multilevel frames for sparse tensor product spaces for au f with an elliptic differential operator a mathcal h rightarrow mathcal h and stochastic data f the m point correlation function mathcal m m u of the random solution u satisfies a deterministic equation with the m fold tensor product operator a m of a sparse tensor products of hierarchic fe spaces in mathcal h are known to allow for approximations to mathcal m m u which converge at essentially the rate as in the case m 1 i e for the deterministic problem they can be realized by wavelet type fe bases von petersdorff and schwab in appl math 51 2 145 180 2006 schwab and todor in computing 71 43 63 2003 if wavelet bases are not available we show here how to achieve the fast computation of sparse approximations of mathcal m m u for galerkin discretizations of a by multilevel frames such as bpx or other multilevel preconditioners of any standard fem approximation for a numerical examples illustrate feasibility and scope of the method

sparse second moment analysis for elliptic problems in stochastic domains we consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations assuming normal perturbations with small amplitude and known mean field and two point correlation function we derive using a second order shape calculus deterministic equations for the mean field and the two point correlation function of the random solution for a model dirichlet problem which are 3rd order accurate in the boundary perturbation size using a variational boundary integral equation formulation on the unperturbed nominal boundary and a wavelet discretization we present and analyze an algorithm to approximate the random solution s mean and its two point correlation function at essentially optimal order in essentially mathcal o n work and memory where n denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain

efficient inversion of the galerkin matrix of general second order elliptic operators with nonsmooth coefficients this article deals with the efficient approximate inversion of finite element stiffness matrices of general second order elliptic operators with l coefficients it will be shown that the inverse stiffness matrix can be approximated by hierarchical matrices h matrices furthermore numerical results will demonstrate that it is possible to compute an approximate inverse with almost linear complexity

numerical analysis of dynamic problems an alternative integration abstract several applications of the well known fourth order runge kutta r k method for the time integration of the equations of motion in structural dynamics are presented the numerical solutions are compared with the analytical solution in order to examine the solution behaviour on successively refined meshes an interesting merit of the method lies in the fact that it can be implemented on personal computers

multistep numerical integrators for time dependent vibrating systems abstract by means of the weighted residual method a general form of linear multistep methods is developed for time dependent vibrating systems and a feasible two step method is then obtained finally two examples are analyzed to show its applications to the initial value problems of time dependent systems and the steady state solution of periodically time dependent ones to the authors knowledge this is the first time that numerical methods for direct integration of the motion equations of time dependent vibrating systems have been deeply investigated

covariance regularity and mathcal h h matrix approximation for rough random fields in an open bounded domain mathrm d subset mathbb r n d rn with smooth boundary partial mathrm d d or on a smooth closed and compact riemannian n manifold mathcal m subset mathbb r n 1 m rn 1 we consider the linear operator equation a u f au f where a is a boundedly invertible strongly elliptic pseudodifferential operator of order r in mathbb r r r with analytic coefficients covering all linear second order elliptic pdes as well as their boundary reductions here f in l 2 omega h t f l2 ht is an h t ht valued random field with finite second moments with h t ht denoting the isotropic sobolev space of not necessarily integer order t modelled on the domain mathrm d d or manifold mathcal m m respectively we prove that the random solution s covariance kernel k u a 1 otimes a 1 k f ku a 1 a 1 kf on mathrm d times mathrm d d d resp mathcal m times mathcal m m m is an asymptotically smooth function provided that the covariance function k f kf of the random data is a schwartz distributional kernel of an elliptic pseudodifferential operator as a consequence numerical mathcal h h matrix calculus allows the deterministic approximation of singular covariances k u ku of the random solution u a 1 f in l 2 omega h t r u a 1f l2 ht r in mathrm d times mathrm d d d text resp mathcal m times mathcal m resp m m with work versus accuracy essentially equal to that for the mean field approximation with splines of fixed order mathrm d d text resp mathcal m resp m overcoming the curse of dimensionality in this case

combination technique based k th moment analysis of elliptic problems with random diffusion we consider the efficient deterministic solution of elliptic boundary value problems with random diffusion matrix assuming random perturbations with known k moments we derive to leading order in the random perturbation s amplitude deterministic equations for k moments of the random solution the solution s k th moment satisfies a k fold tensor product boundary value problem on the k fold product domain which can efficiently be discretized in sparse tensor product spaces by defining the complement spaces via galerkin projections the related system of linear equations decouples and can be solved by standard multilevel finite element solvers numerical results for k 2 are presented to validate and quantify our theoretical findings

higher derivative explicit one step methods for non linear dynamic problems part i design and theory a new higher order explicit one step scheme is presented for use in the solution of a certain class of non linear problems in structural dynamics the generalized scheme covers some recently developed multistep algorithms as well as some new one step methods with improved numerical properties a weighted residual approach is used which contains new aspects for design of single step algorithms accuracy and stability are investigated for linear and non linear systems algorithms with higher accurary than the central difference method are developed and optimized to minimize the reduction in the stability limit

improving the run time and quality of nested dissection ordering when performing sparse matrix factorization the ordering of matrix rows and columns has a dramatic impact on the factorization time this paper describes an approach to the reordering problem that produces significantly better orderings than prior methods the algorithm is a hybrid of nested dissection and minimum degree ordering and combines an assortment of different algorithmic advances new or improved algorithms are described for graph compression multilevel partitioning and separator improvement when these techniques are combined the resulting orderings average 39 better than minimum degree over a suite of test matrices while requiring roughly 2 7 times the run time of liu s multiple minimum degree

alternative integration methods for problems in structural dynamics abstract runge kutta methods for the time integration of the equations of motion in structural dynamics are presented the methods belong to a class called sdirk singly diagonally implicit runge kutta methods the computational cost needed per step is comparable to newmark methods with damping the methods proposed are l stable which means that they instantly damp out the higher modes in the solution the numerical dissipation can be controlled by a parameter both second and third order methods are presented some characteristics of the methods are compared with the newmark method with damping

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

domain decomposition based ℌ lu preconditioners hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approxi mation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based h matrices this new approach yields h lu fac torizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decom position based h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

a unified set of single step algorithms part 3 the beta m method a generalization of the newmark scheme introduced herein is a generalization of newmark s time marching integration scheme called the m method like the sspj method introduced in parts 1 and 2 of this series the m method provides a gcneral single step scheme applicable to any set of initial value problems the method is specialized by specifying the method order m along with rn integration parameters 0 1 m 1 for a particular choice of m the integration parameters provide a subfamily of methods which control accuracy and stability as well as offering options for explicit or implicit algorithms for the most part attention is focused on the application to structural dynamic equations most well known methods e g newmark wilson houbolt etc are shown to be special cases within the m family hence one computationally efficient and surprisingly simple algorithm unifies old and new methods stability and accuracy analyses are presented for method orders m 2 3 and 4 to determine optimal parameters for implicit and explicit schemes along with numerical verification

variable order variable step algorithms for second order systems part 2 the codes we develop a new class of multistage algorithms for second order systems of ordinary differential equations ode s based on methods by zienkiewicz et al 13 our new methods are designed to provide a set of efficient methods which can be used with local error estimators based on embedding techniques in a companion paper we provide a set of subroutines implementing the algorithms derived and a discussion of numerical experience with the resulting codes

mathcal h matrix based second moment analysis for rough random fields and finite element discretizations we consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method the solution s two point correlation can efficiently be approximated by means of an mathcal h matrix in particular if the correlation length is rather short or the correlation kernel is nonsmooth since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the mathcal h matrix format we can solve the correspondent mathcal h matrix equation in essentially linear time by using the mathcal h matrix arithmetic numerical experiments for three dimensional finite element discretizations for several correlation lengths and different smoothness are provided they validate the presented method and demonstrate that the computation times do not increase for nonsmooth or shortly correlated data

bounded subquotients of pseudodifferential operator modules recently there have been several papers on the action of the virasoro lie algebra on the projective decompositions of the modules of pseudodifferential operators on the circle we use their results to prove that a wide class of the uniserial completely indecomposable bounded modules of the virasoro lie algebra may be realized as subquotients of such modules of pseudodifferential operators this gives easy proofs of the existence of many previously known uniserial modules and moreover yields some hitherto undiscovered

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

existence of ℋ matrix approximants to the inverse fe matrix of elliptic operators with l coefficients this article deals with the existence of blockwise low rank approximants so called matrices to inverses of fem matrices in the case of uniformly elliptic operators with l coefficients unlike operators arising from boundary element methods for which the matrix theory has been extensively developed the inverses of these operators do not benefit from the smoothness of the kernel function however it will be shown that the corresponding green functions can be approximated by degenerate functions giving rise to the existence of blockwise low rank approximants of fem inverses numerical examples confirm the correctness of our estimates as a side product we analyse the matrix property of the inverse of the fe mass matrix

approximate iterations for structured matrices important matrix valued functions f a are e g the inverse a 1 the square root sqrt a and the sign function their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices a and f a often f a possesses this structure only approximately however intermediate matrices arising during the evaluation may lose the structure of the initial matrix this would make the computations inefficient and even infeasible however the main result of this paper is that an iterative fixed point like process for the evaluation of f a can be transformed under certain general assumptions into another process which preserves the convergence rate and benefits from the underlying structure it is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique we demonstrate our results by verifying all requirements in the case of the iterative computation of a 1 and sqrt a

a new family of explicit time integration methods for linear and non linear structural dynamics a new family of explicit single step time integration methods with controllable high frequency dissipation is presented for linear and non linear structural dynamic analyses the proposed methods are second order accurate and completely explicit with a diagonal mass matrix even when the damping matrix is not diagonal in the linear structural dynamics or the internal force vector is a function of velocities in the non linear structural dynamics stability and accuracy of the new explicit methods are analysed for the linear undamped damped cases furthermore the new methods are compared with other explicit methods

space of second order linear differential operators as a module over the lie algebra of vector fields abstract the space of linear differential operators on a smooth manifoldmhas a natural one parameter family of diff m and vect m module structures defined by their action on the space of tensor densities it is shown that in the case of second order differential operators the vect m module structures are equivalent for any degree of tensor densities except for three critical values 0 1 2 1 a second order analogue of the lie derivative appears as an intertwining operator between the spaces of second order differential operators on tensor densities

a sparse h matrix arithmetic general complexity estimates in a preceding paper hackbusch computing 62 1999 89 108 a class of matrices h matrices has been introduced which are data sparse and allow an approximate matrix arithmetic of almost linear complexity several types of h matrices have been analysed in hackbusch computing 62 1999 89 108 and hackbusch and khoromskij preprint mpi no 22 leipzig 1999 computing 64 2000 21 47 which are able to approximate integral nonlocal operators in fem and bem applications in the case of quasi uniform unstructured meshes in the present paper the general construction of h matrices on rectangular and triangular meshes is proposed and analysed first the reliability of h matrices in bem is discussed then we prove the optimal complexity of storage and matrix vector multiplication in the case of rather arbitrary admissibility parameters 1 and for nite elements up to the order 1 dened on quasi uniform rectangular triangular meshes in r d d 1 2 3 the almost linear complexity of the matrix addition multiplication and inversion of h matrices is also veried c 2000 elsevier science b v all rights reserved msc 65f05 65f30 65f50

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

a modified euler method for dynamic analyses a modified version of the well known euler method for numerical integration is presented and its application in analyses of the dynamic response of structures is discussed this modified method was first introduced in 1981 and it has been reported that it was discovered by accident when first introduced however valuable information related to the accuracy stability and merits of the method were not provided it is shown that the modified euler method has features that make it an attractive approach for applications in dynamic analyses of structures accuracy and stability analyses are made for the method by considering free vibrational responses of linear undamped single degree of freedom systems the method is explicit and extremely easy to use however it is conditionally stable applications of the method are also made which illustrate its usefulness and inherent simplicity furthermore comparisons of the modified euler method and the newmark beta method are reported which elucidate the relative merits of these methods

hierarchical matrix approximation for the uncertainty quantification of potentials on random domains abstract computing statistical quantities of interest of the solution of partial differential equations on random domains is an important and challenging task in engineering we consider the computation of these quantities by the perturbation approach especially we discuss how third order accurate expansions of the mean and the correlation can numerically be computed these expansions become even fourth order accurate for certain types of boundary variations the correction terms are given by the solution of correlation equations in the tensor product domain which can efficiently be computed by means of h matrices they have recently been shown to be an efficient tool to solve correlation equations with rough data correlations that is with low sobolev smoothness or small correlation length in almost linear time numerical experiments in three dimensions for higher order ansatz spaces show the feasibility of the proposed algorithm the application to a non smooth domain is also included