a likelihood ratio test for stationarity of rating transitions abstract we study the time stationarity of rating transitions modelled by a time continuous discrete state markov process and derive a likelihood ratio test for multiple markov processes from a multiplicative intensity model maximum likelihood parameter estimates can be written as martingale transform of the processes counting transitions between the rating states so that the profile partial likelihood ratio is asymptotically 2 distributed an application to an internal rating data set reveals highly significant instationarity

layoffs and duration dependence in a model of turnover abstract a semi markov model of turnover resulting from optimizing behavior on the part of workers is developed duration dependence in the transition out of employment is modelled as resulting from a layoff probability which decreases with tenure some preliminary results based on dime data are presented these indicate that a declining hazard rate in the transition out of employment is a significant feature of the data

analytical bounds for block approximate factorization methods abstract we derive simple analytical upper bounds on the spectral condition number associated with the preconditioning of large sparse linear systems by the so called modified block incomplete factorizations the results prove to handle a large class of discrete second order elliptic partial differential equations and support the conclusion that sparse approximate block matrix factorizations provide powerful preconditioners

information measures for generalized gamma family the objective of this paper is to integrate the generalized gamma gg distribution into the information theoretic literature we study information properties of the gg distribution and provide an assortment of information measures for the gg family which includes the exponential gamma weibull and generalized normal distributions as its subfamilies the measures include entropy representations of the log likelihood ratio aic and bic discriminating information between gg and its subfamilies a minimum discriminating information function power transformation information and a maximum entropy index of fit to histogram we provide the full parametric bayesian inference for the discrimination information measures we also provide bayesian inference for the fit of gg model to histogram using a semi parametric bayesian procedure referred to as the maximum entropy dirichlet med the gg information measures are computed for duration of unemployment and duration of ceo tenure

conditioning analysis of positive definite matrices by approximate factorizations abstract the conditioning analysis of positive definite matrices by approximate lu factorizations is usually reduced to that of stieltjes matrices or even to more specific classes of matrices by means of perturbation arguments like spectral equivalence we show in the present work that a wider class which we call almost stieltjes matrices can be used as reference class and that it has decisive advantages for the conditioning analysis of finite element approximations of large multidimensional steady state diffusion problems

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

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

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

on sparse block factorization iterative methods a synthetic formalism is proposed for the description of sparse block factorization iterative methods it is used to develop existence convergence and comparison theorems and to compare block factorization schemes against point factorization schemes

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

testing for unobserved heterogeneity in exponential and weibull duration models abstract we examine the use of the likelihood ratio lr statistic to test for unobserved heterogeneity in duration models based on mixtures of exponential or weibull distributions we consider both the uncensored and censored duration cases the asymptotic null distribution of the lr test statistic is not the standard chi square as the standard regularity conditions do not hold instead there is a nuisance parameter identified only under the alternative and a null parameter value on the boundary of the parameter space as in cho and white 2007a we accommodate these and provide methods delivering consistent asymptotic critical values we conduct a number of monte carlo simulations comparing the level and power of the lr test statistic to an information matrix im test due to chesher 1984 and lagrange multiplier lm tests of kiefer 1985 and sharma 1987 our simulations show that the lr test statistic generally outperforms the im and lm tests we also revisit the work of van den berg and ridder 1998 on unemployment durations and of ghysels et al 2004 on interarrival times between stock trades and as it turns out affirm their original informal inferences

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

a general approach to lagrange multiplier model diagnostics abstract a diagnostic test can be formulated by taking the current model as the null hypothesis and a particular form of misspecification as the alternative the general approach of this paper is to calculate the score for some alternative and base a test on its distribution under the null this distribution is derived in two theorems corresponding to different models such a test is a langrange multiplier test and is asymptotically optimal these tests can be formulated as ways of looking at the residuals for particular types of non randomness the tests are simple to calculate even when the alternative may be very complicated examples are given of tests for omitted variables and block recursive structures

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

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

specification diagnostics based on laguerre alternatives for econometric models of duration abstract score tests of the null hypothesis of exponentially distributed durations conditional on regressors against alternatives in a family of approximations to arbitrary distributions for non negative random variables are developed the test statistics take a simple easily calculated and interpreted form the alternatives considered are expansions of various orders in laguerre polynomials an economic model generating exponential unemployment duration is presented the statistics are applied in a look at unemployment durations in the denver income maintenance experiment data

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

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

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

dynamically relaxed block incomplete factorizations for solving two and three dimensional problems to efficiently solve second order discrete elliptic pdes by krylov subspace like methods one needs to use some robust preconditioning techniques relaxed incomplete factorizations rilu are powerful candidates unfortunately their efficiency critically depends on the choice of the relaxation parameter omega whose optimal value is not only hard to estimate but also strongly varies from one problem to another these methods interpolate between the popular ilu and its modified variant milu concerning the pointwise schemes a new variant of rilu that dynamically computes variable omega omega i has been introduced recently like its ancestor rilu and unlike standard methods it is robust with respect to both existence and performance on top of that it breaks the problem dependence of omega opt a block version of this dynamically relaxed method is proposed and compared with classical pointwise and blockwise methods as well as with some existing dynamic variants showing that with the new blockwise preconditioning technique anisotropies are handled more effectively

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

topics in data assimilation stochastic processes abstract stochastic models with varying degrees of complexity are increasingly widespread in the oceanic and atmospheric sciences one application is data assimilation i e the combination of model output with observations to form the best picture of the system under study for any given quantity to be estimated the relative weights of the model and the data will be adjusted according to estimated model and data error statistics so implementation of any data assimilation scheme will require some assumption about errors which are considered to be random for dynamical models some assumption about the evolution of errors will be needed stochastic models are also applied in studies of predictability the formal theory of stochastic processes was well developed in the last half of the twentieth century one consequence of this theory is that methods of simulation of deterministic processes cannot be applied to random processes without some modification in some cases the rules of ordinary calculus must be modified the formal theory was developed in terms of mathematical formalism that may be unfamiliar to many oceanic and atmospheric scientists the purpose of this article is to provide an informal introduction to the relevant theory and to point out those situations in which that theory must be applied in order to model random processes correctly

dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty abstract we estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations within the context of dynamically orthogonal do field equations transient nonlinear dynamics irregular data and non stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation to efficiently quantify uncertainties in such systems it is essential to vary the dimensionality of the stochastic subspace with time an objective here is to provide criteria to do so working directly with the original equations of the dynamical system under study and its do representation we first analyze the scaling of the computational cost of these do equations with the stochastic dimensionality and show that unlike many other stochastic methods the do equations do not suffer from the curse of dimensionality subsequently we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i stability arguments and ii bayesian data updates we then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2d stochastic fluid flows specifically a double gyre wind driven circulation and a lid driven cavity flow in a basin in these two applications we focus on the growth of uncertainty due to internal instabilities in deterministic flows we consider a range of flow conditions described by varied reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions

taking advantage of the potentialities of dynamically modified block incomplete factorizations recently modified block incomplete factorizations with dynamic diagonal perturbations have been introduced as preconditioning techniques to solve large linear systems and were successfully tested on isotropic and moderately anisotropic two dimensional partial differential equations pdes an analytic study is performed on the basis of improved versions of results published elsewhere displaying why dynamic methods should be preferred to standard block methods in particular the empirical observation that the optimal choice of the involved parameter does not significantly vary from one problem to another is theoretically confirmed it is also explained why in the case of strong anisotropy lack of attention when adding diagonal perturbations may result in very poor performance rules to surmount this inconvenience are discussed and tested on discretized two dimensional pdes

a stochastic model of indopacific sea surface temperature anomalies abstract it is often desirable to represent a rapidly varying physical process as stochastic forcing of some slower dynamical system a review of this approximation is presented the white noise limit of rapidly varying processes reduces the dynamical description of the affected slower systems to stochastic differential equations the properties of which are summarized as an application of stochastic differential equations we review the evidence that indopacific sea surface temperature anomalies sstas can be represented as a stable linear process driven by spatially coherent stochastic forcing an inversemodeling approach is used that is the relevant parameters of the best fit stable linear process are obtained from observations and given these parameters the assumptions of stability and linearity are subsequently tested an optimal initial structure for growth predicted by the model is found to occur approximately seven months before most of the major warm and cold el nino la nina events in the data record the optimal structure preferentially occurs in the boreal spring with the mature phase of the extreme event occurring in the subsequent late fall winter the best model to fit the observations including the dependence of the dynamical system on the annual cycle is driven by stochastic forcing with periodic statistics these statistics are inferred using a time dependent fluctuation dissipation relation the variation of the stochastic forcing with the annual cycle is not the same as that of the optimal initial structure yet when a linear numerical model is driven with the derived stochastic forcing the optimal structure including its variation with the annual cycle is recovered this implies that deterministic dynamics play an important role in setting up the optimal structure and therefore a mature phase of el nino may be predicted before the optimal structure actually appears

a specification test for univariate and multivariate proportional hazards models this paper applies white s 1982 econometrica 50 1 25 information matrix im test for correct model specification to proportional hazards models of univariate and multivariate censored survival data several alternative estimators of the test statistic are presented and their size performance examined white also suggested an estimator of the parameter covariance matrix that was robust to certain forms of model misspecification this has been subsequently proposed by others e g royall 1986 international statistical review 54 221 226 and applied by huster brookmeyer and self 1989 biometrics 45 145 156 as part of an independence working model iwm approach to multivariate censored survival data we illustrate how the im test can be used for both univariate data and as part of the iwm approach to multivariate data

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

on the conditioning analysis of block approximate factorization methods abstract the paper is devoted to the conditioning analysis of modified block incomplete factorizations of a given stieltjes matrix we obtain new results improve other theories and compare all existing upper bounds through numerical experiments applied to discrete elliptic pdes our results show that an o h 1 spectral bound can be achieved for a large class of problems

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

estimating parameters in stochastic systems a variational bayesian approach this work is concerned with approximate inference in dynamical systems from a variational bayesian perspective when modelling real world dynamical systems stochastic differential equations appear as a natural choice mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics hence inference in such processes has drawn much attention here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational bayesian algorithm the paper begins by showing that the new extension of this variational algorithm can be used for state estimation smoothing and converges to the original algorithm however the main focus is on estimating the hyper parameters of these systems i e drift parameters and diffusion coefficients the new approach is validated on a range of different systems which vary in dimensionality and non linearity these are the ornstein uhlenbeck process the exact likelihood of which can be computed analytically the univariate and highly non linear stochastic double well and the multivariate chaotic stochastic lorenz 63 3d model as a special case the algorithm is also applied to the 40 dimensional stochastic lorenz 96 system in our investigation we compare this new approach with a variety of other well known methods such as the hybrid monte carlo dual unscented kalman filter full weak constraint 4d var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases in particular we show that we are able to estimate parameters in both the drift deterministic and the diffusion stochastic part of the model evolution equations using our new methods

joint and separate score tests for state dependence and unobserved heterogeneity abstract the paper compares separate conditional and joint score tests of duration dependence and unobserved heterogeneity when the null is the exponential model and the alternative is the heterogeneous weibull model the score tests based on the conditional score function include the neyman c test as a special case an examination of the non null distribution of the joint test explains when all score tests have low power in the presence of multiple misspecifications monte carlo experiments show that the conditional score tests are superior to the standard separate tests which confound unobserved heterogeneity and duration dependence

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

on a simple test for neglected heterogeneity in panel studies summary this paper deals with the performance of a simple test for the detection of unmeasured heterogeneity in regression models for panel data it is assumed that the unmeasured heterogeneity can be modeled by an individual specific random variable that does not vary over time conditionally on the individual specific effects and the regressor variables the distribution of the dependent variable is assumed to belong to a linear exponential family this class of distributions includes linear models for continuous dependent variables as well as nonlinear models for discrete outcomes the proposed test does not assume a special form of the distribution of the heterogeneity and the test statistic can be calculated with available computer programs the performance of the test is investigated in a simulation experiment

on axelsson s perturbations abstract a nice perturbation technique was introduced by axelsson and further developed by gustafsson to prove that factorization iterative methods are able under appropriate conditions to reach a convergence rate larger by an order of magnitude than that of classical schemes gustafsson observed however that the perturbations introduced to prove this result seemed actually unnecessary to reach it in practice in the present work on the basis of eigenvalue bounds recently obtained by the author we offer an alternative approach which brings a partial confirmation of gustafsson s conjecture

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

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

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

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

conditioning analysis of modified block incomplete factorizations abstract we improve the conditioning analysis of modified block incomplete factorizations of stieltjes matrices letting n denote the number of diagonal blocks our results show that the spectral condition number is bounded by n for a large class of two dimensional pdes

specification diagnostics for duration models a martingale approach abstract this paper develops specification diagnostics for discrete time duration models which are based on the martingale theory of stochastic processes the methods can be applied to parametric grouped data duration models with time varying covariates right censoring and which account for unobserved heterogeneity a residual process is defined such that under the correct model specification the process is approximately a martingale difference sequence chi square tests based on these residuals are derived results from monte carlo simulations are presented

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

existence and conditioning properties of sparse approximate block factorizations a particular class of sparse approximate block lu factorizations is investigated sufficient conditions are determined for the existence of b lp 1 u for a given coefficient matrix a bounds on the spectral condition number of b 1 a are obtained for symmetric b and a when applied to discrete multidimensional elliptic partial differential equations these results lead under reasonable assumptions to o h 1 spectral condition numbers

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

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

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

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

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

lower eigenvalue bounds for pencils of matrices abstract eigenvalue bounds are obtained for pencils of matrices a vb where a is a stieltjes matrix and b is positive definite under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods where b represents the approximate factorization of a the upper bounds obtained depend on the connectivity structure of the matrices involved which enters through matrix graph considerations in addition a more classical argument is used to obtain a lower bound potential applications of these results include a partial confirmation of gustafsson s conjecture concerning the nonnecessity of axelsson s perturbations

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

numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows the quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions such situations are common in multiscale intermittent and non homogeneous fluid and ocean flows the dynamically orthogonal do field equations provide an adaptive methodology to predict the probability density functions of such flows the present work derives efficient computational schemes for the do methodology applied to unsteady stochastic navier stokes and boussinesq equations and illustrates and studies the numerical aspects of these schemes semi implicit projection methods are developed for the mean and for the do modes and time marching schemes of first to fourth order are used for the stochastic coefficients conservative second order finite volumes are employed in physical space with new advection schemes based on total variation diminishing methods other results include i the definition of pseudo stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic ii symmetric advection schemes for the stochastic velocities iii the use of generalized inversion to deal with singular subspace covariances or deterministic modes and iv schemes to maintain orthonormal modes at the numerical level to verify our implementation and study the properties of our schemes and their variations a set of stochastic flow benchmarks are defined including asymmetric dirac and symmetric lock exchange flows lid driven cavity flows and flows past objects in a confined channel different reynolds number and grashof number regimes are employed to illustrate robustness optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations

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

a survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods such as preconditioned conjugate gradient methods in recent years various estimates have been derived for perturbed modified block incomplete factorizations we survey and extend some of these and derive new estimates in particular we derive upper and lower estimates of individual eigenvalues and of condition number this includes a discussion that the condition number of preconditioned second order elliptic difference matrices is o h 1 some of the methods are applied to compute certain parameters involved in the computation of the preconditioner

vectorizable preconditioners for elliptic difference equations in three space dimensions abstract until now research in vectorizable preconditioners has mainly focused on problems in two space dimensions employing all degrees of freedom we here present preconditioners for 3d matrices based on point line and plane factorizations both in vectorizable and for reasons of comparisons in recursive versions by a limit analysis and accompanying numerical tests of elliptic difference equations with anisotropy we find that methods based on line block factorizations are to be preferred in general among the vectorizable methods a line method where euler expansion of the factors of the pivot blocks is used is the most robust with respect to the problem parameters furthermore this method turns out to approach the best of the recursive methods in efficiency in terms of operation counts it has the added bonus of almost maximal vectorlength

a comment on a simple test for neglected heterogeneity in panel studies a recently proposed test for unmeasured heterogeneity hamerle 1990 biometrics 46 193 199 has the merit of being relatively convenient to compute due to the extremely simple form of the variance estimator employed it is argued that this variance estimator is inconsistent in that it asymptotically overestimates the true variance this renders the test procedure as described by hamerle asymptotically undersized

generalised residuals and heterogeneous duration models with applications to the weilbull model this paper deals with models for the duration of an event that are misspecified by the neglect of random multiplicative heterogeneity in the hazard function this type of misspecification has been widely discussed in the literature e g heckman and singer 1982 lancaster and nickell 1980 but no study of its effect on maximum likelihood estimators has been given this paper aims to provide such a study with particular reference to the weibull regression model which is by far the most frequently used parametric model e g heckman and borjas 1980 lancaster 1979 in this paper we define generalised errors and residuals in the sense of cox and snell 1968 1971 and show how their use materially simplifies the analysis of both true and misspecified duration models we show that multiplicative heterogeneity in the hazard of the weibull model has two errors in variables interpretations we give the exact asymptotic inconsistency of m l estimation in the weibull model and give a general expression for the inconsistency of m l estimators due to neglected heterogeneity for any duration model to o 2 where 2 is the variance of the error term we also discuss the information matrix test for neglected heterogeneity in duration models and consider its behaviour when 2 0

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