitable linear combination of the mentioned eigenvectors examples from well known tensegrities including prismatic truncated tetrahedron expandable octahedron and truncated icosahedron tensegrities are studied using the present method and the results obtained are compared with those documented in the literature to verify the efficiency of the present method

a fast bie iteration method for an arbitrary body in a flow of incompressible inviscid fluid the paper is concerned with the new iteration algorithm to solve boundary integral equations arising in boundary value problems of mathematical physics the stability of the algorithm is demonstrated on the problem of a flow around bodies placed in the incompressible inviscid fluid with a discrete numerical treatment we approximate the exact matrix by a certain toeplitz one and then apply a fast algorithm for this matrix on each iteration step we illustrate the convergence of this iteration scheme by a number of numerical examples both for hard and soft boundary conditions it appears that the method is highly efficient for hard boundaries being much less efficient for soft boundaries

a fast dual boundary element method for 3d anisotropic crack problems in the present paper a fast solver for dual boundary element analysis of 3d anisotropic crack problems is formulated implemented and tested the fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix the admissible low rank blocks are computed by adaptive cross approximation aca the performance of aca against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated focusing on the balance between the kernel representation accuracy and the accuracy required for aca the system solution is computed by a preconditioned gmres and the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format the effectiveness of the proposed technique for anisotropic crack problems has been numerically demonstrated highlighting the accuracy as well as the significant reduction in memory storage and analysis time in particular it has been numerically shown that the computational cost grows almost linearly with the number of degrees of freedom obtaining up to solution speedups of order 10 for systems of order 104 moreover the sensitivity of the performance of the numerical scheme to materials with different degrees of anisotropy has been assessed copyright 2009 john wiley sons ltd

attractors for differential equations with variable delays using the relatively new concept of a pullback attractor we present some results on the existence of attractors for differential equations with variable delay we give a variety of examples to which our result applies

inexact data sparse boundary element tearing and interconnecting methods the boundary element tearing and interconnecting beti methods have recently been introduced as boundary element counterparts of the well established finite element tearing and interconnecting feti methods in this paper we present inexact data sparse versions of the beti methods which avoid the elimination of the primal unknowns and dense matrices however instead of symmetric and positive definite systems we finally have to solve twofold saddle point problems the proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor moreover the solvers are robust with respect to large coefficient jumps

combining kronecker product approximation with discrete wavelet transforms to solve dense function related linear systems a new solution technique is proposed for linear systems with large dense matrices of a certain class including those that come from typical integral equations of potential theory this technique combines kronecker product approximation and wavelet sparsification for the kronecker product factors the user is only required to supply a procedure for computation of each entry of the given matrix the main sources of efficiency are the incomplete cross approximation procedure adapted from the mosaic skeleton method of the second author and data sparse preconditioners the incomplete lu decomposition with dynamic choice of the fill in structure with a prescribed threshold and the inverse kronecker product preconditioner constructed for the sum of kronecker products of sparsified finger like matrices computed by the discrete wavelet transform in some model but quite representative examples the new technique allowed us to solve dense systems with more than 1 million unknowns in a few minutes on a personal computer with 1 gbyte operative memory

using the tensor train approach to solve the ground state eigenproblem for hydrogen molecules we consider the born oppenheimer approximation of the schrodinger equation for two hydrogen atoms in the case of large separation distances we show that the feschbach schur perturbation method can be used to solve the problem for the difference between a given separation and infinite separation this leads to a simplified problem which can be solved iteratively we show that this iteration converges for sufficiently large separation distances and we solve the arising sequence of six dimensional pdes with a finite element method in combination with low rank tensor techniques to make the computations tractable in particular we show how the discretized problems can be represented and solved in the tensor train format since the storage and computational complexity of this format scale linearly in the dimension a very large number of grid points can be employed which leads to accurate approximations of the ground state energy and ground state wavefunction various numerical experiments show the performa

quasioptimality of maximum volume cross interpolation of tensors we consider a cross interpolation of high dimensional arrays in the tensor train format we prove that the maximum volume choice of the interpolation sets provides the quasioptimal interpolation accuracy that differs from the best possible accuracy by the factor which does not grow exponentially with dimension for nested interpolation sets we prove the interpolation property and propose greedy cross interpolation algorithms we justify the theoretical results and measure speed and accuracy of the proposed algorithm with numerical experiments

black box approximation of tensors in hierarchical tucker format we derive and analyse a scheme for the approximation of order d tensors a rn1 nd in the hierarchical h tucker format a dimension multilevel variant of the tucker format and strongly related to the tt tensor train format for a fixed rank parameter k the storage complexity of a tensor in h tucker format is odk3 k i 1dni and we present a heuristic algorithm that finds an approximation to a tensor in the h tucker format in odk4 log d k2 i 1dni by inspection of only odk3 log d k2 i 1dni entries under mild assumptions tensors in the h tucker format are reconstructed for general tensors we derive error bounds that are based on the approximability of matrices matricizations of the tensor by few outer products of its rows and columns the construction parallelizes with respect to the order d and we also propose an adaptive approach that aims at finding the rank parameter for a given target accuracy e automatically

error bound of the multilevel adaptive cross approximation mlaca an error bound of the multilevel adaptive cross approximation mlaca which is a multilevel version of the adaptive cross approximation singular value decomposition aca svd is rigorously derived for compressing an off diagonal submatrix of the method of moments mom impedance matrix with a binary tree the l level mlaca includes l 1 steps and each step includes 2 l aca svd decompositions if the relative frobenius norm error of the aca svd used in the mlaca is smaller than varepsilon the rigorous proof in this communication shows that the relative frobenius norm error of the l level mlaca is smaller than 1 varepsilon l 1 1 in practical applications the error bound of the mlaca can be approximated as varepsilon l 1 because varepsilon is always ll 1 the error upper bound can be used to control the accuracy of the mlaca to ensure an error of the l level mlaca smaller than varepsilon for different l the aca svd threshold can be set to 1 varepsilon tfrac 1 l 1 1 which approximately equals varepsilon l 1 for practical applications