computational methods in linear algebra abstract this paper is the written version of a talk given at the workshop on computational seismology salishan lodge oregon march 24 26 1981 state of the art and recent developments in computational linear algebra including linear systems least squares techniques the singular value decomposition and eigenvalue problems are reviewed briefly

improved frequency domain elastic wave modeling using weighted averaging difference operators we develop a new finite difference scheme that reduces the number of grid points per wavelength required in frequency domain elastic modeling our approach computes weighted averages of the spatial second order derivative and the mass acceleration terms using a 25 point computational stencil by determining the weighting coefficients to minimize numerical dispersion and numerical anisotropy we reduce the number of grid points to 3 3 per shear wavelength with a resulting error in velocities smaller than 1 our choice of grid points reduces the computer memory needed to store the complex impedance matrix to 4 of that for a conventional second order scheme and to 54 of that for a combined second order scheme the 25 point weighted averaging scheme of this paper makes it possible to accurately simulate realistic models numerical examples show that this technique can achieve the same accurate solutions with fewer grid points than those from previous frequency domain second order schemes our technique can be extended directly to 3 d elastic modeling the computational efficiency will be even greater than that realized for 2 d models

a graph theoretic model of symmetric givens operations and its implications abstract symmetric givens operations a gag t g a givens rotation matrix are a basic tool in many matrix computations especially for eigenvalue eigenvector computations a graph theoretic model of these operations is given for symmetric matrices analogous to the graph theoretic model of cholesky factorization using this model it is shown that unless there is accidental cancellation it is impossible to reduce a range of different matrix classes to tridiagonal form in o n 2 givens operations these classes include arrowhead matrices pentadiagonal matrices and cyclic tridiagonal matrices

minimum fill in inapproximability and almost tight lower bounds abstract given an n n sparse symmetric matrix with m nonzero entries performing gaussian elimination may turn some zeroes into nonzero values so called fill ins the minimum fill in problem asks whether it is possible to perform the elimination with at most k fill ins we exclude the existence of polynomial time approximation schemes for this problem assuming p np and the existence of 2 o n 1 time approximation schemes for any positive assuming the exponential time hypothesis we also give a 2 o k 1 2 n o 1 parameterized lower bound all these results come as corollaries of a new reduction from vertex cover to the minimum fill in problem which might be of its own interest all previous reductions for similar problems start from some kind of graph layout problems and hence have limited use in understanding their fine grained complexity

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

the diameter of the first nodal line of a convex domains the main goal of this paper is to prove that the first nodal line for the dirichlet problem in a convex planar domain has diameter less than an absolute constant times the inradius of the domain more precisely we locate the nodal line to within a distance comparable to the inradius near the zero of an ordinary differential equation which is associated to the domain in a natural way we also derive estimates for the first and second eigenvalues in terms of the corresponding eigenvalues of the ordinary differential equation and construct an approximate first eigenfunction two examples a rectangle and a circular sector illustrate the two extreme possibilities for the location of the nodal line for the rectangle r x y o 1 we have the second eigenfunction

optimizing quadratic forms of adjacency matrices of trees and related eigenvalue problems abstract let a be an adjacency matrix of a tree t with n vertices conditions are determined for the existence of a fixed permutation matrix p that maximizes the quadratic form x t p t apx over all nonnegative vectors x with entries arranged in nondecreasing order this quadratic form problem is completely solved and its answer leads to a corresponding solution for the problem of determining conditions for the existence of a fixed permutation matrix p that maximizes the largest eigenvalue of matrices of the form pdp t a over all real diagonal matrices d with nondecreasing diagonal entries it is shown that there is a tree with six vertices for which neither of the problems has a solution and all other trees with six or fewer vertices have solutions for both problems by duality the results also apply to the analogous problem of minimizing the smallest eigenvalue of matrices of the form pdp t a

node and element resequencing using the laplacian of a finite element graph part i general concepts and algorithm a finite element graph feg is defined here as a nodal graph g a dual graph g or a communication graph g associated with a generic finite element mesh the laplacian matrix l g l g or l g used for the study of spectral properties of an feg is constructed from usual vertex and edge connectivities of a graph an automatic algorithm based on spectral properties of an feg g g or g is proposed to reorder the nodes and or elements of the associated finite element mesh the new algorithm is called spectral peg resequencing sfr this algorithm uses global information in the graph it does not depend on a pseudoperipheral vertex in the resequencing process and it does not use any kind of level structure of the graph moreover the sfr algorithm is of special advantage in computing environments with vector and parallel processing capabilities nodes or elements in the mesh can be reordered depending on the use of an adequate graph representation associated with the mesh if g is used then the nodes in the mesh are properly reordered for achieving profile and wavefront reduction of the finite element stiffness matrix if either g or g is used then the elements in the mesh are suitably reordered for a finite element frontai solver a unified approach involving fegs and finite element concepts is presented conclusions are inferred and possible extensions of this research are pointed out in part ii of this work 1 the computational implementation of the sfr algorithm is described and several numerical examples are presented the examples emphasize important theoretical numerical and practical aspects of the new resequencing method

interval completion is fixed parameter tractable we present an algorithm with runtime o k 2k n 3m for the following np complete problem m garey and d johnson computers and intractability a guide to the theory of np completeness w h freeman and co san francisco 1979 problem gt35 given an arbitrary graph g on n vertices and m edges can we obtain an interval graph by adding at most k new edges to g this resolves the long standing open question h kaplan r shamir and r e tarjan siam j comput 28 1999 pp 1906 1922 r g downey and m r fellows parameterized complexity springer verlag new york 1999 m serna and d thilikos bull eur assoc theory comput sci eatcs 86 2005 pp 41 65 g gutin s szeider and a yeo in proceedings iwpec 2006 lecture notes in comput sci 4169 springer verlag berlin 2006 pp 60 71 first posed by kaplan shamir and tarjan of whether this problem was fixed parameter tractable the problem has applications in profile minimization for sparse matrix computations j a george and j w h liu computer solution of large sparse positive definite systems prentice hall englewood cliffs nj 1981 r e tarjan in sparse matrix computations j r bunch and d j rose eds academic press 1976 pp 3 22 and our results show tractability for the case of a small number k of zero elements in the envelope our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph and combines this with a greedy algorithm when graphs of a certain structure are reached by the search

a symmetric formulation of coupled bem fem in solving responses of submerged elastic structures for large degrees of freedom this work presents a symmetric formulation for the coupling of boundary element method bem with finite element method fem to compute responses of submerged elastic structures in a heavy acoustic medium the acoustic loading derived from bem is formulated in a symmetric complex matrix the symmetry of the acoustic loading matrix is proven by an acoustic reciprocal principle which relate two sets of arbitrary surface normal velocities and surface pressures in terms of surface integrals the structural equation is represented by fem the fact that the acoustic loading and the structural equation are symmetric allows us to take the storage and computational advantages of the banded matrix which has been widely used in conventional finite element methods consequently the computational efficiency of the proposed coupled fem bem is significantly increased the proposed method involving 20 000 degrees of freedom can be implemented on a personal computer numerical results are verified by analytical solutions of a constant thickness spherical shell subject to a point alternating force as well as the previously published variational solution of a constant thickness spheroidal shell excited by a concentrated force

a direct method for sparse least squares problems with lower and upper bounds a direct method is developed for solving linear least squares problems mathop min left ax b right 2 limits x wherea is large and sparse and the solution is subject to lower and upper boundsl x u the problem is initially transformed to upper triangular form by a sparseqr factorization an active set algorithm is then used the key step is the stable updating of ther factor associated with the columns ofa corresponding to the free variables when theq factor is not available for this a new method is developed which uses the semi normal equations and iterative refinement

large scale estimation of variance and covariance components this paper concerns matrix computations within algorithms for variance and covariance component estimation hemmerle and hartley technometrics 15 1973 pp 819 831 showed how to compute the objective function and its derivatives for maximum likelihood estimation of variance components using matrices with dimensions of the order of the number of coefficients rather than that of the number of observations their approach was extended by corbeil and searle technometrics 18 1976 pp 31 38 for restricted maximum likelihood estimation a similar reduction in dimension is possible using expectation maximization em algorithms in most cases variance components are assumed to be strictly positive we advocate the use of a modification that is numerically stable even if variance component estimates are small in magnitude for problems in which the number of coefficients is large fellner proc statistical computing section american statistical association 1984 pp 150 154 comm statist simulat

a unified set of resequencing algorithms a general algorithm for resequencing equations and finite element meshes is presented this algorithm incorporates in a unified form the most successful resequencing schemes namely the sloan gibbs king gk and gibbs poole stockmeyer gps algorithms as well it provides a context within which the development of new algorithms may take place with a minimum of change it is easily programmed and has been shown to speed up the execution of the original gk algorithm

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

the incomplete factorization multigraph algorithm we present a new family of multigraph algorithms ilu mg based upon an incomplete sparse matrix factorization using a particular ordering and allowing a limited amount of fill in while much of the motivation for multigraph comes from multigrid ideas ilu mg is distinctly different from algebraic multilevel methods the graph of the sparse matrix a is recursively coarsened by eliminating vertices using a graph model similar to gaussian elimination incomplete factorizations are obtained by allowing only the fill in generated by the vertex parents associated with each vertex multigraph is numerically compared with algebraic multigrid on some examples arising from discretizations of partial differential equations on unstructured grids

substructuring preconditioners for the q_1 mortar element method the q 1 mortar element method is a non conforming finite element method with q 1 elements based on domain decomposition for the laplace equation it yields an ill conditioned linear system for solving the linear system the so called preconditioned conjugate gradient method in a subspace is used preconditioners are proposed and estimates on condition numbers and arithmetical complexity are given finally numerical experiments are presented

a hybrid algorithm for reducing matrix bandwidth a hybrid algorithm for reducing the bandwidth of symmetric matrices is described in terms of a finite element grid the new algorithm produces significantly lower bandwidths than either the commonly used gibbs poole stockmeyer gps or cuthill mckee cm algorithms with run times comparable to the original cm algorithm the new hybrid algorithm uses the gps algorithm as a preprocessor to provide a good initial node numbering for the author s node shuffling algorithm which accounts for variable degrees of freedom per node dof node the hybrid algorithm was tested on the 30 benchmark problems that were compiled by everstine and on 10 supplemental problems with variable dof node bandwidths and cpu times are presented for all problems

cholesky decomposition of a positive semidefinite matrix with known kernel the cholesky decomposition of a symmetric positive semidefinite matrix a is a useful tool for solving the related consistent system of linear equations or evaluating the action of a generalized inverse especially when a is relatively large and sparse to use the cholesky decomposition effectively it is necessary to identify reliably the positions of zero rows or columns of the factors and to choose these positions so that the nonsingular submatrix of a of the maximal rank is reasonably conditioned the point of this note is to show how to exploit information about the kernel of a to accomplish both tasks the results are illustrated by numerical experiments

a cache aware algorithm for pdes on hierarchical data structures based on space filling curves competitive numerical algorithms for solving partial differential equations have to work with the most efficient numerical methods like multigrid and adaptive grid refinement and thus with hierarchical data structures unfortunately in most implementations hierarchical data typically stored in trees cause a nonnegligible overhead in data access to overcome this quandary numerical efficiency versus efficient implementation our algorithm uses space filling curves to build up data structures which are processed linearly in fact the only kind of data structure used in our implementation is stacks thus data access becomes very fast even faster than the common access to nonhierarchical data stored in matrices and in particular cache misses are reduced considerably furthermore the implementation of multigrid cycles and or higher order discretizations as well as the parallelization of the whole algorithm become very easy and straightforward on these data structures

a fast direct solver for elliptic partial differential equations on adaptively refined meshes we present a new class of fast direct solvers for elliptic partial differential equations on adaptively refined meshes these solvers rely on a combination of standard fast solvers for uniform grids and potential theory unlike standard iterative approaches they have a well defined operation count they also preserve the order of accuracy of the uniform grid solver despite the presence of coarse fine interfaces

how good is recursive bisection the most commonly used p way partitioning method is recursive bisection rb it first divides a graph or a mesh into two equal sized pieces by a good bisection algorithm and then recursively divides the two pieces ideally we would like to use an optimal bisection algorithm because the optimal bisection problem that partitions a graph into two equal sized subgraphs to minimize the number of edges cut is np complete practical rb algorithms use more efficient heuristics in place of an optimal bisection algorithm most such heuristics are designed to find the best possible bisection within allowed time we show that the rb method even when an optimal bisection algorithm is assumed may produce a p way partition that is very far way from the optimal one our negative result is complemented by two positive ones first we show that for some important classes of graphs that occur in practical applications such as well shaped finite element and finite difference meshes rb is within a constant factor of the optimal one almost always second we show that if the balance condition is relaxed so that each block in the p way partition is bounded by 2n p where n is the number of vertices of the graph then a modified rb finds an approximately balanced p way partition whose cost is within an o log p factor of the cost of the optimal p way partition

a robust and efficient ilu that incorporates the growth of the inverse triangular factors in this paper we present a new ilu decomposition which is based on an existing sparse direct solver in contrast to many incomplete lu decompositions this ilu incorporates information about the inverse factors l 1 and u 1 which have direct influence on the dropping strategy we demonstrate in several large scale examples that this implementation constructs a robust preconditioner

log determinant relaxation for approximate inference in discrete markov random fields graphical models are well suited to capture the complex and non gaussian statistical dependencies that arise in many real world signals a fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nodes this paper proposes a novel method applicable to discrete valued markov random fields mrfs on arbitrary graphs for approximately solving this marginalization problem the foundation of our method is a reformulation of the marginalization problem as the solution of a low dimensional convex optimization problem over the marginal polytope exactly solving this problem for general graphs is intractable for binary markov random fields we describe how to relax it by using a gaussian bound on the discrete entropy and a semidefinite outer bound on the marginal polytope this combination leads to a log determinant maximization problem that can be solved efficiently by interior point methods thereby providing approximations to the exact marginals we show how a slightly weakened log determinant relaxation can be solved even more efficiently by a dual reformulation when applied to denoising problems in a coupled mixture of gaussian model defined on a binary mrf with cycles we find that the performance of this log determinant relaxation is comparable or superior to the widely used sum product algorithm over a range of experimental conditions

efficient recursions for general factorisable models let n s valued categorical variables be jointly distributed according to a distribution known only up to an unknown normalising constant for an unnormalised joint likelihood expressible as a product of factors we give an algebraic recursion which can be used for computing the normalising constant and other summations a saving in computation is achieved when each factor contains a lagged subset of the components combining in the joint distribution with maximum computational efficiency as the subsets attain their minimum size if each subset contains at most r 1 of the n components in the joint distribution we term this a lag r model whose normalising constant can be computed using a forward recursion in o sr 1 computations as opposed to o sn for the direct computation we show how a lag r model represents a markov random field and allows a neighbourhood structure to be related to the unnormalised joint likelihood we illustrate the method by showing how the normalising constant of the ising or autologistic model can be computed

a pipelined givens method for computing the qr factorization of a sparse matrix this paper discusses an extension of the pipelined givens method for computing the qr factorization of a real m x n matrix to the case in which the matrix is sparse when restricted to one process the algorithm performs the same computation as the serial sparse givens algorithm of george and heath our implementation is compatible with the data structures used in sparspak the pipelined algorithm is well suited to parallel computers having globally shared memory and low overhead synchronization primitives such as the denelcor hep for which computational results are presented we point out certain synchronization problems that arise in the adaptation to the sparse setting and discuss the effect on parallel speedup of accessing a serial data file 15 references 7 figures

nodal sets of thin curved layers abstract this paper is concerned with the location of nodal sets of eigenfunctions of the dirichlet laplacian in thin tubular neighbourhoods of hypersurfaces of the euclidean space of arbitrary dimension in the limit when the radius of the neighbourhood tends to zero it is known that spectral properties of the laplacian are approximated well by an effective schrodinger operator on the hypersurface with a potential expressed solely in terms of principal curvatures by applying techniques of elliptic partial differential equations we strengthen the known perturbation results to get a convergence of eigenfunctions in holder spaces this enables us in particular to conclude that every nodal set has a non empty intersection with the boundary of the tubular neighbourhood

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

implicit multigrid algorithms for incompressible turbulent flows on unstructured grids an implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described the foundation of the code is a backward euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned generalized minimal residual gmres technique for the gmres calculations several techniques are investigated for forming the matrix vector product convergence acceleration is achieved through a multigrid scheme that uses nonnested coarse grids that are generated using a technique described in the present paper convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four element airfoil viscous results which are compared with experimental data include the turbulent flow over a naca 4412 airfoil a three element airfoil for which mach number effects are investigated and three dimensional flow over a wing with a partial span flap

locating the first nodal set in higher dimensions we extend the two dimensional results of jerison 2000 on the location of the nodal set of the first neumann eigenfunction of a convex domain to higher dimensions if a convex domain in n is contained in a long and thin cylinder 0 n x b 0 with nonempty intersections with x 1 0 and x 1 n then the first nonzero eigenvalue is well approximated by the eigenvalue of an ordinary differential equation by a bound proportional to whose coefficients are expressed in terms of the volume of the cross sections of the domain also the first nodal set is located within a distance comparable to e near the zero of the corresponding ordinary differential equation

locating the first nodal linein the neumann problem the location of the nodal line of the first nonconstant neumann eigenfunction of a convex planar domain is specified to within a distance comparable to the inradius this is used to prove that the eigenvalue of the partial differential equation is approximated well by the eigenvalue of an ordinary differential equation whose coefficients are expressed solely in terms of the width of the domain a variant of these estimates is given for domains that are thin strips and satisfy a lipschitz condition

decomposed block cholesky factorization in the karmarkar algorithm solving a class of super large lp problems abstract with the introduction of the karmarkar algorithm hope for optimally solving super large lp problems often encountered in the communication industry had risen high however a general code of the original karmarkar algorithm takes a long computation time to solve such problems employing proper factorization schemes substantially reduces the computation time of the karmarkar algorithm for solving super large linear programming lp problems in this paper a modified version of the karmarkar algorithm is described and the complexities involved in applying the algorithm to a class of super large lp problems are identified a factorization scheme decomposed block cholesky factorization is proposed in order to reduce the complexities of a matrix inversion operation that occurs within every iteration of the karmarkar algorithm the factorization scheme capitalizes on the generally well behaved block structure of the constraint matrix of the super large lp problems the computation time required for solving a class of super large lp problems using this scheme is estimated to be reduced by a factor of two orders of magnitude the proposed factorization scheme is even more promising in solving the super large lp problems on parallel processors using the karmarkar algorithm

the adiabatic limit of schrodinger operators on fibre bundles we consider schrodinger operators h delta g varepsilon v on a fibre bundle m mathop rightarrow limits pi b with compact fibres and a metric g varepsilon that blows up directions perpendicular to the fibres by a factor varepsilon 1 gg 1 we show that for an eigenvalue lambda of the fibre wise part of h satisfying a local gap condition and every n in mathbb n there exists a subspace of l 2 m that is invariant under h up to errors of order varepsilon n 1 the dynamical and spectral features of h on this subspace can be described by an effective operator on the fibre wise lambda eigenspace bundle mathcal e rightarrow b giving detailed asymptotics for h

a survey of the advances in the exploitation of the sparsity in the solution of large problems abstract the numerical treatment of many mathematical models which arise for example in physics chemistry biology or economics leads very often to huge algebraic problems so that it is difficult both with regard to the storage needed and with regard to the computing time spent to handle these problems even on the big modern computers however the matrices that occur in the algebraic problems are fortunately sparse many of their elements are equal to zero the exploitation of the sparsity leads to savings in both storage and computer time so that problems which can not be handled numerically when the zero elements are stored in the computer memory and when the arithmetic operations involving zero elements are performed become tractable if the sparsity is exploited in a proper way there are two basic groups of storage schemes for exploiting the sparsity if a scheme of the first group is in use then the non zero elements have permanent locations in the computer memory during the whole computational process therefore the schemes of the first group are called static a non zero element may be moved from one location to another when a scheme from the second group is applied such schemes are called dynamic the advantages and the limitations of the schemes from these two groups are discussed the advances achieved after 1980 in the efforts to improve the performance of the schemes belonging to both groups are given in a systematic way some questions that are still open are briefly discussed the advances achieved in some other stages in the exploitation of the sparsity which are not directly connected with the storage schemes used are outlined in the last section

massively parallel structured direct solver for equations describing time harmonic qp polarized waves in tti media abstractwe considered the discretization and approximate solutions of equations describing time harmonic qp polarized waves in 3d inhomogeneous anisotropic media the anisotropy comprises general tilted transversely isotropic symmetries we are concerned with solving these equations for a large number of different sources we considered higher order partial differential equations and variable order finite difference schemes to accommodate anisotropy on the one hand and allow higher order accuracy to control sampling rates for relatively high frequencies on the other hand we made use of a nested dissection based domain decomposition in a massively parallel multifrontal solver combined with hierarchically semiseparable matrix compression techniques the higher order partial differential operators and the variable order finite difference schemes require the introduction of separators with variable thickness in the nested dissection the development of these and their integration with the multifrontal

finite element network approximation of conductivity in particle composites a new finite element method computes conductivity in some unstructured particle reinforced composite material the 2 phase material under consideration is composed of a poorly conducting matrix material filled by highly conducting circular inclusions which are randomly dispersed the mathematical model is a poisson type problem with discontinuous coefficients the discontinuities are huge in contrast and quantity the proposed method generalizes classical continuous piecewise affine finite elements to special computational meshes which encode the particles in a network structure important geometric parameters such as the volume fraction are preserved exactly the computational complexity of the method is almost proportional to the number of inclusions this is minimal in the sense that the representation of the underlying geometry via the positions and radii of the inclusions is of the same complexity the discretization error is proportional to the distance of neighboring inclusions and independent of the conductivity contrast in the medium

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

the factorability of symmetric matrices and some implications for statistical linear models abstract general conditions where a symmetric matrix is factorable by cholesky decomposition are described while numerical stability is a remaining issue whenever the cholesky decomposition is used to factor indefinite matrices the existence of such factors is demonstrated for matrix structures that are commonly found in statistics kalman filtering for example is rediscovered in the cholesky decomposition of an indefinite matrix moreover the cholesky decomposition uniquely defines the likelihood function in linear statistical models and this includes situations when the variance matrix is singular or when the cholesky decomposition does not run to completion alternative methods of likelihood evaluation which may involve for example the bunch parlett factorization are available only when the cholesky decomposition exists suggestions are made for computing an adaptive precision cholesky decomposition when numerical stability is an issue

refining an approximate inverse direct methods have made remarkable progress in the computational efficiency of factorization algorithms during the last three decades the advances in graph theoretic algorithms have not received enough attention from the iterative methods community for example we demonstrate how symbolic factorization algorithms from direct methods can accelerate the computation of a factored approximate inverse preconditioner for very sparse preconditioners however a reformulation of the algorithm with outer products can exploit even more zeros to good advantage we also explore the possibilities of improving cache efficiency in the application of the preconditioner through reorderings the article ends by proposing a block version of the algorithm for further gains in efficiency and robustness

a helmholtz iterative solver for 3d seismic imaging problems a preconditioned iterative solver for the 3d frequency domain wave equation applied to seismic problems is evaluated the preconditioner corresponds to an approximate inverse of a heavily damped wave equation deduced from the undamped wave equation the approximate inverse is computed with one multigrid cycle numerical results show that the method is robust and that the number of iterations increases roughly linearly with frequency when the grid spacing is adapted to keep a constant number of discretization points per wavelength to evaluate the relevance of this iterative solver the number of floating point operations required for two imaging problems are roughly evaluated this rough estimate indicates that the time domain migration approach is more than one order of magnitude faster the full wave form tomography based on a least squares formulation and a scale separation approach has the same complexity in both domains

cache oblivious matrix multiplication using an element ordering based on a peano curve one of the keys to tap the full performance potential of current hardware is the optimal utilization of cache memory cache oblivious algorithms are designed to inherently benefit from any underlying hierarchy of caches but do not need to know about the exact structure of the cache in this paper we present a cache oblivious algorithm for matrix multiplication the algorithm uses a block recursive structure and an element ordering that is based on peano curves in the resulting code index jumps can be totally avoided which leads to an asymptotically optimal spatial and temporal locality of the data access

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

a parallel multigrid based preconditioner for the 3d heterogeneous high frequency helmholtz equation we investigate the parallel performance of an iterative solver for 3d heterogeneous helmholtz problems related to applications in seismic wave propagation for large 3d problems the computation is no longer feasible on a single processor and the memory requirements increase rapidly therefore parallelization of the solver is needed we employ a complex shifted laplace preconditioner combined with the bi cgstab iterative method and use a multigrid method to approximate the inverse of the resulting preconditioning operator a 3d multigrid method with 2d semi coarsening is employed we show numerical results for large problems arising in geophysical applications

the average parallel complexity of cholesky factorization abstract we analyze the average parallel complexity of the solution of large sparse positive definite linear systems more precisely using probabilistic techniques we study the cholesky factorization with the application of the minimum degree algorithm main results are the estimation of the evolution of sparsity during the factorization and a characterization of the elimination tree in terms of depth and number of leaves we also conjecture that the number of parallel steps needed to perform the factorization is linear with respect to the matrix size

greedy coarsening strategies for nonsymmetric problems the solution of large scale linear systems in computational science and engineering requires efficient solvers and preconditioners often the most effective such techniques are those based on multilevel splittings of the problem in this paper we consider the problem of partitioning both symmetric and nonsymmetric matrices based solely on algebraic criteria a new algorithm is proposed that combines attractive features of two previous techniques proposed by the authors it offers rigorous guarantees of certain properties of the partitioning yet is naturally compatible with the threshold based dropping known to be effective for incomplete factorizations numerical results show that the new partitioning scheme leads to improved results for a variety of problems the effects of further matrix reordering within the fine scale block are also considered

inverse free recursive multiresolution algorithms for a data approximation problem we present inverse free recursive multiresolution algorithms for data approximation problems based on energy functionals minimization during the multiresolution process a linear system needs to be solved at each different resolution level which can be solved with direct or iterative methods numerical results are reported using the sparse cholesky factorization for two applications one concerning the localization of regions in which the energy of a given surface is mostly concentrated and another one regarding noise reduction of a given dataset in addition for large scale data approximation problems that require a very fine resolution we discuss the use of the preconditioned conjugate gradient pcg iterative method coupled with a specialized monolithic preconditioner for which one preconditioner is built for the highest resolution level and then the corresponding blocks of that preconditioner are used as preconditioners for the forthcoming lower levels

lower eigenvalue bounds for singular pencils of matrices abstract beauwens procedure for obtaining lower eigenvalue bounds for regular pencils of matrices a b is simplified and extended to the singular case the theory is then compared through a particular perturbed modified incomplete factorization with notay s generalization of another approach initiated by gustafsson and developed by axelsson and barker and by wilmet

block diagonalization analysis of symmetric plates abstract this paper presents a block diagonalization method to solve stiffness equations of isotropic symmetric plates by means of a suitable local coordinate transformation chosen based on group theory the stiffness matrix is decomposed into a block diagonal form the stiffness equation in the local coordinate is solved block by block thus realizing numerical efficiency and greatly reducing the requisite amount of computer memory the efficiency has been further upgraded with the aid of the concept of augmented orbit this method is applied to a square isotropic plate subject to asymmetric loads to show its usefulness

three level bddc in two dimensions balancing domain decomposition by constraints bddc methods are non overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems they are similar to the balancing neumann neumann algorithm however in bddc methods a small number of continuity constraints are enforced across the interface and these constraints form a new coarse global component an important advantage of using such constraints is that the schur complements that arise in the computation will all be strictly positive definite the matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation however this problem can ultimately become a bottleneck if the number of subdomains is very large in this paper two three level bddc methods are introduced for solving the coarse problem approximately in two dimensional space while still maintaining a good convergence rate estimates of the condition numbers are provided for the two three level bddc methods and numerical experiments are also discussed copyright 2006 john wiley sons ltd

on large scale generalized inverses in solving two by two block linear systems abstract the goal is to analyze a role of generalized inverses in the solution of two by two block linear systems by means of an algorithm combining the schur complement reduction with the null space method the efficient procedures for computing actions of generalized inversions to rectangular large scale matrices are presented with a special attention to the moore penrose inverse a regularization technique for matrices with the known null space basis is introduced that enables to get generalized inverses without necessity to recognize zero pivots the theoretical results are confirmed by numerical examples

the h path distribution of the lifetime of conditioned brownian motion for non smooth domains consider the h paths of doob and let d be the lifetime of brownian motion killed at d where d is a bounded lipschitz or nta domain in n then log p x h tau d t sim lambda d t as t to infty where d is the first positive eigenvalue of 1 2 on d

an algorithm to compute a sparse basis of the null space leta be a realm n matrix with full row rankm in many algorithms in engineering and science such as the force method in structural analysis the dual variable method for the navier stokes equations or more generally null space methods in quadratic programming it is necessary to compute a basis matrixb for the null space ofa hereb isn r r n m of rankr withab 0 in many instancesa is large and sparse and often banded the purpose of this paper is to describe and test a variation of a method originally suggested by topcu and called the turnback algorithm for computing a banded basis matrixb two implementations of the algorithm are given one using gaussian elimination and the other using orthogonal factorization by givens rotations the fortran software was executed on an ibm 3081 computer with an fps 164 attached array processor at the triangle universities computing center and on a cyber 205 vector computer test results on a variety of structural analysis problems including two and three dimensional frames plane stress plate bending and mixed finite element problems are discussed these results indicate that both implementations of the algorithm yielded a well conditioned banded basis matrixb whena is well conditioned however the orthogonal implementation yielded a better conditionedb for large illconditioned problems

spectral gap for convex planar domains a sharp lower bound is proved for the spectral gap of convex domains in the plane dirichlet boundary conditions are imposed on the laplacian

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

distributing the kalman filter for large scale systems this paper presents a distributed kalman filter to estimate the state of a sparsely connected large scale n dimensional dynamical system monitored by a network of n sensors local kalman filters are implemented on nl dimensional subsystems nl lt n obtained by spatially decomposing the large scale system the distributed kalman filter is optimal under an lth order gauss markov approximation to the centralized filter we quantify the information loss due to this lth order approximation by the divergence which decreases as l increases the order of the approximation l leads to a bound on the dimension of the subsystems hence providing a criterion for subsystem selection the approximated centralized riccati and lyapunov equations are computed iteratively with only local communication and low order computation by a distributed iterate collapse inversion dici algorithm we fuse the observations that are common among the local kalman filters using bipartite fusion graphs and consensus averaging algorithms the proposed algorithm achieves full distribution of the kalman filter nowhere in the network storage communication or computation of n dimensional vectors and matrices is required only nl lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors in other words knowledge of the state is itself distributed

accurate viscoelastic modeling by frequency domain finite differences using rotated operators the viscoelastic wave equation is an integro differential equation that requires special methods when using time domain numerical finite difference methods in the frequency domain the integral terms are easily represented by complex valued elastic media properties there are further significant advantages to using the frequency domain if the forward or the inverse problem requires modeling or inverting a large number of prestack source gathers numerical modeling is expensive for seismic data because of the large number of wavelenghths typically separating sources from receivers which results in a need for a large number of grid points a major obstacle to using frequency domain methods is the consequent storage requirements to reduce these we maximize the accuracy and simultaneously minimize the spatial extent of the numerical operators we achieve this by extending earlier published methods introduced for the viscoacoustic case to the viscoelastic case this requires the formulation of two new nume

rebuttal of donnelly s paper on the spectral gap the spectral gap conjecture of m van den berg 2 formula 65 asserts that 2 1 3 for all convex euclidean domains of diameter 1 where 1 and 2 denote the first two eigenvalues of the dirichlet laplacian notice that equality holds for the 1 dimensional unit interval which can be regarded also as a degenerate n dimensional rectangular box the gap estimate is conjectured to hold more generally for schrodinger operators with convex potentials under dirichlet boundary conditions see the work of s t yau and collaborators 9 11 this schrodinger gap conjecture was proved some time ago in 1 dimension by r lavine 8 and more recently in all dimensions by b andrews and j clutterbuck 1 the proof in this journal by h donnelly 3 of the original gap conjecture in 2 dimensions for the dirichlet laplacian with zero potential is not correct the editors of mathematische zeitschrift have asked us to describe the flaws in the proof in order to clarify the state of the literature donnelly s approach to the problem is a natural one first perform a shape optimization to rule out a non degenerate minimizing domain and then analyze the spectral gap for a sequence of domains degenerating to an interval with the help of results by d jerison 5 for some history on this approach and on the gap conjecture more generally see the report on the aim meeting low eigenvalues of laplace and schrodinger operators 10 especially page 12 of the open problems list the error lies in the proof of the shape optimization step as we now explain donnelly wishes to prove that no minimizing domain can exist for

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

lu decomposition and numerical structure for solving large sparse nonsymmetric linear systems abstract in this work the solution of a large sparse linear system of equations with an arbitrary sparsity pattern is obtained by using lu decomposition method as well as numerical structure approach the lu decomposition method is based on doolittle s method while the numerical structure approach is based on cramer s rule the numerical structure approach produces direct solution without facing fill in problems as encountered in lu decomposition in order to reduce the fill ins in the decomposition the powers of a boolean matrix obtained from the coefficient matrix a are taken so that the fill ins in the structure of a can be known in advance the position of fill ins in a are thus determined in the best choice manner that is it is very effective and memory wise cheap we also outline a method by using numerical structure with reduced computation efforts finally experiments are performed on eight examples to compare the efficiency of the proposed methods the results obtained are reported in a table it is found that the lu decomposition method is much better than numerical structure the usefulness of numerical structure approach is also discussed

algebraic two level preconditioners for the schur complement method the solution of elliptic problems is challenging on parallel distributed memory computers since their green s functions are global to address this issue we present a set of preconditioners for the schur complement domain decomposition method they implement a global coupling mechanism through coarse space components similar to the one proposed in bramble pasciak and shatz math comp 47 1986 pp 103 134 the definition of the coarse space components is algebraic they are defined using the mesh partitioning information and simple interpolation operators these preconditioners are implemented on distributed memory computers without introducing any new global synchronization in the preconditioned conjugate gradient iteration the numerical and parallel scalability of those preconditioners are illustrated on two dimensional model examples that have anisotropy and or discontinuity phenomena

multiply rooted multiscale models for large scale estimation divide and conquer or multiscale techniques have become popular for solving large statistical estimation problems the methods rely on defining a state which conditionally decorrelates the large problem into multiple subproblems each more straightforward than the original however this step cannot be carried out for asymptotically large problems since the dimension of the state grows without bound leading to problems of computational complexity and numerical stability in this paper we propose a new approach to hierarchical estimation in which the conditional decorrelation of arbitrarily large regions is avoided and the problem is instead addressed piece by piece the approach possesses promising attributes it is not a local method the estimate at every point is based on all measurements it is numerically stable for problems of arbitrary size and the approach retains the benefits of the multiscale framework on which it is based a broad class of statistical models a stochastic realization theory an algorithm to calculate statistical likelihoods and the ability to fuse local and nonlocal measurements

algorithm for random close packing of spheres with periodic boundary conditions the isotropic algorithm is constructed for random close packing of equisized spheres with triply periodic boundary conditions all previously published packing methods with periodic boundaries were kinetics determined i e they contained a densification rate as an arbitrary parameter in contrast the present algorithm is kinetic independent and demonstrates an unambiguous convergence to the experimental results to suppress crystallization the main principles of our algorithm are 1 to form a contact network at an early stage and 2 retain contacts throughout the densification as far as possible the particles are allowed to swell by the numerical solution of the differential equations of densification the rhs of these equations is calculated efficiently from a linear system by a combination of conjugate gradient iterations and exact sparse matrix technology when an excessive contact occurs and one of the existing bonds should be broken to continue the densification an efficient criterion based on multidimensional simplex geometry is used for searching the separating bond the algorithm has a well defined termination point resulting in a perfect contact network with the average coordination number six for particle number n 1 and a system of normal reactions between the spheres maintaining the structure these forces are the counterpart of the algorithm and can be used to calculate small elastic particle deformations in a granular medium extensive calculations are presented for 50 n 5 400 and demonstrate very good agreement with the experimental packing density about 0 637 and structure

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

elastic viscoplastic differential equations on a manifold modelling of in plane stretching of sheet metal the computational method for large deformation plasticity called differential equations on a manifold dem has previously been shown to be effective for axisymmetric and plane strain sheet metal forming problems the method has now been formulated for in plane stretching problems incorporated into a computer code and applied to several problems the code s performance is robust and accurate as evidenced by a comparison with other published results incorporation of an automatic mesh generator significantly reduces data preparation time

a parallel multithreaded sparse triangular linear system solver abstract we propose a parallel sparse triangular linear system solver based on the spike algorithm sparse triangular systems are required to be solved in many applications often they are a bottleneck due to their inherently sequential nature furthermore typically many successive systems with the same coefficient matrix and with different right hand side vectors are required to be solved the proposed solver decouples the problem at the cost of extra arithmetic operations as in the banded case compared to the banded case there are extra savings due to the sparsity of the triangular coefficient matrix we show the parallel performance of the proposed solver against the state of the art parallel sparse triangular solver in intel s math kernel library mkl on a multicore architecture we also show the effect of various sparse matrix reordering schemes numerical results show that the proposed solver outperforms mkl s solver in 80 of cases by a factor of 2 47 on average

a structural diagnosis of some ic orderings we present a novel analysis of the potential effectiveness of a matrix ordering for the incomplete cholesky factorization ic in terms of just the sparsity structure by looking at the structure of the approximate inverse implicitly created by ic we can help to explain the success of reverse cuthill mckee orderings the problems ic 0 has under red black orderings that disappear when extra fill is included and where fill must be added to make fill reducing orderings such as minimum degree effective

modified block approximate factorization strategies two variants of modified incomplete block matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations pdes even in the case of quasi singular problems

the size of the first eigenfunction of a convex planar domain the goal of this paper is to estimate the size of the first eigenfunction u uniformly for all convex domains in particular we will locate the place where u achieves its maximum to within a distance comparable to the inradius uniformly for arbitrarily large diameter in addition we will estimate the location of other level sets of u by showing that u is well approximated by the first eigenfunction of a naturally associated ordinary differential schrodinger operator we intend to show in a separate paper that the estimates here are best possible in order of magnitude the present paper depends on the ideas and results of our earlier work j and gj where detailed estimates for the zero set of the second eigenfunction or first nodal line are obtained the paper j also contains some estimates for the first eigenfunction and lowest eigenvalue but the techniques of gj and new techniques introduced here are essential to the best possible estimates for the first eigenfunction presented here the maximum of the first eigenfunction occurs at the point of largest displacement of a vibrating drum with fixed edges when it vibrates at its fundamental or first resonant frequency the first nodal line is the stationary curve of the drum at the second resonant frequency the maximum is harder to find experimentally than the nodal line because it is a single point its location has less influence on the eigenvalue or dirichlet integral so it is also harder to locate mathematically another way to describe the difficulty is as follows to find the maximum of the first eigenfunction we will need to estimate its first directional derivative derivatives of the first eigenfunction are roughly speaking analogous to the second eigenfunction because they are solutions to an eigenfunction equation moreover convexity properties of the first eigenfunction imply that the zero set of the derivative divides the region into two connected components but the derivatives are harder to estimate than a second eigenfunction because they do not vanish at the boundary

calculs de complexite relatifs a une methode de dissection emboitee nous presentons une numerotation de type nested dissection des inconnues d un systeme lineaireax b pour des ensembles de matricescreuses symetriques definies positives correspondant a des famille de graphes non orientes a degre borne et admettant un sqrt n thereme de separation comparativement aux methodes et resultats de rose 9 l algorithme presente est plus simple mais les theoremes de complexite moins generaux en raison de l hypothese restrictive de degre borne en outre les demonstrations font appel en permanence a la structure d arbre sur la famille des separateurs qui constitue par ailleurs une partition de l ensemble des sommets du graphe initial nous presentons ensuite le schema general d implementation dans le cadre du code d elements finis modulef pourdes problemes plans d elements finis et nous donnons quelques mesures comparatives avec la numerotation plus classique qui tend a minimiser le profil de la matrice a nested dissection ordering is given for solving any system of linear equationsax b for the family ofsparse symmetric positive definite matrices corresponding to the class of undirected graphs withbounded degree and satisfyinga sqrt n separator theorem if we compare the methods and results presented by rose 9 our algorithm is more simple but the complexity results are less general because of the restriction of bounded degree besides our proofs use continually the arborescent structure on the family of separators which is by another way a partition of the set of vertices for the initial graph then the general implementation scheme in the finite element package modulef fortwo dimensional finite element problems is presented and numerical comparisons between our ordering and the standard envelope method are given

stochastic preconditioning for diagonally dominant matrices this paper presents a new stochastic preconditioning approach for large sparse matrices for the class of matrices that are rowwise and columnwise irreducibly diagonally dominant we prove that an incomplete rm ldl t factorization in a symmetric case or an incomplete ldu factorization in an asymmetric case can be obtained from random walks and used as a preconditioner it is argued that our factor matrices have better quality i e better accuracy size trade offs than preconditioners produced by existing incomplete factorization methods therefore a resulting preconditioned krylov subspace iterative solver requires less computation than traditional methods to solve a set of linear equations with the same error tolerance the advantage increases for larger and denser matrices these claims are verified by numerical tests and we provide techniques that can potentially extend the theory to non diagonally dominant matrices

robust stability analysis of sparsely interconnected uncertain systems in this paper we consider robust stability analysis of large scale sparsely interconnected uncertain systems by modeling the interconnections among the subsystems with integral quadratic constraints we show that robust stability analysis of such systems can be performed by solving a set of sparse linear matrix inequalities we also show that a sparse formulation of the analysis problem is equivalent to the classical formulation of the robustness analysis problem and hence does not introduce any additional conservativeness the sparse formulation of the analysis problem allows us to apply methods that rely on efficient sparse factorization techniques and our numerical results illustrate the effectiveness of this approach compared to methods that are based on the standard formulation of the analysis problem

effective preconditioning through minimum degree ordering interleaved with incomplete factorization in this paper we study a kind of effective preconditioning technique which interleaves the incomplete cholesky ic factorization with an approximate minimum degree ordering an ic factorization algorithm derived from ikj version gaussian elimination is proposed and some details on implementation are presented then we discuss the ways to compute the degrees of the unnumbered nodes exactly and approximately using the concept of element absorbing when used in conjunction with conjugate gradient algorithm the new preconditioners usually lead to fast convergence the numerical experiments show that the interleaving of symbolic ordering and numerical ic factorization will generate better preconditioners than those generated by the ic factorization without ordering or with purely symbolic ordering ahead of the factorization

the evolution of the minimum degree ordering algorithm over the past fifteen years the implementation of the minimum degree algorithm has received much study and many important enhancements have been made to it this paper describes these various enhancements their historical development and some experiments showing how very effective they are in improving the execution time of the algorithm a shortcoming is also presented that exists in all of the widely used implementations of the algorithm namely that the quality of the ordering provided by the implementations is surprisingly sensitive to the initial ordering for example changing the input ordering can lead to an increase or decrease of as much as a factor of three in the cost of the subsequent numerical factorization this sensitivity is caused by the lack of an effective tie breaking strategy and the authors experiments illustrate the importance of developing such a strategy