equidistribution of expanding translates of curves and diophantine approximation on matrices we study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space g gamma of a semisimple algebraic group g we define two families of algebraic subvarieties of the associated partial flag variety g p which give the obstructions to non divergence and equidistribution we apply this to prove that for lebesgue almost every point on an analytic curve in the space of m times n real matrices whose image is not contained in any subvariety coming from these two families dirichlet s theorem on simultaneous diophantine approximation cannot be improved the proof combines geometric invariant theory ratner s theorem on measure rigidity for unipotent flows and linearization technique

asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds extending the earlier results for analytic curve segments in this article we describe the asymptotic behaviour of evolution of a finite segment of a c n smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n manifold in particular we show that if the curve satisfies certain natural geometric conditions the pushforward of the parameter measure on the curve under the ge odesic flow converges to the normalized canonical riemannian measure on the tangent bundle in the limit we also study the limits of geodesic evolution of shrinking segments we use ratner s classification of ergodic invariant measures for unipo tent flows on homogeneous spaces of so n 1 and an observation re lating local growth properties of smooth curves and dynamics of linear sl 2 r actions

equidistribution of expanding translates of curves and dirichlet s theorem on diophantine approximation we show that for almost all points on any analytic curve on k which is not contained in a proper affine subspace the dirichlet s theorem on simultaneous approximation as well as its dual result for simultaneous approximation of linear forms cannot be improved the result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in k 1 the proof involves ratner s theorem on ergodic properties of unipotent flows on homogeneous spaces

limiting distributions of curves under geodesic flow on hyperbolic manifolds we consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic n manifold under the geodesic flow suppose that the curve is not contained in a stable leaf of the flow it is shown that under the geodesic flow the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold moreover if this immersed submanifold is a proper subset then a lift of the curve to the universal covering space t 1 h n is mapped into a proper subsphere of the ideal boundary sphere h n under the visual map this proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in h n covering the closed immersed submanifold in particular if the visual map does not send a lift of the curve into a proper subsphere of h n then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bun dle of the manifold with respect to the normalized natural rie mannian measure the proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of so n 1

stability of parallel triangular system solvers several parallel algorithms have been proposed for the solution of triangular systems the stability of four of them is analysed here a fan in algorithm a block elimination method a method based on a factorized power series expansion of the matrix inverse and a method based on a divide and conquer matrix inversion technique new forward error and residual bounds are derived including an improvement on the bounds of sameh and brent for the fan in algorithm a forward error bound is identified that holds not only for all the methods described here but for any triangular equation solver that does not rely on algebraic cancellation among the implications of the bound is that any such method is extremely accurate for certain special types of triangular systems

imf an incomplete multifrontal lu factorization for element structured sparse linear systems we propose an incomplete multifrontal lu factorization imf preconditioner that extends supernodal multifrontal methods to incomplete factorizations it can be used as a preconditioner in a krylov subspace method to solve large scale sparse linear systems with an element structure e g those arising from a finite element discretization of a partial differential equation the fact that the element matrices are dense is exploited to increase the computational performance and the robustness of the factorization through efficient partial pivoting imf is compared with the multilevel arms2 the level of fill in ilu and the threshold based ilutp preconditioners our experiments indicate that imf is competitive with arms2 on saddle point problems arising in the solution of the steady state navier stokes equation experiments with element structured matrices arising from structural engineering applications found in the university of florida sparse matrix collection illustrate the robustness of imf final

the accuracy of solutions to triangular systems triangular systems play a fundamental role in matrix computations it has been prominently stated in the literature but is perhaps not widely appreciated that solutions to triangular systems are usually computed to high accuracy higher than the traditional condition numbers for linear systems suggest this phenomenon is investigated by use of condition numbers appropriate to the componentwise backward error analysis of triangular systems results of wilkinson are unified and extended among the conclusions are that the conditioning of a triangular system depends on the right hand side as well as the coefficient matrix that use of pivoting in lu qr and cholesky factorisations can greatly improve the conditioning of a resulting triangular system and that a triangular matrix may be much more or less ill conditioned than its transpose

the impact of high performance computing in the solution of linear systems trends and problems we review the influence of the advent of high performance computing on the solution of linear equations we will concentrate on direct methods of solution and consider both the case when the coefficient matrix is dense and when it is sparse we will examine the current performance of software in this area and speculate on what advances we might expect in the early years of the next century

particle filters for state estimation of jump markov linear systems jump markov linear systems jmls are linear systems whose parameters evolve with time according to a finite state markov chain in this paper our aim is to recursively compute optimal state estimates for this class of systems we present efficient simulation based algorithms called particle filters to solve the optimal filtering problem as well as the optimal fixed lag smoothing problem our algorithms combine sequential importance sampling a selection scheme and markov chain monte carlo methods they use several variance reduction methods to make the most of the statistical structure of jmls computer simulations are carried out to evaluate the performance of the proposed algorithms the problems of on line deconvolution of impulsive processes and of tracking a maneuvering target are considered it is shown that our algorithms outperform the current methods

stability of the partitioned inverse method for parallel solution of sparse triangular systems several authors have recently considered a parallel method for solving sparse triangular systems with many right hand sides the method employs a partition into sparse factors of the product form of the inverse of the coefficient matrix it is shown here that while the method can be unstable stability is guaranteed if a certain scalar that depends on the matrix and the partition is small and that this scalar is small when the matrix is well conditioned moreover when the partition is chosen so that the factors have the same sparsity structure as the coefficient matrix the backward error matrix can be taken to be sparse

maximum likelihood sequence estimation of cpm signals transmitted over rayleigh flat fading channels a method for the sequential updating of log likelihood functions for maximum likelihood sequence estimation is presented it is shown that in a general case this method can be implemented using kalman filtering techniques for the special case of rayleigh flat fading and continuous phase modulation cpm signaling this approach is shown to lead to an attractive receiver structure this type of receiver called the linear predictive receiver can be implemented in the form of the viterbi algorithm with the trellis updates being computed using a bank of finite pulse response fir filter and square operations simulation results are presented that demonstrate the superiority of the linear predictive receiver over receivers employing differential detection in the presence of fast fading it is shown that the optimal linear predictive receiver does not possess an irreducible error rate for a class of rayleigh fading channels used to model terrestrial mobile channels

approximate conditional mean particle filtering for linear nonlinear dynamic state space models we consider linear systems whose state parameters are separable into linear and nonlinear sets and evolve according to some known transition distribution and whose measurement noise is distributed according to a mixture of gaussians in doing so we propose a novel particle filter that addresses the optimal state estimation problem for the aforementioned class of systems the proposed filter referred to as the approximate conditional mean particle filter acm pf is a combination of the approximate conditional mean filter and the sequential importance sampling particle filter the algorithm development depends on approximating a mixture of gaussians distribution with a moment matched gaussian in the weight update recursion a condition indicating when this approximation is valid is given in order to evaluate the performance of the proposed algorithm we address the blind signal detection problem for an impulsive flat fading channel and the tracking of a maneuvering target in the presence of glint noise extensive computer simulations were carried out for computationally intensive implementations large number of particles the proposed algorithm offers performance that is comparable to other state of the art particle filtering algorithms in the scenario where computational horsepower is heavily constrained it is shown that the proposed algorithm offers the best performance amongst the considered algorithms for these specific examples

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

optimal parallel solution of sparse triangular systems this paper considers the parallel solution of a sparse system lx b with triangular matrix l which is often a performance bottleneck in parallel computation when many systems with the same matrix are to be solved the parallel efficiency can be improved by representing the inverse of l as a product of a few sparse factors the factorization with the smallest number of factors is constructed subject to the requirement that no new nonzero elements are created applications are to iterative solvers with triangular preconditioners to structural analysis or to power systems applications experimental results on the connection machine show the method to be highly valuable

uniformly distributed orbits of certain flows on homogeneous spaces is non compact for a co compact lattice this result was strengthened by f irstenberg 1 lj proving that every orbit is uniformly distributed with respect to a g invariant measure for non uniform lattices in sl 2 r using a classification of invariant measures obtained by dani in 21 dani and smillie 31 proved that every non periodic orbit is uniformly distributed there are also various results obtained on orbit closures and invariant measures etc of larger subgroups consisting of unipotent elements especially the horospherical subgroups recently there was a spurt in the area initiated by margulis proof cf 151 see also 7 of oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows the reader is referred to the survey articles by dani 41 and margulis 14j for various related developments we now note some conjectures expected to hold for orbits of a unipotent flow namely the u action on

simple linear time algorithms to test chordality of graphs test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs chordal graphs arise naturally in the study of gaussian elimination on sparse symmetric matrices acyclic hypergraphs arise in the study of relational data bases rose tarjan and lueker siam j c

mixture kalman filters in treating dynamic systems sequential monte carlo methods use discrete samples to represent a complicated probability distribution and use rejection sampling importance sampling and weighted resampling to complete the on line filtering task we propose a special sequential monte carlo method the mixture kalman filter which uses a random mixture of the gaussian distributions to approximate a target distribution it is designed for on line estimation and prediction of conditional and partial conditional dynamic linear models which are themselves a class of widely used non linear systems and also serve to approximate many others compared with a few available filtering methods including monte carlo methods the gain in efficiency that is provided by the mixture kalman filter can be very substantial another contribution of the paper is the formulation of many non linear systems into conditional or partial conditional linear form to which the mixture kalman filter can be applied examples in target tracking and digital communications are given to demonstrate the procedures proposed

parallelizable approximate solvers for recursions arising in preconditioning abstract for the recursions used in the modified incomplete lu preconditioner namely the construction of incomplete decomposition the forward elimination and the back substitution parallelizable approximate solvers are introduced the analysis shows that the solutions of the recursions depend only weakly on their initial conditions which may be interpreted to indicate that the inexact solution is close in some sense to the exact one the method is based on a domain decomposition approach suitable for parallel implementations on message passing architectures it requires a small number of communication steps per preconditioned iteration independent of the number of subdomains and the size of the problem the overlapping subdomains are either squares suitable for mesh connected arrays of processors or defined by the data flow rule of the recursions suitable for line connected arrays of simd or vector processors numerical examples show that in both cases the overhead in the number of preconditioned iterations required for convergence is small in comparison with the speedup gained

partitioning a chordal graph into transitive subgraphs for parallel sparse triangular solution abstract a recent approach for solving sparse triangular systems of equations on massively parallel computers employs a factorization of the triangular coefficient matrix to obtain a representation of its inverse in product form the number of general communication steps required by this approach is proportional to the number of factors in the factorization the triangular matrix can be symmetrically permuted to minimize the number of factors over suitable classes of permutations and thereby the complexity of the parallel algorithm can be minimized algorithms for minimizing the number of factors over several classes of permutations have been considered in earlier work let f l lt denote the symmetric filled matrix corresponding to a cholesky factor l and let gf denote the adjacency graph of f we consider the problem of minimizing the number of factors over all permutations which preserve the structure of gf the graph model of this problem is to partition the vertices gf into the fewest transitively closed subgraphs over all perfect elimination orderings while satisfying a certain precedence relationship the solution to this chordal graph partitioning problem can be described by a greedy scheme which eliminates a largest permissible subgraph at each step further the subgraph eliminated at each step can be characterized in terms of lengths of chordless paths in the current elimination graph this solution relies on several results concerning transitive perfect elimination orderings introduced in this paper we describe a partitioning algorithm with o v e time and space complexity

filtering via simulation auxiliary particle filters this article analyses the recently suggested particle approach to filtering time series we suggest that the algorithm is not robust to outliers for two reasons the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution here we tackle the first of these problems

following a moving target monte carlo inference for dynamic bayesian models markov chain monte carlo mcmc sampling is a numerically intensive simulation technique which has greatly improved the practicality of bayesian inference and prediction however mcmc sampling is too slow to be of practical use in problems involving a large number of posterior target distributions as in dynamic modelling and predictive model selection alternative simulation techniques for tracking moving target distributions known as particle filters which combine importance sampling importance resampling and mcmc sampling tend to suffer from a progressive degeneration as the target sequence evolves we propose a new technique based on these same simulation methodologies which does not suffer from this progressive degeneration

a clique tree algorithm for partitioning a chordal graph into transitive subgraphs a partitioning problem on chordal graphs that arises in the solution of sparse triangular systems of equations on parallel computers is considered roughly the problem is to partition a chordal graph g into the fewest transitively orientable subgraphs over all perfect elimination orderings of g subject to a certain precedence relationship on its vertices in earlier work a greedy scheme that solved the problem by eliminating a largest subset of vertices at each step was described and an algorithm implementing the scheme in time and space linear in the number of edges of the graph was provided here a more efficient greedy scheme obtained by representing the chordal graph in terms of its maximal cliques which eliminates a subset of the leaf cliques of the current graph at each step is described several new results about minimal vertex separators in chordal graphs and in particular the concept of a em critical separator of a leaf clique are employed to prove that the new scheme solves the partitioning problem we provide an algorithm implementing the scheme in time and space linear in the size of the clique tree

fine grained parallel incomplete lu factorization this paper presents a new fine grained parallel algorithm for computing an incomplete lu factorization all nonzeros in the incomplete factors can be computed in parallel and asynchronously using one or more sweeps that iteratively improve the accuracy of the factorization unlike existing parallel algorithms the amount of parallelism is large irrespective of the ordering of the matrix and matrix ordering can be used to enhance the accuracy of the factorization rather than to increase parallelism numerical tests show that very few sweeps are needed to construct a factorization that is an effective preconditioner

adaptive bayesian multiuser detection for synchronous cdma with gaussian and impulsive noise we consider the problem of simultaneous parameter estimation and data restoration in a synchronous cdma system in the presence of either additive gaussian or additive impulsive white noise with unknown parameters the impulsive noise is modeled by a two term gaussian mixture distribution bayesian inference of all unknown quantities is made from the superimposed and noisy received signals the gibbs sampler a markov chain monte carlo procedure is employed to calculate the bayesian estimates the basic idea is to generate ergodic random samples from the joint posterior distribution of all unknown and then to average the appropriate samples to obtain the estimates of the unknown quantities adaptive bayesian multiuser detectors based on the gibbs sampler are derived for both the gaussian noise synchronous cdma channel and the impulsive noise synchronous cdma channel a salient feature of the proposed adaptive bayesian multiuser detectors is that they can incorporate the a priori symbol probabilities and they produce as output the a posteriori symbol probabilities that is they are soft input soft output algorithms hence these methods are well suited for iterative processing in a coded system which allows the adaptive bayesian multiuser detector to refine its processing based on the information from the decoding stage and vice versa a receiver structure termed the adaptive turbo multiuser detector

flows on homogeneous spaces and diophantine approximation on manifolds we present a new approach to metric diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces this approach yields a new proof of a conjecture of mahler originally settled by v g sprindzuk in 1964 we also prove several related hypotheses of baker and sprindzuk formulated in 1970s the core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices

expanding translates of curves and dirichlet minkowski theorem on linear forms we show that a multiplicative form of dirichlet s theorem on simulta neous diophantine approximation as formulated by minkowski cannot be improved for almost all points on any analytic curve on r k which is not contained in a proper affine subspace such an investigation was initiated by davenport and schmidt in the late sixties the diophantine problem is then settled via showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in r k 1 gets equidistributed in the limit we use ratner s theorem on unipotent flows linearization techniques and a new observation about intertwined linear dynamics of various sl m r s contained in sl k 1 r

fuzzy particle filtering for uncertain systems in this paper we propose a novel fuzzy particle filtering method for online estimation of nonlinear dynamic systems with fuzzy uncertainties this approach uses a sequential fuzzy simulation to approximate the possibilities of the state intervals in the state space and estimates the state by fuzzy expected value operator to solve the degeneracy problem of the fuzzy particle filter one corresponding resampling technique is introduced in addition we compare the fuzzy particle filter with ordinary particle filter in both aspects of the theoretical basis and algorithm design and demonstrate that the proposed filter outperforms standard particle filters especially when the number of the particles is small the numerical simulations of two continuous state nonlinear systems and a jump markov system are employed to show the effectiveness and robustness of the proposed fuzzy particle filter

a blind particle filtering detector of signals transmitted over flat fading channels a new particle filtering detector pfd is proposed for blind signal detection over flat rayleigh fading channels whose model coefficients are unknown the detector employs a hybrid importance function and a mixture kalman filter it also incorporates an auxiliary particle filtering strategy with a smoothing kernel in the resampling step further by considering practical information of communication systems and the physical interpretation of the adopted second order autoregressive ar channel model a fully blind particle filtering implementation is developed the structure of the proposed pfd can be easily adapted to other system requirements simulations are provided that demonstrate the performance of the new pfd

particle filtering for demodulation in fading channels with non gaussian additive noise an efficient particle filtering algorithm is developed to solve the problem of demodulation of m ary modulated signals under conditions of fading channels in the presence of non gaussian additive noise simulations for mdpsk signals are presented the results show that the algorithm outperforms the current methods

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