the multifrontal method for sparse matrix solution theory and practice this paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems the method is formulated in terms of frontal matrices update matrices and an assembly tree formal definitions of these notions are given based on the sparse matrix structure various advances to the basic method are surveyed they include the role of matrix reorderings the use of supernodes and other implementatjon techniques the use of the method in different computational environments is also described

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

quantifying the effect of the surrogate marker by information gain statistical validation of a surrogate marker has been studied for more than a decade recently alonso et al 2004 biometrics 60 724 728 proposed a quantity called the likelihood reduction factor lrf to evaluate the validity of a surrogate marker however as pointed out in the present article the lrf may not correctly validate a surrogate marker therefore a new quantity the proportion of information gain pig using the kullback leibler information is proposed simulations show that under some model assumptions the pig precisely reflects the role of a surrogate marker

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

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

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

fuzzy uniform spaces in this paper we introduce the notion of fuzzy uniform spaces in section 2 we give some basic results and show how a fuzzy topology is derived from a fuzzy uniformity in sections 3 and 4 we prove that the notions which are introduced are good extensions and we prove that the category of uniform spaces is nicely injected in the category of fuzzy uniform spaces finally in section 5 we prove some basic results

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 bayesian approach to surrogacy assessment using principal stratification in clinical trials a surrogate marker s is a variable that can be measured earlier and often more easily than the true endpoint t in a clinical trial most previous research has been devoted to developing surrogacy measures to quantify how well s can replace t or examining the use of s in predicting the effect of a treatment z however the research often requires one to fit models for the distribution of t given s and z it is well known that such models do not have causal interpretations because the models condition on a postrandomization variable s in this article we directly model the relationship among t s and z using a potential outcomes framework introduced by frangakis and rubin 2002 biometrics 58 21 29 we propose a bayesian estimation method to evaluate the causal probabilities associated with the cross classification of the potential outcomes of s and t when s and t are both binary we use a log linear model to directly model the association between the potential outcomes of s and t through the odds ratios the quantities derived from this approach always have causal interpretations however this causal model is not identifiable from the data without additional assumptions to reduce the nonidentifiability problem and increase the precision of statistical inferences we assume monotonicity and incorporate prior belief that is plausible in the surrogate context by using prior distributions we also explore the relationship among the surrogacy measures based on traditional models and this counterfactual model the method is applied to the data from a glaucoma treatment study

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

a generalised form of compactness in fuzzy topological spaces the n compactness of wang 13 extended in 151 provides the first theory of compactness which is applicable to arbitrary fuzzy subsets many of the standard properties associated with compactness are valid for n compactness including the product theorem furthermore n compactness reduces to ordinary compactness in the topologically generated case on closer inspection one finds evidence that a concept weaker than n compactness might also be desirable for example n compactness does not reduce to any of the standard fuzzified versions of compactness 1 2 6 73 in the case where the fuzzy subset under consideration is the whole space in particular when applied to the whole space n compactness is substantially stronger than fuzzy compactness as fuzzy compactness seems to be a good working concept for the whole space an extension of this notion to arbitrary fuzzy subsets might well have applications not accessible to the much stronger n compactness as a consequence of the definition an n compact fuzzy set is forced to attain a maximum value as a result it is possible to have fuzzy sets which are never n compact even if the fuzzy topology has only a finite number of open fuzzy sets our intuition based on the case of general topology would expect every fuzzy set to be compact in such a case in this paper we introduce a weaker generalisation of compactness which we call f compactness it is shown that ficompactness is inherited by closed fuzzy subsets continuous images finite suprema and products it reduces to compactness or to fuzzy compactness in all cases where

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

criteria for the validation of surrogate endpoints in randomized experiments the validation of surrogate endpoints has been studied by prentice 1989 statistics in medicine 8 431 440 and freedman graubard and schatzkin 1992 statistics in medicine 11 167 178 we extended their proposals in the cases where the surrogate and the final endpoints are both binary or normally distributed letting t and s be random variables that denote the true and surrogate endpoint respectively and z be an indicator variable for treatment prentice s criteria are fulfilled if z has a significant effect on t and on s if s has a significant effect on t and if z has no effect on t given s freedman relaxed the latter criterion by estimating pe the proportion of the effect of z on t that is explained by s and by requiring that the lower confidence limit of pe be larger than some proportion say 0 5 or 0 75 this condition can only be verified if the treatment has a massively significant effect on the true endpoint a rare situation we argue that two other quantities must be considered in the validation of a surrogate endpoint re the effect of z on t relative to that of z on s and gamma z the association between s and t after adjustment for z a surrogate is said to be perfect at the individual level when there is a perfect association between the surrogate and the final endpoint after adjustment for treatment a surrogate is said to be perfect at the population level if re is 1 a perfect surrogate fulfills both conditions in which case s and t are identical up to a deterministic transformation fieller s theorem is used for the estimation of pe re and their respective confidence intervals logistic regression models and the global odds ratio model studied by dale 1986 biometrics 42 909 917 are used for binary endpoints linear models are employed for continuous endpoints in order to be of practical value the validation of surrogate endpoints is shown to require large numbers of observations

using surrogate failure time data to increase cost effectiveness in clinical trials summary in clinical trials with failure time outcomes the cost of follow up for the endpoint of primary interest is sometimes expensive in many of these settings one or more secondary or surrogate failure time outcomes are also available which either occur much earlier or are much less expensive to follow than the primary outcome this paper presents a method for using these secondary outcomes to increase the power of detecting a treatment effect on the primary outcome only without introducing bias the proposed method requires that patients be randomized to either extended or limited follow up a test statistic is constructed by forming a minimum variance linear combination of a linear rank statistic based on the primary outcome and a mean zero weighted sum of linear rank statistics based on secondary outcomes from both follow up arms large sample distribution theory for this statistic is developed while moderate sample behaviour is explored through a simulation study

an efficient helmholtz solver for acoustic transversely isotropic media abstractthe acoustic approximation even for anisotropic media is widely used in current industry imaging and inversion algorithms mainly because p waves constitute most of the energy recorded in seismic exploration the resulting acoustic formulas tend to be simpler resulting in more efficient implementations and they depend on fewer medium parameters however conventional solutions of the acoustic wave equation with higher order derivatives suffer from s wave artifacts thus we separate the quasi p wave propagation in anisotropic media into the elliptic anisotropic operator free of the artifacts and the nonelliptic anisotropic components which form a pseudodifferential operator we then develop a separable approximation of the dispersion relation of nonelliptic anisotropic components specifically for transversely isotropic media finally we iteratively solve the simpler lower order elliptical wave equation for a modified source function that includes the nonelliptical terms represented in the

computationally efficient three dimensional acoustic finite difference frequency domain seismic modeling in vertical transversely isotropic media with sparse direct solver abstractthe computational burden of frequency domain full waveform inversion fwi of wide aperture fixed spread data is conventionally reduced by limiting the inversion to a few discrete frequencies in this framework frequency domain seismic modeling is performed efficiently for multiple sources by solving the linear system resulting from the discretization of the time harmonic wave equation with the massively parallel sparse direct solver frequency domain seismic modeling based on the sparse direct solver dsfdm requires specific design finite difference stencils of compact support to minimize the computational cost of the lower upper decomposition of the impedance matrix in terms of memory demand and floating point operations a straightforward adaptation of such finite difference stencil originally developed for the isotropic acoustic wave equation is proposed to introduce vertical transverse isotropy vti in the modeling without any extra computational cost the method relies on a fourth ord

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

stable p wave modeling for reverse time migration in tilted ti media we present an approach for p wave modeling in inhomogeneous transversely isotropic media with tilted symmetry axis tti media suitable for anisotropic reverse time migration the proposed approach is based on wave equations derived from first principles the equations of motion and hooke s law under the acoustic ti approximation consequently no assumptions are made about the spatial variation of medium parameters a rotation of the stress and strain tensors to a local coordinate system aligned with the ti symmetry axis makes it possible to benefit from the simple and sparse form of the ti elastic tensor in that system the resulting wave equations can be formulated either as a set of five first order or as a set of two second order partial differential equations for the constant density case the second order tti wave equations involve mixed and nonmixed second order spatial derivatives with respect to global nonrotated coordinates we propose a numerical implementation of these equations using high order centered finite differences to minimize modeling artifacts related to the use of centered first derivative operators we use discrete second derivative operators for the nonmixed second order spatial derivatives and repeated discrete first derivative operators for the mixed derivatives such a combination of finite difference operators leads to a stable wave propagator provided that the operators are designed properly in practice stability is achieved by slightly weighting down terms that contain mixed derivatives this has a minor practically negligible effect on the kinematics of wave propagation the stability of the presented scheme in inhomogeneous tti models with rapidly varying anisotropic symmetry axis direction is demonstrated with numerical examples

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

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

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

an acoustic wave equation for anisotropic media a wave equation derived using the acoustic medium assumption for p waves in transversely isotropic ti media with a vertical symmetry axis vti media yields a good kinematic approximation to the familiar elastic wave equation for vti media the wavefield solutions obtained using this vti acoustic wave equation are free of shear waves which significantly reduces the computation time compared to the elastic wavefield solutions for exploding reflector type applications from this vti acoustic wave equation the eikonal and transport equations that describe the ray theoretical aspects of wave propagation in a ti medium are derived these equations based on the acoustic assumption shear wave velocity 0 are much simpler than their elastic counterparts yet they yield an accurate description of traveltimes and geometrical amplitudes numerical examples prove the usefulness of this acoustic equation in simulating the kinematic aspects of wave propagation in complex ti models

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

cauchy filters and prefilters abstract a suitable characterisation of cauchy filters in a uniform space inspires the definition of a cauchy prefilter in a fuzzy uniform space this is shown to be a good extension of the standard notion and the theory of cauchy prefilters in a fuzzy uniform space is developed

the relationship between a fuzzy uniformity and its family of a level uniformities abstract we show that a fuzzy uniformity d is uniquely determined by its family of level uniformities d 0 1 on the other hand if a family d 0 1 of uniformities on a set x is given and the family satisfies certain conditions then there exists a unique fuzzy uniformity d on x whose family of level uniformities is the given family

surrogate marker evaluation from an information theory perspective the last 20 years have seen lots of work in the area of surrogate marker validation partly devoted to frame the evaluation in a multitrial framework leading to definitions in terms of the quality of trial and individual level association between a potential surrogate and a true endpoint buyse et al 2000 biostatistics 1 49 67 a drawback is that different settings have led to different measures at the individual level here we use information theory to create a unified framework leading to a definition of surrogacy with an intuitive interpretation offering interpretational advantages and applicable in a wide range of situations our method provides a better insight into the chances of finding a good surrogate endpoint in a given situation we further show that some of the previous proposals follow as special cases of our method we illustrate our methodology using data from a clinical study in psychiatry

microlocal analysis of seismic inverse scattering in anisotropic elastic media seismic data is modeled in the high frequency approximation using the techniques of microlocal analysis we consider general anisotropic elastic media our methods are designed to allow for the formation of caustics the data is modeled in two ways first we give a microlocal treatment of the kirchhoff approximation where the medium is assumed to be piecewise smooth and reflection and transmission occur at interfaces second we give a refined view on the born approximation based upon a linearization of the scattering process in the medium parameters around a smooth background medium the joint formulation of born and kirchhoff scattering allows us to take into account general scatterers as well as the nonlinear dependence of reflection coefficients on the medium parameters the latter allows the treatment of scattering up to grazing angles the outcome of the analysis is a characterization of the singular part of seismic data we obtain a set of pseudodifferential operators that annihilate the data in the process we construct a fourier integral operator and a reflectivity function such that the data can be represented by this operator acting on the reflectivity function in our construction this fourier integral operator becomes invertible we give the conditions for invertibility for general acquisition geometry the result is also of interest for inverse scattering in acoustic media c 2002 john wiley sons inc

strictly minimal fuzzy neighborhood division rings abstract the aim of this article is on the one hand to determine suitable classes of fuzzy neighborhood division rings that can be used as scalar domains for linear fuzzy neighborhood spaces so as to accommodate 1 a a nonvanishing linear transformation t is continuous on the space if and only if the kernel of t is n hyperclosed 2 b a nonvanishing linear transformation t is continuous on the space if and only if it has n hyperclosed graph 3 c every finite dimensional linear space admits exactly one admissible fuzzy neighborhood system and on the other hand to examine several elegant features of those classes of fuzzy neighborhood division rings

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

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

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

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

generalised filters 1 we continue the study started in 8 of generalised filters prime prefilters have played a central role in the theory of lowen fuzzy uniform spaces and lowen discovered a characterisation of the set of all minimal prime prefilters finer than a given prefilter in terms of ultrafilters we define the notion of a prime generalised filter and describe the set of all minimal prime g filters finer than a given g filter in terms of ultrafilters the relationship between prime prefilters and prime g filters is revealed the behaviour of the images and preimages of g filters are investigated

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

boundedness in uniform spaces and fuzzy uniform spaces abstract the notion of a weak cauchy ultrafilter in a uniform space is defined and a useful characterisation theorem obtained it is then shown that a subset is bounded iff every ultrafilter on the set is weak cauchy the properties of bounded sets can then be established using this theorem turning to fuzzy uniform spaces the notion of a weak cauchy prime prefilter is defined and the theory of these prefilters is developed the notion of a bounded fuzzy set is defined and it is shown that it is a good extension of the standard notion the theory of bounded fuzzy sets is then obtained and it is seen to be a generalisation of the theory of bounded sets in a uniform space

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

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

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

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

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

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

acoustic inverse scattering via helmholtz operator factorization and optimization we present a joint acoustic seismic inverse scattering and finite frequency reflection tomography program formulated as a coupled set of optimization problems in terms of inhomogeneous helmholtz equations we use a higher order finite difference scheme for these helmholtz equations to guarantee sufficient accuracy we adapt a structured approximate direct solver for the relevant systems of algebraic equations which addresses storage requirements through compression to yield a complexity for computing the gradients or images in the optimization problems that consists of two parts viz the cost for all the matrix factorizations which is roughly o rn for example o rnlogn when d 2 times the number of frequencies and the cost for all solutions by substitution which is roughly o n for example o nlog rlogn when d 2 times the number of frequencies times the number of sources events where n n d if n is the number of grid samples in any direction and r is a parameter depending on the preset accuracy and the problem at hand with this complexity the multi frequency approach to inverse scattering and finite frequency tomography becomes computationally feasible with large data sets in dimensions d 2 and 3

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

an entropy based nonparametric test for the validation of surrogate endpoints we present a nonparametric test to validate surrogate endpoints based on measure of divergence and random permutation this test is a proposal to directly verify the prentice statistical definition of surrogacy the test does not impose distributional assumptions on the endpoints and it is robust to model misspecification our simulation study shows that the proposed nonparametric test outperforms the practical test of the prentice criterion in terms of both robustness of size and power we also evaluate the performance of three leading methods that attempt to quantify the effect of surrogate endpoints the proposed method is applied to validate magnetic resonance imaging lesions as the surrogate endpoint for clinical relapses in a multiple sclerosis trial copyright 2012 john wiley sons ltd

on the propagation of polarization sets for systems of real principal type we define and study the polarization sets for vector valued distributions systems of real principal type are defined and we prove that for such systems of pseudodifferential operators we have a unique propagation of the polarization sets for the solutions

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

an algorithm adapting encoded simultaneous source full waveform inversion to marine streamer acquisition data abstractin recent years the encoded simultaneous source full waveform inversion essfwi algorithm which can reduce the computational cost of the individual source full waveform inversion fwi has been studied by many researchers although the essfwi technique can be successfully applied to fixed spread acquisition data the application of this technique to marine streamer data is difficult because of the acquisition geometry the difference in the acquisition geometry between the observed and modeled data produces missing trace parts in a stacked seismogram and unwanted events in the residual seismogram to avoid these unwanted residuals the correlation based essfwi technique has been suggested by several researchers however the correlation based approach unpredictably modifies the residual seismograms which may result in distortion of the inverted velocity model we developed an alternative approach to properly cope with the acquisition geometry problem our algorithm modified the offset limited

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

3d finite difference frequency domain modeling of visco acoustic wave propagation using a massively parallel direct solver a feasibility study we present a finite difference frequency domain method for 3d visco acoustic wave propagation modeling in the frequency domain the underlying numerical problem is the resolution of a large sparse system of linear equations whose right hand side term is the source this system is solved with a massively parallel direct solver we first present an optimal 3d finite difference stencil for frequency domain modeling the method is based on a parsimonious staggered grid method differential operators are discretized with second order accurate staggered grid stencils on different rotated coordinate systems to mitigate numerical anisotropy an antilumped mass strategy is implemented to minimize numerical dispersion the stencil incorporates 27 grid points and spans two grid intervals dispersion analysis shows that four grid points per wavelength provide accurate simulations in the 3d domain to assess the feasibility of the method for frequency domain full waveform inversion we computed simulations in the 3d seg eage overthrust model for frequencies 5 7 and 10 hz results confirm the huge memory requirement of the factorization several hundred figabytes but also the cpu efficiency of the resolution phase few seconds per shot heuristic scalability analysis suggests that the memory complexity of the factorization is o 35n 4 for a n 3 grid our method may provide a suitable tool to perform frequency domain full waveform inversion using a large distributed memory platform further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time and frequency domain modeling of wave propagation to perform 3d full waveform inversion

fast 3d frequency domain full waveform inversion with a parallel block low rank multifrontal direct solver application to obc data from the north sea wide azimuth long offset obc obn surveys provide a suitable framework to perform computationally efficient frequency domain full waveform inversion fwi with a few discrete frequencies frequency domain seismic modeling is performed efficiently with moderate computational resources for a large number of sources with a sparse multifrontal direct solver gauss elimination techniques for sparse matrices approximate solutions of the time harmonic wave equation are computed using a block low rank blr approximation leading to a significant reduction in the operation count and in the volume of communication during the lu factorization as well as offering a great potential for reduction in the memory demand moreover the sparsity of the seismic source vectors is exploited to speed up the forward elimination step during the computation of the monochromatic wavefields the relevance and the computational efficiency of the frequency domain fwi performed in the visco acoustic vti approximation is shown with a real 3d obc case study from the north sea the fwi subsurface models show a dramatic resolution improvement relative to the initial model built by reflection traveltime tomography the amplitude errors introduced in the modeled wavefields by the blr approximation for different low rank thresholds have a negligible footprint in the fwi results with respect to a standard multifrontal sparse direct factorization and without compromise on the accuracy of the imaging the blr approximation can bring a reduction of the lu factor size by a factor up to three this reduction is not yet exploited to reduce the effective memory usage ongoing work the flop reduction can be larger than a factor of 10 and can bring a factor of time reduction of around three moreover this reduction factor tends to increase with frequency namely with the matrix size frequency domain visco acoustic vti fwi can be viewed as an efficient tool to build an initial model for elastic fwi of 4 c obc data

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