efficient low rank approximation of the stochastic galerkin matrix in tensor formats in this article we describe an efficient approximation of the stochastic galerkin matrix which stems from a stationary diffusion equation the uncertain permeability coefficient is assumed to be a log normal random field with given covariance and mean functions the approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats it will be shown that under additional assumptions the approximation error depends only on the smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate hermite polynomials

fft based homogenization for microstructures discretized by linear hexahedral elements summary the fft based homogenization method of moulinec suquet has recently emerged as a powerful tool for computing the macroscopic response of complex microstructures for elastic and inelastic problems in this work we generalize the method to problems discretized by trilinear hexahedral elements on cartesian grids and physically nonlinear elasticity problems we present an implementation of the basic scheme that reduces the memory requirements by a factor of four and of the conjugate gradient scheme that reduces the storage necessary by a factor of nine compared with a naive implementation for benchmark problems in linear elasticity the solver exhibits mesh and contrast independent convergence behavior and enables the computational homogenization of complex structures for instance arising from computed tomography computed tomography ct imaging techniques there exist 3d microstructures involving pores and defects for which the original fft based homogenization scheme does not converge in contrast for the proposed scheme convergence is ensured also the solution fields are devoid of the spurious oscillations and checkerboarding artifacts associated to conventional schemes we demonstrate the power of the approach by computing the elasto plastic response of a long fiber reinforced thermoplastic material with 172 106 displacement degrees of freedom copyright 2016 john wiley sons ltd

solving stochastic systems with low rank tensor compression abstract for parametrised equations which arise for example in equations dependent on random parameters the solution naturally lives in a tensor product space the application which we have in mind is a stochastic linear elliptic partial differential equation spde usual spatial discretisation leads to a potentially large linear system for each point in the parameter space approximating the parametric dependence by a galerkin ansatz the already large number of unknowns for a fixed parameter value is multiplied by the dimension of the galerkin subspace for the parametric dependence and thus can be very large therefore we try to solve the total system approximately on a smaller submanifold which is found adaptively through compression during the solution process by alternating iteration and compression we show that for general linearly converging stationary iterative schemes and general adaptation processes which can be seen as a modification or perturbation of the iteration the interlaced procedure still converges our proposed modification can be used for most stationary solvers for systems on tensor products we demonstrate this on an example of a discretised spde with a simple preconditioned iteration we choose a truncated singular value decomposition svd for the compression and give details of the implementation finishing with some examples

combining galerkin approximation techniques with the principle of hashin and shtrikman to derive a new fft based numerical method for the homogenization of composites we report on the mathematical analysis of two different fft based numerical schemes for the homogenization of composite media within the framework of linear elasticity the basic scheme of moulinec and suquet 1994 1998 9 and 10 and the energy based scheme of brisard and dormieux 2010 13 casting these two schemes as galerkin approximations of the same variational problem allows us to assert their well posedness and convergence more importantly we extend in this work their domains of application by relieving some stringent conditions on the reference material which were previously thought necessary the origins of the flaws of each scheme are identified and a third scheme is proposed which seems to combine the strengths of the basic and energy based schemes while leaving out their weaknesses finally a rule is proposed for handling heterogeneous pixels voxels a situation frequently met when images of real materials are used as input to these schemes

the large n behaviour of the lipkin model and exceptional points the ubiquitous lipkin model is investigated for an interaction parameter beyond the traditional critical point it is argued that a phase transition occurs higher up in the spectrum for such larger interaction where using appropriate scaling of the energies the position of the phase transition becomes independent of the particle number the phase transition is related to near singularities in the complex interaction plane the exceptional points consideration of finite temperature yields the well known physical features associated with phase transitions

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

short note accelerating a fft based solver for numerical homogenization of periodic media by conjugate gradients in this short note we present a new technique to accelerate the convergence of a fft based solver for numerical homogenization of complex periodic media proposed by moulinec and suquet 1 the approach proceeds from discretization of the governing integral equation by the trigonometric collocation method due to vainikko 2 to give a linear system which can be efficiently solved by conjugate gradient methods computational experiments confirm robustness of the algorithm with respect to its internal parameters and demonstrate significant increase of the convergence rate for problems with high contrast coefficients at a low overhead per iteration

flow equations and the strong coupling expansion for the hubbard model applying the method of continuous unitary transformations to a class of hubbard models we reexamine the derivation of thet u expansion for the strong coupling case the flow equations for the coupling parameters of the higher order effective interactions can be solved exactly resulting in a systematic expansion of the hamiltonian in powers oft u valid for any lattice in arbitrary dimension and for general band filling the expansion ensures a correct treatment of the operator products generated by the transformation and only involves the explicit recursive calculation of numerical coefficients this scheme provides a unifying framework to study the strong coupling expansion for the hubbard model which clarifies and circumvents several difficulties inherent to earlier approaches our results are compared with those of other methods and it is shown that the freedom in the choice of the unitary transformation that eliminates interactions between different hubbard bands can affect the effective hamiltonian only at ordert3 u2 or higher

guaranteed upper lower bounds on homogenized properties by fft based galerkin method abstract guaranteed upper lower bounds on homogenized coefficients arising from the periodic cell problem are calculated in a scalar elliptic setting our approach builds on the recent variational reformulation of the moulinec suquet 1994 fast fourier transform fft homogenization scheme by vond ejc et al 2014 which is based on the conforming galerkin approximation with trigonometric polynomials upper lower bounds are obtained by adjusting the primal dual finite element framework developed independently by dvo ak 1993 and wi ckowski 1995 to the fft based galerkin setting we show that the discretization procedure differs for odd and non odd number of grid points thanks to the helmholtz decomposition inherited from the continuous formulation the duality structure is fully preserved for the odd discretizations in the latter case a more complex primal dual structure is observed due to presence of the trigonometric polynomials associated with the nyquist frequencies these theoretical findings are confirmed with numerical examples to conclude the main advantage of the fft based approach over conventional finite element schemes is that the primal and the dual problems are treated on the same basis and this property can be extended beyond the scalar elliptic setting

non perturbative flow equations from continuous unitary transformations we use a novel parametrization of the flowing hamiltonian to show that the flow equations based on continuous unitary transformations as proposed by wegner can be implemented through a nonlinear partial differential equation involving one flow parameter and two system specific auxiliary variables the implementation is non perturbative as the partial differential equation involves a systematic expansion in fluctuations controlled by the size of the system rather than the coupling constant the method is applied to the lipkin model to construct a mapping which maps the non interacting spectrum onto the interacting spectrum to a very high accuracy this function is universal in the sense that the full spectrum for any large number of particles can be obtained from it in a similar way expectation values for a large class of operators can be obtained which also makes it possible to probe the structure of the eigenstates

tensor structured galerkin approximation of parametric and stochastic elliptic pdes we investigate the convergence rate of approximations by finite sums of rank 1 tensors of solutions of multiparametric elliptic pdes such pdes arise for example in the parametric deterministic reformulation of elliptic pdes with random field inputs based for example on the m term truncated karhunen loeve expansion our approach could be regarded as either a class of compressed approximations of these solutions or as a new class of iterative elliptic problem solvers for high dimensional parametric elliptic pdes providing linear scaling complexity in the dimension m of the parameter space it is based on rank reduced tensor formatted separable approximations of the high dimensional tensors and matrices involved in the iterative process combined with the use of spectrally equivalent low rank tensor structured preconditioners to the parametric matrices resulting from a finite element discretization of the high dimensional parametric deterministic problems numerical illustrations for the m dimensional parametric elliptic pdes resulting from spdes on parameter spaces of dimensions m leq100 indicate the advantages of employing low rank tensor structured matrix formats in the numerical solution of such problems

a comparative study on low memory iterative solvers for fft based homogenization of periodic media in this paper we assess the performance of four iterative algorithms for solving non symmetric rank deficient linear systems arising in the fft based homogenization of heterogeneous materials defined by digital images our framework is based on the fourier galerkin method with exact and approximate integrations that has recently been shown to generalize the lippmann schwinger setting of the original work by moulinec and suquet from 1994 it follows from this variational format that the ensuing system of linear equations can be solved by general purpose iterative algorithms for symmetric positive definite systems such as the richardson the conjugate gradient and the chebyshev algorithms that are compared here to the eyre milton scheme the most efficient specialized method currently available our numerical experiments carried out for two dimensional elliptic problems reveal that the conjugate gradient algorithm is the most efficient option while the eyre milton method performs comparably to the chebyshev semi iteration the richardson algorithm equivalent to the still widely used original moulinec suquet solver exhibits the slowest convergence besides this we hope that our study highlights the potential of the well established techniques of numerical linear algebra to further increase the efficiency of fft based homogenization methods

goodness of fit diagnostics for bayesian hierarchical models this article proposes methodology for assessing goodness of fit in bayesian hierarchical models the methodology is based on comparing values of pivotal discrepancy measures pdms computed using parameter values drawn from the posterior distribution to known reference distributions because the resulting diagnostics can be calculated from standard output of markov chain monte carlo algorithms their computational costs are minimal several simulation studies are provided each of which suggests that diagnostics based on pdms have higher statistical power than comparable posterior predictive diagnostic checks in detecting model departures the proposed methodology is illustrated in a clinical application an application to discrete data is described in supplementary material

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

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

a finite element perspective on non linear fft based micromechanical simulations fourier solvers have become efficient tools to establish structure property relations in heterogeneous materials introduced as an alternative to the finite element fe method they are based on fixed point solutions of the lippmann schwinger type integral equation their computational efficiency results from handling the kernel of this equation by the fast fourier transform fft however the kernel is derived from an auxiliary homogeneous linear problem which renders the extension of fft based schemes to nonlinear problems conceptually difficult this paper aims to establish a link between fe based and fft based methods in order to develop a solver applicable to general history dependent and time dependent material models for this purpose we follow the standard steps of the fe method starting from the weak form proceeding to the galerkin discretization and the numerical quadrature up to the solution of nonlinear equilibrium equations by an iterative newton krylov solver no auxiliary linear problem is thus needed by analyzing a two phase laminate with nonlinear elastic elastoplastic and viscoplastic phases and by elastoplastic simulations of a dual phase steel microstructure we demonstrate that the solver exhibits robust convergence these results are achieved by re using the nonlinear fe technology with the potential of further extensions beyond small strain inelasticity considered in this paper

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

on the thermodynamic limit of the lipkin model the thermodynamic limit of the lipkin model is investigated while the limit turns out to be rather elusive the analysis gives strong indications that the limit yields two analytically dissociated operators one for the normal and one for the deformed phase while the lipkin hamiltonian is hermitian and has a second order phase transition in finite dimensions finite particle number both properties seem to be destroyed in the thermodynamic limit

fft based homogenisation accelerated by low rank tensor approximations abstract fast fourier transform fft based methods have turned out to be an effective computational approach for numerical homogenisation in particular fourier galerkin methods are computational methods for partial differential equations that are discretised with trigonometric polynomials their computational effectiveness benefits from efficient fft based algorithms as well as a favourable condition number here these kinds of methods are accelerated by low rank tensor approximation techniques for a solution field using canonical polyadic tucker and tensor train formats this reduced order model also allows to efficiently compute suboptimal global basis functions without solving the full problem it significantly reduces computational and memory requirements for problems with a material coefficient field that admits a moderate rank approximation the advantages of this approach against those using full material tensors are demonstrated using numerical examples for the model homogenisation problem that consists of a scalar linear elliptic variational problem defined in two and three dimensional settings with continuous and discontinuous heterogeneous material coefficients this approach opens up the potential of an efficient reduced order modelling of large scale engineering problems with heterogeneous material

energy based comparison between the fourier galerkin method and the finite element method abstract the fourier galerkin method in short ffth has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom because the method incorporates the fast fourier transform fft in the linear solver it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method fem here we systematically compare these two methods using the energetic norm of local fields which has the clear physical interpretation as being the error in the homogenised properties this enables the comparison of memory and computational requirements at the same level of approximation accuracy we show that the methods effectiveness relies on the smoothness regularity of the solution and thus on the material coefficients thanks to its approximation properties fem outperforms ffth for problems with jumps in material coefficients while ambivalent results are observed for the case that the material coefficients vary continuously in space ffth profits from a good conditioning of the linear system independent of the number of degrees of freedom but generally needs more degrees of freedom to reach the same approximation accuracy more studies are needed for other fft based schemes non linear problems and dual problems which require special treatment in fem but not in ffth

fourier based schemes with modified green operator for computing the electrical response of heterogeneous media with accurate local fields a modified green operator is proposed as an improvement of fou rier based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media contrary to other methods the number of iterations necessary to achieve convergence tends to a finite value when the contrast of properties between the phases becomes infinite furtherm ore it is shown that the method produces much more accurate local field s inside highly conducting and quasi insulating phases as well as in the vicinity of the phases interfaces these good propert ies stem from the discretization of green s function which is consistent with the pixel grid while retaining the local nature of the op erator that acts on the polarization field finally a fast im plementation of the direct scheme of moulinec et al 1994 that allows for parcimonious memory use is proposed

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

eigenvalue distributions from a star product approach we use the well known isomorphism between operator algebras and function spaces equipped with a star product to study the asymptotic properties of certain matrix sequences in which the matrix dimension d tends to infinity our approach is based on the su 2 coherent states which allow for a systematic 1 d expansion of the star product this produces a trace formula for functions of the matrix sequence elements in the large d limit which includes higher order finite d corrections from this a variety of analytic results pertaining to the asymptotic properties of the density of states eigenstates and expectation values associated with the matrix sequence follows it is shown how new and existing results in the settings of collective spin systems and orthogonal polynomial sequences can be readily obtained as special cases in particular this approach allows for the calculation of higher order corrections to the zero distributions of a large class of orthogonal polynomials

moyal implementation of flow equations a non perturbative approach to quantum many body systems we show how wegner s flow equations can be reformulated as ordinary differential equations through the use of the moyal bracket in finite dimensional hilbert spaces the introduction of the moyal bracket leads naturally to the identification of a small expansion parameter namely the inverse of the dimensionality of the space this expansion corresponds to a non perturbative treatment of the coupling constant in the case of infinite dimensional spaces plays the role of the small parameter and the moyal formulation then allows for a semi classical treatment of the flow equation we demonstrate these statements for the lipkin and dicke models as well as for the symmetric x4 and double well potentials

tensor based methods for numerical homogenization from high resolution images we present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images we first introduce specific numerical treatments for the translation of image based homogenization problems into a tensor framework it includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images we then introduce some variants of proper generalized decomposition pgd methods for the construction of tensor decompositions in different tensor formats of the solution of boundary value problems a new definition of pgd is introduced which allows the progressive construction of a tucker decomposition of the solution this tensor format is well adapted to the present application and improves convergence properties of tensor decompositions finally we use a dual based error estimator on quantities of interest which was recently introduced in the context of pgd we exhibit its specificities when it is used for assessing the error on the homogenized properties of the heterogeneous material we also provide a complete goal oriented adaptive strategy for the progressive construction of tensor decompositions of primal and dual solutions yielding to predictions of homogenized quantities with a prescribed accuracy

an fft based galerkin method for homogenization of periodic media in 1994 moulinec and suquet introduced an efficient technique for the numerical resolution of the cell problem arising in homogenization of periodic media the scheme is based on a fixed point iterative solution to an integral equation of the lippmann schwinger type with action of its kernel efficiently evaluated by the fast fourier transform techniques the aim of this work is to demonstrate that the moulinec suquet setting is actually equivalent to a galerkin discretization of the cell problem based on approximation spaces spanned by trigonometric polynomials and a suitable numerical integration scheme for the latter framework and scalar elliptic problems we prove convergence of the approximate solution to the weak solution including a priori estimates for the rate of convergence for sufficiently regular data and the effects of numerical integration moreover we also show that the variational structure implies that the resulting non symmetric system of linear equations can be solved by the conjugate gradient method apart from providing a theoretical support to fast fourier transform based methods for numerical homogenization these findings significantly improve on the performance of the original solver and pave the way to similar developments for its many generalizations proposed in the literature

finite strain fft based non linear solvers made simple computational micromechanics and homogenization require the solution of the mechanical equilibrium of a periodic cell that comprises a generally complex microstructure techniques that apply the fast fourier transform have attracted much attention as they outperform other methods in terms of speed and memory footprint moreover the fast fourier transform is a natural companion of pixel based digital images which often serve as input in its original form one of the biggest challenges for the method is the treatment of geometrically non linear problems partially due to the need for a uniform linear reference problem in a geometrically linear setting the problem has recently been treated in a variational form resulting in an unconditionally stable scheme that combines newton iterations with an iterative linear solver and therefore exhibits robust and quadratic convergence behavior through this approach well known key ingredients were recovered in terms of discretization numerical quadrature consistent linearization of the material model and the iterative solution of the resulting linear system as a result the extension to finite strains using arbitrary constitutive models is at hand because of the application of the fast fourier transform the implementation is substantially easier than that of other finite element methods both claims are demonstrated in this paper and substantiated with a simple code in python of just 59 lines without comments the aim is to render the method transparent and accessible whereby researchers that are new to this method should be able to implement it efficiently the potential of this method is demonstrated using two examples each with a different material model

convergence of iterative methods based on neumann series for composite materials theory and practice iterative fast fourier transform methods are useful for calculating the fields in composite materials and their macroscopic response by iterating back and forth until convergence the differential constraints are satisfied in fourier space and the constitutive law in real space the methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli it is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative fast fourier transform methods we look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity obnosov theoretically some of the methods converge when the inclusions have zero or even negative conductivity however the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations there is little point in iterating beyond this accuracy improves when the grid size is reduced showing that the discrepancy is linked to the discretization finally it is shown that none of the three iterative schemes investigated over performs the others for all possible microstructures and all contrasts

flow equations for hamiltonians contrasting different approaches by using a numerically solvable model to contrast different generators for flow equations for hamiltonians and to discuss the dependence of physical quantities on unitarily equivalent but effectively different initial hamiltonians a numerically solvable model is considered which is structurally similar to impurity models by this we discuss the question of optimization for the first time a general truncation scheme is established that produces good results for the hamiltonian flow as well as for the operator flow nevertheless it is also pointed out that a systematic and feasible scheme for the operator flow on the operator level is missing for this an explicit analysis of the operator flow is given for the first time we observe that truncation of the series of the observable flow after the linear or bilinear terms does not yield satisfactory results for the entire parameter regime as especially close to resonances even high orders of the exact series expansion carry considerable weight

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

flow equations and normal ordering a survey first we give an introduction to the method of diagonalizing or block diagonalizing continuously a hamiltonian and explain how this procedure can be used to analyse the two dimensional hubbard model then we give a short survey on applications of this flow equation on other models finally we outline how symmetry breaking can be introduced by means of a symmetry breaking of the normal ordering not of the hamiltonian