an extended boundary element method formulation for the direct calculation of the stress intensity factors in fully anisotropic materials copyright 2016 john wiley sons ltd we propose a formulation for linear elastic fracture mechanics in which the stress intensity factors are found directly from the solution vector of an extended boundary element method formulation the enrichment is embedded in the boundary element method formulation rather than adding new degrees of freedom for each enriched node therefore a very limited number of new degrees of freedom is added to the problem which contributes to preserving the conditioning of the linear system of equations the stroh formalism is used to provide boundary element method fundamental solutions for any degree of anisotropy and these are used for both conventional and enriched degrees of freedom several numerical examples are shown with benchmark solutions to validate the proposed method copyright 2016 john wiley sons ltd

imposing dirichlet boundary conditions in the extended finite element method this paper is devoted to the imposition of dirichlet type conditions within the extended finite element method x fem this method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces imposing neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh observed numerically in earlier papers on the contrary much less work has been devoted to dirichlet boundary conditions for the x fem or the limiting case of stiff boundary conditions in this paper we introduce a strategy to impose dirichlet boundary conditions while preserving the optimal rate of convergence the key aspect is the construction of the correct lagrange multiplier space on the boundary as an application we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding rtm process while avoiding remeshing the case of inner conditions is also discussed as well as two important practical cases material interfaces and phase transformation front capturing copyright 2006 john wiley sons ltd

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

invertible incline matrices and cramer s rule over inclines inclines are the additively idempotent semirings in which the products are less than or equal to factors thus inclines generalize boolean algebra fuzzy algebra and distributive lattice and the boolean matrices the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices i e the matrices over inclines in this paper the complete description of the invertible incline matrices is given some necessary and sufficient conditions for an incline matrix to be invertible are studied cramer s rule over inclines is presented and the group of invertible incline matrices is investigated the main results in the present paper generalize and develop the corresponding results in the literatures for the boolean matrices the fuzzy matrices and the lattice matrices

variationally consistent extended fe model for 3d planar and curved imperfect interfaces abstract we propose an extended finite element method convergent to the asymptotic solution of a thin interface problem for both planar and curved imperfect interfaces in three dimensions the main advantage over standard cohesive zone models is the bulk mesh size independence with respect to standard extended finite element method in the proposed procedure blending and quadrature sub domains are not required the focus is on the evaluation of the accuracy of the proposed approach in solving three dimensional benchmark tests the numerical results are compared with those available from analytical solutions and spring like interface models

solving the master equation without kinetic monte carlo in multiscale modeling of heterogeneous catalytic processes one crucial point is the solution of a markovian master equation describing the stochastic reaction kinetics usually this is too high dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic monte carlo method in this study we break the curse of dimensionality for the direct solution of the markovian master equation by exploiting the tensor train format for this purpose the performance of the approach is demonstrated on a first principles based reduced model for the co oxidation on the ruo2 110 surface we investigate the complexity for increasing system size and for various reaction conditions the advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness

the geometry of algorithms using hierarchical tensors in this paper the differential geometry of the novel hierarchical tucker format for tensors is derived the set ht k of tensors with fixed tree t and hierarchical rank k is shown to be a smooth quotient manifold namely the set of orbits of a lie group action corresponding to the non unique basis representation of these hierarchical tensors explicit characterizations of the quotient manifold its tangent space and the tangent space of ht k are derived suitable for high dimensional problems the usefulness of a complete geometric description is demonstrated by two typical applications first new convergence results for the nonlinear gauss seidel method on ht k are given notably and in contrast to earlier works on this subject the task of minimizing the rayleigh quotient is also addressed second evolution equations for dynamic tensor approximation are formulated in terms of an explicit projection operator onto the tangent space of ht k in addition a numerical comparison is made between this dynamical approach and the standard one based on truncated singular value decompositions 2013 elsevier inc

indices and periods of incline matrices abstract inclines are the additively idempotent semirings in which products are less than or equal to factors thus inclines generalize boolean algebra fuzzy algebra and distributive lattice and the boolean matrices the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices i e the matrices over inclines this paper studies the power sequence of incline matrices in detail a necessary and sufficient condition for the incline matrix to have index is given and the indices of some incline matrices with indices are estimated it is proved that the period of n n incline matrix with index is a divisor of n and that the set of periods of n n incline matrices with indices is not bounded from above in the sense of a power of n for all n an equivalent condition and some sufficient conditions for the incline matrix to converge in finite steps are established the stability of the orbits of an incline matrix is considered and a theorem in fuzzy sets and systems 81 1996 227 is pointed out being false the results in the present paper include some previous results in the literatures which were obtained for the boolean matrices the fuzzy matrices and the lattice matrices among their special cases

eigenvalues and eigenvectors for matrices over distributive lattices abstract let l be a complete and completely distributive lattice a vector is said to be an eigenvector of a square matrix a over the lattice l if a for some l the elements are called the associated eigenvalues in this paper we characterize the eigenvalues and the eigenvectors and also the roots of the characteristic equation of a

inclines and incline matrices a survey inclines are additively idempotent semirings in which products are less than or equal to either factor thus they generalize boolean and fuzzy algebra we outline results obtained in a book with cao cao kim roush incline algebra and applications john wiley sons new york 1984 on algebraic structure of inclines themselves matrices over inclines topological and convergence results and some applications this survey paper represents our talk at the 10th annual ilas conference at auburn university june 2002

eigen fuzzy sets and fuzzy relations abstract when r is a fuzzy relation between the elements of a finite set x the fuzzy sets a of x such that r a a max min composition are called eigen fuzzy sets the main result of this paper is the determination of the greatest eigen fuzzy set associated with a given fuzzy relation and we give three methods illustrated by an example we then state that the greatest eigen fuzzy set associated with r the transitive closure of r is exactly the one associated with r finally we describe how to obtain all fuzzy relations keeping invariant a given fuzzy set

eigen fuzzy number sets in this paper we extend certain results of sanchez resolution of eigen fuzzy sets equations fuzzy sets and systems 1 1978 69 75 to the context of fuzzy numbers a slight modification of the dubois prade definition of fuzzy numbers is used

on enrichment functions in the extended finite element method this paper presents mathematical derivation of enrichment functions in the extended finite element method for numerical modeling of strong and weak discontinuities the proposed approach consists in combining the level set method with characteristic functions as well as domain decomposition and reproduction technique we start with the simple case of a triangular linear element cut by one interface across which displacement field suffers a jump the main steps towards the derivation of enrichment functions are as follows 1 extension of the subfields separated by the interface to the whole element domain and definition of complementary nodal variables 2 construction of characteristic functions for describing the geometry and physical field 3 determination of the sets of basic nodal variables 4 domain decompositions according to step 3 and then reproduction of the physical field in terms of characteristic functions and nodal variables and 5 comparison of the piecewise interpolations formulated at steps 3 and 4 with the standard extended finite element method form which yields enrichment functions in this process the physical meanings of both the basic and complementary nodal variables are clarified which helps to impose dirichlet boundary conditions enrichment functions for weak discontinuities are constructed from deeper insights into the structure of the functions for strong discontinuities relationships between the two classes of functions are naturally established improvements upon basic enrichment functions for weak discontinuities are performed so as to achieve satisfactory convergence and accuracy from numerical viewpoints a simple and efficient treatment on the issue of blending elements is also proposed with implementation details for validation purposes applications of the derived functions to heterogeneous problems with imperfect interfaces are presented

an improved stable xfem is xfem with a novel enrichment function for the computational modeling of cohesive cracks abstract owing to the solution space enriched with a priori known knowledge through the partition of unity the extended or generalized finite element method xfem gfem has been widely applied to the modeling of localized failure in solids as far as cohesive cracks are concerned in this work on the one hand the xfem enriched with heaviside function or the shifted one with respect to nodal values is of optimal accuracy but is prone to the issue of ill conditioned or even singular system matrix for arbitrary crack propagation on the other hand the stable xfem s xfem circumvents the ill conditioning issues by introducing the residual heaviside function with respect to its linear interpolant at the cost of losing accuracy e g spurious stress locking for discontinuities with non uniform displacement jumps aiming to reconcile the incompatibility between accuracy of the solution and conditioning of the system matrix this work addresses a simple yet effective improved stable xfem is xfem with a novel enrichment function both the xfem and s xfem are recovered as its particular cases with different approximation schemes for the bridging scale which incorporates information from both the continuous coarse scale and the discontinuous fine scale in the proposed is xfem as the enrichment basis functions are not contained in the standard polynomial space conditioning of the resulting system matrix is nearly insensitive to the mesh discontinuity configuration even if the crack path gets close to element nodes furthermore the is xfem with a small stabilization parameter is almost of the same accuracy as the xfem particularly pathological numerical results polluted by spurious stress locking exhibited in the s xfem are not observed across a fully softened discontinuity with non uniform displacement jumps as only the enrichment function is modified the proposed is xfem together with appropriate crack propagation criterion and tracking algorithm can be implemented easily in the existing xfem codes several representative numerical simulations of element and structure benchmark tests are presented to validate the proposed is xfem regarding both accuracy of the solution and conditioning of the system matrix

solution of linear systems and matrix inversion in the tt format tensors arise naturally in high dimensional problems in chemistry financial mathematics and many other areas the numerical treatment of such problems is difficult due to the curse of dimensionality the number of unknowns and the computational complexity grow exponentially with the dimension of the problem to break the curse of dimensionality low parametric representations or formats have to be used in this paper we make use of the tt format tensor train format which is one of the most effective stable representations of high dimensional tensors basic linear algebra operations in the tt format are now well developed our goal is to provide a black box type of solver for linear systems where both the matrix and the right hand side are in the tt format an efficient dmrg density matrix renormalization group method is proposed and several tricks are employed to make it work the numerical experiments confirm the effectiveness of our approach

higher order extended finite elements with harmonic enrichment functions for complex crack problems summary in this paper we analyze complex crack problems in elastic media using harmonic enrichment functions in a higher order extended finite element implementation the numerically computed enrichment function of a crack is the solution of the laplace equation with discontinuous dirichlet boundary condition along the crack and its interaction with branches or other cracks is realized by imposing vanishing neumann boundary conditions along those cracks the classical finite element displacement approximation is enriched by adding the enrichment function of a crack through the framework of partition of unity a nested subgrid mesh is used in the laplace solve with a rasterized approximation of a crack which simplifies the numerical integration no partitioning of finite elements is required harmonic enrichment functions readily permit the extension to handle multiple interacting and branched cracks without any special treatment around the junction points several numerical examples are presented that arm the accuracy and eectiveness of the method when applied to complex crack

a tensor train accelerated solver for integral equations in complex geometries abstract we present a framework using the quantized tensor train qtt decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions we describe how the qtt decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations for a broad range of problems computational and storage costs of the inversion scheme are extremely modest o log n and once the inverse is computed it can be applied in o n log n we analyze the qtt ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as fmm and hss we prove that the qtt ranks are bounded for translation invariant systems and argue that this behavior extends to non translation invariant volume and boundary integrals for volume integrals the qtt decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers we present results demonstrating the remarkable performance of the qtt based solver when applied to both translation and non translation invariant volume integrals in 3d for boundary integral equations we demonstrate that using a qtt decomposition to construct preconditioners for a krylov subspace method leads to an efficient and robust solver with a small memory footprint we test the qtt preconditioners in the iterative solution of an exterior elliptic boundary value problem laplace formulated as a boundary integral equation in complex multiply connected geometries

an application of incline matrices in dynamic analysis of generalized fuzzy bidirectional associative memories this paper studies an application of the incline matrix theory in the dynamic analysis of incline valued fuzzy bidirectional associative memories l fbams for short it is proved that the strong convergence and the strong stability of an l fbam are equivalent to the existence of indices and the convergence in finite steps of the product matrices of connection incline matrices of the l fbam respectively it is shown that the convergence index the period of limit cycles the stable states and equilibria of a strongly convergent l fbam are expressed by the indices the periods and the standard eigenvectors of the product matrices of connection incline matrices of the l fbam respectively

generalizing the column row matrix decomposition to multi way arrays abstract in this paper we provide two generalizations of the cur matrix decomposition y cur also known as pseudo skeleton approximation method 1 to the case of n way arrays tensors these generalizations which we called fiber sampling tensor decomposition types 1 and 2 fstd1 and fstd2 provide explicit formulas for the parameters of a rank r r r tucker representation the core tensor of size r r r and the matrix factors of sizes i n r n 1 2 n based only on some selected entries of the original tensor fstd1 uses p n 1 p r n mode fibers of the original tensor while fstd2 uses exactly r fibers in each mode as matrix factors as suggested by the existence theorem provided in oseledets et al 2008 2 with a core tensor defined in terms of the entries of a subtensor of size r r r for n 2 our results are reduced to the already known cur matrix decomposition where the core matrix is defined as the inverse of the intersection submatrix i e u w 1 additionally we provide an adaptive type algorithm for the selection of proper fibers in the fstd1 model which is useful for large scale applications several numerical results are presented showing the performance of our fstd1 adaptive algorithm compared to two recently proposed approximation methods for 3 way tensors

a corrected xfem approximation without problems in blending elements the extended finite element method xfem enables local enrichments of approximation spaces standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured in elements that blend the enriched areas with the rest of the domain problems arise in general these blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate a modification of the xfem approximation is proposed in this work the enrichment functions are modified such that they are zero in the standard elements unchanged in the elements with all their nodes being enriched and varying continuously in the blending elements all nodes in the blending elements are enriched the modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements the corrected xfem is applied to problems in linear elasticity and optimal convergence rates are achieved copyright 2007 john wiley sons ltd

tt cross approximation for multidimensional arrays abstract as is well known a rank r matrix can be recovered from a cross of r linearly independent columns and rows and an arbitrary matrix can be interpolated on the cross entries other entries by this cross or pseudo skeleton approximation are given with errors depending on the closeness of the matrix to a rank r matrix and as well on the choice of cross in this paper we extend this construction to d dimensional arrays tensors and suggest a new interpolation formula in which a d dimensional array is interpolated on the entries of some tt cross tensor train cross the total number of entries and the complexity of our interpolation algorithm depend on d linearly so the approach does not suffer from the curse of dimensionality we also propose a tt cross method for computation of d dimensional integrals and apply it to some examples with dimensionality in the range from d 100 up to d 4000 and the relative accuracy of order 10 10 in all constructions we capitalize on the new tensor decomposition in the form of tensor trains tt decomposition

a computational approach to handle complex microstructure geometries in multiscale analysis of components there is usually a need to solve microstructures with complex geometries in this paper we use the extended finite element method x fem to solve scales involving complex geometries the x fem allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach for material interfaces this is achieved by introducing a new enrichment strategy although the mesh does not need to conform to the physical surfaces it needs to be fine enough to capture the geometry of these surfaces a simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement numerical experiments on the periodic homogenization of two phase complex cells demonstrate the accuracy and simplicity of the x fem c 2003 elsevier b v all rights reserved

on strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method we consider a problem stemming from recent models of phase transitions in stimulus responsive hydrogels wherein a sharp interface separates swelled and collapsed phases extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly fit the interface geometry are emphasized attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface we show that as the reciprocal interfacial mobility vanishes it plays the role of a penalty parameter enforcing a pure dirichlet constraint eventually triggering oscillations in the interfacial velocity we also examine alternative formulations employing a lagrange multiplier to enforce this constraint it is shown that the most convenient choice of basis for the lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms suggesting a lack of stability in the discrete formulation under such conditions neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux fortunately the adaptation and use of local domain integral methodologies considerably improves the flux evaluations several example problems are presented to compare and contrast the various techniques and methods

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

numerically determined enrichment functions for the extended finite element method and applications to bi material anisotropic fracture and polycrystals strain singularities appear in many linear elasticity problems a very fine mesh has to be used in the vicinity of the singularity in order to obtain acceptable numerical solutions with the finite element method fem special enrichment functions describing this singular behavior can be used in the extended finite element method x fem to circumvent this problem these functions have to be known in advance but their analytical form is unknown in many cases li et al described a method to calculate singular strain fields at the tip of a notch numerically a slight modification of this approach makes it possible to calculate singular fields also in the interior of the structural domain we will show in numerical experiments that convergence rates can be significantly enhanced by using these approximations in the x fem the convergence rates have been compared with the ones obtained by the fem this was done for a series of problems including a polycrystalline structure copyright 2010 john wiley sons ltd

fast integration and weight function blending in the extended finite element method two issues in the extended finite element method xfem are addressed efficient numerical integration of the weak form when the enrichment function is self equilibrating and blending of the enrichment the integration is based on transforming the domain integrals in the weak form into equivalent contour integrals it is shown that the contour form is computationally more efficient than the domain form especially when the enrichment function is singular and or discontinuous a method for alleviating the errors in the blending elements is also studied in this method the enrichment function is pre multiplied by a smooth weight function with compact support to allow for a completely smooth transition between enriched and unenriched subdomains a method for blending step function enrichment with singular enrichments is described it is also shown that if the enrichment is not shifted properly the weighted enrichment is equivalent to the standard enrichment an edge dislocation and a crack problem are used to benchmark the technique the influence of the variables that parameterize the weight function is analyzed the resulting method shows both improved accuracy and optimum convergence rates and is easily implemented into existing xfem codes

a generalized finite element method for polycrystals with discontinuous grain boundaries summary we present a generalized finite element method for the analysis of polycrystals with explicit treatment of grain boundaries grain boundaries and junctions understood as loci of possible displacement discontinuity are inserted into finite elements by exploiting the partition of unity property of finite element shape functions consequently the finite element mesh does not need to conform to the polycrystal topology the formulation is outlined and a numerical example is presented to demonstrate the potential and accuracy of the approach the proposed methodology can also be used for branched and intersecting cohesive cracks and comparisons are made to a related approach int j numer meth engng 2000 48 1741 copyright 2006 john wiley sons ltd

standard eigenvectors of incline matrices abstract inclines are the additively idempotent semirings in which products are less than or equal to either factor thus inclines generalize boolean algebra fuzzy algebra and distributive lattice this paper studies the standard eigenvectors of incline matrices i e matrices over inclines in detail as the main results for an incline matrix with index and for a lattice matrix the structures of all the standard eigenvectors are given and the greatest standard eigenvector is determined also it is proved that for an incline matrix with index the set of all the standard eigenvectors of it coincides with the set of all the standard eigenvectors of its transitive closure the power convergence of an incline matrix with index is characterized in terms of the sets of standard eigenvectors the results in the present paper include some previous results for the boolean matrices the fuzzy matrices and the lattice matrices among their special cases

modeling quasi static crack growth with the extended finite element method part i computer implementation abstract the extended finite element method x fem is a numerical method for modeling strong displacement as well as weak strain discontinuities within a standard finite element framework in the x fem special functions are added to the finite element approximation using the framework of partition of unity for crack modeling in isotropic linear elasticity a discontinuous function and the two dimensional asymptotic crack tip displacement fields are used to account for the crack this enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces and hence quasi static crack propagation simulations can be carried out without remeshing in this paper we discuss some of the key issues in the x fem and describe its implementation within a general purpose finite element code the finite element program dynaflow is considered in this study and the implementation for modeling 2 d cracks in isotropic and bimaterial media is described in particular the array allocation for enriched degrees of freedom use of geometric based queries for carrying out nodal enrichment and mesh partitioning and the assembly procedure for the discrete equations are presented we place particular emphasis on the design of a computer code to enable the modeling of discontinuous phenomena within a finite element framework

t eigen fuzzy sets abstract this paper concerns the study of the solutions of sup t equations such that atr a where t is a triangular norm and r is a fuzzy relation by generalizing a result from sanchez 12 we determine the greatest t eigen fuzzy set of a fuzzy relation a t eigen fuzzy set of r being a fuzzy set a which is a solution of atr a we also study some algebraic properties for the class of the t eigen sets of a fuzzy relation and for the set of fuzzy relations leaving invariant a given fuzzy set finally properties related with the transitive closure are established

on nilpotent incline matrices inclines are additively idempotent semirings in which products are less than or equal to either factor thus they generalize boolean algebra fuzzy algebra and distributive lattice this paper studies the nilpotent incline matrices in detail it is proved that an incline matrix is nilpotent if and only if it has index and the zero vector is its unique standard eigenvector the nilpotent matrices over an incline without nilpotent elements are characterized in terms of principal minors main diagonals nilpotent indices and adjoint matrices also some properties of the reduction of nilpotent matrices over an additively residuated incline without nilpotent elements are established the results obtained here generalize the corresponding ones on fuzzy matrices and lattice matrices shown in the references

maximum order index of matrices over commutative inclines an answer to an open problem abstract this paper proves that the maximum order index of n n matrices over an arbitrary commutative incline equals n 1 2 1 this is an answer to an open problem compute the maximum order index of a member of m n l proposed by cao kim and roush in a monograph incline algebra and applications 1984 where m n l is the set of all n n matrices over an incline l

eigenvectors in bottleneck algebra abstract let b be a nonempty linearly ordered set without maximum and minimum and max min a vector x is said to be an eigenvector of a square matrix a if a x x the aim of the present paper is to characterize the eigenvectors by means of the associated graph of the matrix and to give bounds for the set of all eigenvectors we define the lower and the upper basic eigenvectors and derive efficient methods for computing them

breaking the curse of dimensionality or how to use svd in many dimensions for d dimensional tensors with possibly large d 3 an hierarchical data structure called the tree tucker format is presented as an alternative to the canonical decomposition it has asymptotically the same and often even smaller number of representation parameters and viable stability properties the approach involves a recursive construction described by a tree with the leafs corresponding to the tucker decompositions of three dimensional tensors and is based on a sequence of svds for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial spatial dimensions it is shown how this format can be applied to the problem of multidimensional convolution convincing numerical examples are given

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

an extended finite element level set method to study surface effects on the mechanical behavior and properties of nanomaterials we present a new approach based on coupling the extended finite element method xfem and level sets to study surface and interface effects on the mechanical behavior of nanostructures the coupled xfem level set approach enables a continuum solution to nanomechanical boundary value problems in which discontinuities in both strain and displacement due to surfaces and interfaces are easily handled while simultaneously accounting for critical nanoscale surface effects including surface energy stress elasticity and interface decohesion we validate the proposed approach by studying the surface stress driven relaxation of homogeneous and bi layer nanoplates as well as the contribution from the surface elasticity to the effective stiffness of nanobeams for each case we compare the numerical results with new analytical solutions that we have derived for these simple problems for the problem involving the surface stress driven relaxation of a homogeneous nanoplate we further validate the proposed approach by comparing the results with those obtained from both fully atomistic simulations and previous multiscale calculations based upon the surface cauchy born model these numerical results show that the proposed method can be used to gain critical insights into how surface effects impact the mechanical behavior and properties of homogeneous and composite nanobeams under generalized mechanical deformation copyright 2010 john wiley sons ltd

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

on the eigenproblem of matrices over distributive lattices abstract let l be a complete and completely distributive lattice a vector is said to be an eigenvector of a square matrix a over the lattice l if a for some in l the elements are called the associated eigenvalues in this paper we obtain the maximum eigenvector of a for a given eigenvalue and give some properties of the maximum matrix m in t the set of matrices with a given eigenvector and eigenvalue we also consider the structure of matrices which possess a given primitive eigenvector and show in particular that for any given in l there is a matrix namely m having as a maximal primitive eigenvector associated with the eigenvalue

low rank tensor train coefficient array estimation for tensor on tensor regression the tensor on tensor regression can predict a tensor from a tensor which generalizes most previous multilinear regression approaches including methods to predict a scalar from a tensor and a tensor from a scalar however the coefficient array could be much higher dimensional due to both high order predictors and responses in this generalized way compared with the current low candecomp parafac cp rank approximation based method the low tensor train tt approximation can further improve the stability and efficiency of the high or even ultrahigh dimensional coefficient array estimation in the proposed low tt rank coefficient array estimation for tensor on tensor regression we adopt a tt rounding procedure to obtain adaptive ranks instead of selecting ranks by experience besides an ell 2 constraint is imposed to avoid overfitting the hierarchical alternating least square is used to solve the optimization problem numerical experiments on a synthetic data set and two real life data sets demonstrate that the proposed method outperforms the state of the art methods in terms of prediction accuracy with comparable computational complexity and the proposed method is more computationally efficient when the data are high dimensional with small size in each mode

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

algebraic wavelet transform via quantics tensor train decomposition in this paper we show that recently introduced quantics tensor train qtt decomposition can be considered as an algebraic wavelet transform with adaptively determined filters the main algorithm for obtaining qtt decomposition can be reformulated as a method seeking good subspaces or good bases and considered as a parameterized transformation of an initial tensor into a sparse tensor this interpretation allows us to introduce a modification of the tensor train svd tt svd algorithm to make it work in cases where the original algorithm does not work it results in the new wavelet like transforms called wavelet tensor train wtt transform properties of wtt transforms are studied numerically and a theoretical conjecture on the number of vanishing moments is proposed it is shown that wtt transforms are orthogonal by construction and the efficiency of wtt is compared with and often outperforms daubechies wavelet transforms on certain classes of function related vectors and matrices

low rank tensor recovery via iterative hard thresholding abstract we study extensions of compressive sensing and low rank matrix recovery matrix completion to the recovery of low rank tensors of higher order from a small number of linear measurements while the theoretical understanding of low rank matrix recovery is already well developed only few contributions on the low rank tensor recovery problem are available so far in this paper we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions namely the higher order singular value decomposition hosvd the tensor train format tt and the general hierarchical tucker decomposition ht we provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank we show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format these bounds are extended to partial fourier maps combined with random sign flips of the tensor entries finally we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from gaussian random measurements tensor completion recovery of missing entries and fourier measurements for third order tensors

interface material failure modeled by the extended finite element method and level sets multi phase materials are characterized by the fact that they possess a specific heterogeneous microstructure this feature often necessitates complex formulations in order to obtain realistic mechanical behavior with macroscopic material models furthermore such complex models cause substantial difficulties when the corresponding material parameters need to be identified and their physical meaning interpreted in multiscale analyses usually the microstructure of a material point is resolved directly thus the material behavior at the material point can be modeled in an elegant natural and simple way the present contribution aims at a detailed geometric modeling of multi phase materials as well as at a local mechanical modeling of material interfaces and interfacial failure the geometry of the material distribution within the microstructure is described through level set functions the mechanical modeling of material interfaces and interfacial cracks is accomplished by the extended finite element method x fem the combination of both the level set method and the x fem allows one to model such internal features of the microstructure without the adaptation of the mesh thus expensive meshing procedures can be avoided and considerable flexibility is obtained with respect to the variation of the material design in the x fem the finite element approximation is enriched by appropriate functions through the concept of partition of unity remarkably the level set method cannot only be applied to describe the geometry of the material interfaces and cracks respectively but also to support and even to develop the corresponding enrichment function

on the construction of blending elements for local partition of unity enriched finite elements for computational efficiency partition of unity enrichments are preferably localized to the sub domains where they are needed it is shown that an appropriate construction of the elements in the blending area the regionwhere the enriched elements blend to unenriched elements is often crucial for good performance of local partition of unity enrichments an enhanced strain formulation is developed which leads to good performance the optimal rate of convergence is achieved for polynomial enrichments it is shown that a proper choice of the finite element shape functions and partition of unity shape functions also improves the accuracy and convergence the methods are illustrated by several examples the examples deal primarily with the signed distance function enrichment for treating discontinuous derivatives inside an element but other enrichments are also considered results show that both methods provide optimal rates of convergence

higher order principal component analysis for the approximation of tensors in tree based low rank formats this paper is concerned with the approximation of tensors using tree based tensor formats which are tensor networks whose graphs are dimension partition trees we consider hilbert tensor spaces of multivariate functions defined on a product set equipped with a probability measure this includes the case of multidimensional arrays corresponding to finite product sets we propose and analyse an algorithm for the construction of an approximation using only point evaluations of a multivariate function or evaluations of some entries of a multidimensional array the algorithm is a variant of higher order singular value decomposition which constructs a hierarchy of subspaces associated with the different nodes of the tree and a corresponding hierarchy of interpolation operators optimal subspaces are estimated using empirical principal component analysis of interpolations of partial random evaluations of the function the algorithm is able to provide an approximation in any tree based format with either a prescribed rank or a prescribed relative error with a number of evaluations of the order of the storage complexity of the approximation format under some assumptions on the estimation of principal components we prove that the algorithm provides either a quasi optimal approximation with a given rank or an approximation satisfying the prescribed relative error up to constants depending on the tree and the properties of interpolation operators the analysis takes into account the discretization errors for the approximation of infinite dimensional tensors for a tensor with finite and known rank in a tree based format the algorithm is able to recover the tensor in a stable way using a number of evaluations equal to the storage complexity of the representation of the tensor in this format several numerical examples illustrate the main results and the behavior of the algorithm for the approximation of high dimensional functions using hierarchical tucker or tensor train tensor formats and the approximation of univariate functions using tensorization

mesh independent modelling of cracks by using higher order shape functions a mesh independent crack modelling approach based on displacement approximation with higher order shape functions is proposed the heaviside step function based local enrichment method known as extended finite element method is modified by replacing the step function with a higher order shape functions approximation polynomial b spline approximation functions are used in the present paper an advantage of the proposed method is that its implementation only involves integration of the products of original shape functions and their derivatives and does not require modification of the integration domains a volume integral based expression is proposed to calculate the effective surface area of the crack modelled by using an approximate step function it is shown to give the actual crack surface area in the limit of the approximate step function approaching the heaviside function the convergence and accuracy of the method is illustrated in examples of transverse and oblique with respect to loading direction crack problems in rectangular plates uniaxial tension of a unidirectional composite with an open hole is considered hoop stress relaxation due to longitudinal splitting is successfully modelled by the method proposed and compared to direct modelling by using ansys software published in 2002 by john wiley sons ltd

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

on generalized fuzzy matrices with periods in this paper generalized fuzzy matrices are considered as matrices over a special type of semiring which is called path algebra some properties and characterizations for generalized fuzzy matrices with periods are established and an expression for the transitive closure of a generalized fuzzy matrix with a period as a sum of its powers is shown reduction of generalized fuzzy matrices is defined and some properties for reduction of generalized fuzzy matrices with periods are obtained also the structure of all standard eigenvectors for a generalized fuzzy matrix with a period is determined

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

on the powers of matrices over a distributive lattice abstract in this paper the index and the period for a lattice matrix are estimated some necessary and sufficent conditions for the convergence of the powers of a lattice matrix are obtained also some equivalent conditions for a nilpotent lattice matrix are discussed

the semigroup of incline hall matrices this paper investigates the semigroup of incline hall matrices an incline hall matrix means an incline matrix which is greater than or equal to an incline invertible matrix it is proved that any incline hall matrix has index some necessary and sufficient conditions for an incline hall matrix to be regular in the semigroup of incline hall matrices are given and green s relations on the such semigroup are described also the sandwich semigroup of incline hall matrices is considered the main results in the present paper generalize some previous results on boolean matrices fuzzy matrices and lattice matrices

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

fuzzy relational inequalities and equations fuzzy quasi orders closures and openings of fuzzy sets the paper deals with systems of fuzzy relation equations known in the literature as eigen fuzzy set equations as well as with systems of related fuzzy relation inequalities our main results are algorithms for computing the greatest and the smallest solutions to the considered systems of fuzzy relation inequalities and algorithms for computing the greatest solutions to the considered systems of fuzzy relation equations the systems are studied in the framework of complete residuated lattices as the structure of membership values by means of fuzzy quasi orders and closures and openings on a collection of fuzzy sets

convergence eigen fuzzy sets and stability analysis of relational matrices abstract if stable control applications are formulated with a max product composition then instability in the relational matrix under any conditions is undesirable this paper reviews the stability analysis of relational matrices combined with the max min composition 19 and then presents an analysis of the stability of relational matrices combined with the max product composition this analysis includes results defining the convergence properties of the relational matrix determination of the eigen fuzzy sets of the stable matrices and algorithmic solutions for several important cases a method is also presented to stabilize unstable relational matrices for some control applications