indefinite overlapping schwarz methods for time dependent stokes problems abstract a class of indefinite overlapping schwarz methods is introduced and studied for time dependent stokes problems discretized with either mixed finite elements or mixed spectral elements the methods proposed are based on the solution of local time dependent stokes problems on overlapping subdomains and on the solution of a coarse time dependent stokes problem defined on the coarse subdomain mesh the resulting preconditioner is accelerated by an appropriate krylov space method producing a very efficient scalable and parallel solver

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

adaptive sparse polynomial chaos expansion based on least angle regression polynomial chaos pc expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable so called polynomial chaos basis the number of terms to be computed grows dramatically with the size of the input random vector which makes the computational cost of classical solution schemes may it be intrusive i e of galerkin type or non intrusive unaffordable when the deterministic finite element model is expensive to evaluate to address such problems the paper describes a non intrusive method that builds a sparse pc expansion first an original strategy for truncating the pc expansions based on hyperbolic index sets is proposed then an adaptive algorithm based on least angle regression lar is devised for automatically detecting the significant coefficients of the pc expansion beside the sparsity of the basis the experimental design used at each step of the algorithm is systematically complemented in order to avoid the overfitting phenomenon the accuracy of the pc metamodel is checked using an estimate inspired by statistical learning theory namely the corrected leave one out error as a consequence a rather small number of pc terms are eventually retained sparse representation which may be obtained at a reduced computational cost compared to the classical full pc approximation the convergence of the algorithm is shown on an analytical function then the method is illustrated on three stochastic finite element problems the first model features 10 input random variables whereas the two others involve an input random field which is discretized into 38 and 30 500 random variables respectively

uniform preconditioners for a parameter dependent saddle point problem with application to generalized stokes interface equations we consider an abstract parameter dependent saddle point problem and present a general framework for analyzing robust schur complement preconditioners the abstract analysis is applied to a generalized stokes problem which yields robustness of the cahouet chabard preconditioner motivated by models for two phase incompressible flows we consider a generalized stokes interface problem application of the general theory results in a new schur complement preconditioner for this class of problems the robustness of this preconditioner with respect to several parameters is treated results of numerical experiments are given that illustrate robustness properties of the preconditioner

stability estimates and structural spectral properties of saddle point problems for a general class of saddle point problems sharp estimates for babuska s inf sup stability constants are derived in terms of the constants in brezzi s theory in the finite dimensional hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented which are helpful for the convergence analysis of common krylov subspace methods the theoretical results are applied to two model problems from optimal control with time periodic state equations numerical experiments with the preconditioned minimal residual method are reported

stopping time resampling for sequential monte carlo methods summary motivated by the statistical inference problem in population genetics we present a new sequential importance sampling with resampling strategy the idea of resampling is key to the recent surge of popularity of sequential monte carlo methods in the statistics and engineering communities but existing resampling techniques do not work well for coalescent based inference problems in population genetics we develop a new method called stopping time resampling which allows us to compare partially simulated samples at different stages to terminate unpromising partial samples and to multiply promising samples early on to illustrate the idea we first apply the new method to approximate the solution of a dirichlet problem and the likelihood function of a non markovian process then we focus on its application in population genetics all our examples show that the new resampling method can significantly improve the computational efficiency of existing sequential importance sampling methods

a class of fast exact bayesian filters in dynamical models with jumps we address the statistical filtering problem in dynamical models with jumps when a particular application is adequately modeled by linear and gaussian probability density functions with jumps a usual method consists in approximating the optimal bayesian estimate in the sense of the minimum mean square error mmse in a linear and gaussian jump markov state space system jmss practical solutions include algorithms based on numerical approximations or on sequential monte carlo smc methods in this paper we propose a class of alternative methods which consists in building statistical models which locally similarly model the problem of interest but in which the computation of the mmse estimate can be be computed exactly without numerical nor smc approximations and at a computational cost which is linear in the number of observations

iterative techniques for time dependent stokes problems abstract in this paper we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent stokes equations using a mixed finite element spatial discretization at each time step a two by two block system corresponding to a perturbed stokes problem must be solved although there are a number of techniques for iteratively solving this type of block system to be effective they require a good preconditioner for the resulting pressure operator schur complement in contrast to the time independent stokes equations where the pressure operator is well conditioned the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced and or the reynolds number is increased in this paper we shall describe and analyze preconditioners for the resulting pressure systems these preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and reynolds number parameter k

efficient computation of rcs from scatterers of uncertain shapes we propose a way of accounting for the lack of detailed knowledge about material shapes in computational time domain electromagnetics we use legendre gauss lobatto stroud 2 and stroud 3 quadrature formulas to solve the resulting stochastic equation and we show the efficiency of the proposed method over statistical monte carlo simulations we also show how the radar cross section rcs in scattering is affected by the uncertainty in shape of the objects and by the direction of the incident field

error analysis of some finite element methods for the stokes problem we prove the optimal order of convergence for some two dimensional finite element methods for the stokes equations first we consider methods of the taylor hood type the triangular p3 p2 element and the qk qk 1 i k 2 family of quadrilateral elements then we introduce two new low order methods with piecewise constant approximations for the pressure the analysis is performed using our macroelement technique which is reviewed in a slightly altered form

on line inference for multiple changepoint problems we propose an on line algorithm for exact filtering of multiple changepoint problems this algorithm enables simulation from the true joint posterior distribution of the number and position of the changepoints for a class of changepoint models the computational cost of this exact algorithm is quadratic in the number of observations we further show how resampling ideas from particle filters can be used to reduce the computational cost to linear in the number of observations at the expense of introducing small errors and propose two new optimum resampling algorithms for this problem one a version of rejection control allows the particle filter to automatically choose the number of particles required at each time step the new resampling algorithms substantially out perform standard resampling algorithms on examples we consider and we demonstrate how the resulting particle filter is practicable for segmentation of human gc content

block boundary value methods for the solution of ordinary differential equations block boundary value methods are considered and applied to solve initial value problems results on their convergence and stability properties are presented and their intermediate position between the class of multistep and runge kutta methods is established

a note on efficient preconditioner of implicit runge kutta methods with application to fractional diffusion equations abstract this paper is concerned with the construction of efficient preconditioning method for systems arising from implicit runge kutta discretizations of time dependent pdes a polynomial preconditioner is proposed based on the kronecker product splitting iteration method introduced by chen bit 54 607 721 2014 some useful properties on the spectrum of the preconditioned matrix are established numerical experiments concerning discrete solution of fractional diffusion equations are presented to show the effectiveness of the proposed polynomial preconditioner

on approximate maximum likelihood methods for blind identification how to cope with the curse of dimensionality we discuss approximate maximum likelihood methods for blind identification and deconvolution these algorithms are based on particle approximation versions of the expectation maximization em algorithm we consider three different methods which differ in the way the posterior distribution of the symbols is computed the first algorithm is a particle approximation method of the fixed interval smoothing the two filter smoothing and the novel joined two filter smoothing involve an additional backward information filter because the state space is finite it is furthermore possible at each step to consider all the offsprings of any given particle it is then required to construct a novel particle swarm by selecting among all these offsprings particle positions and computing appropriate weights we propose here a novel unbiased selection scheme which minimizes the expected loss with respect to general distance functions we compare these smoothing algorithms and selection schemes in a monte carlo experiment we show a significant performance increase compared to the expectation maximization viterbi algorithm emva a fixed lag smoothing algorithm and the block constant modulus algorithm cma

a gaussian mixture filter for range only tracking range only tracking problems arise in extended data collection for inverse synthetic radar applications robotics navigation and other areas for such problems the conditional density of the state variable given the measurement history is multi modal or exhibits curvature even in seemingly benign scenarios for this reason the use of extended kalman filter ekf and other nonlinear filtering techniques based on gaussian approximations can result in inaccurate estimates we introduce a new filter for such tracking problems in two dimensions called the gaussian mixture range only filter gmrof which generates gaussian mixture approximations to the conditional densities the filter equations are derived by analytic techniques based on the specific nonlinearities of range only tracking a slight modification of the standard measurement process model noise before nonlinearity is used to simplify the moment calculations implementation requires at each step the fitting of a low order gaussian mixture to a simple exponentiated trigonometric function of a scalar variable simulations involving scenarios from earlier comparative studies indicate that the gmrof consistently outperformed the ekf and achieved the accuracy of particle filters while significantly reducing the computational cost

a preconditioner for optimal control problems constrained by stokes equation with a time harmonic control in this article we construct an efficient preconditioner for solving the algebraic systems arising from discretized optimal control problems with time periodic stokes equations based on a preconditioning technique for stationary stokes constrained optimal control problems considered in an earlier paper by the authors a simplified analysis of the derivation of the preconditioner and its properties is presented the preconditioner is fully parameter independent and the condition number of the corresponding preconditioned matrix is bounded by 2 the so constructed preconditioner is favourably compared with another robust preconditioner for the same problem

a circulant preconditioner for the systems of lmf based ode codes in this paper a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations odes is investigated most ode codes based on implicit formulas at each integration step need the solution of one or more unsymmetric linear systems that are often large and sparse here the boundary value methods a class of implicit methods for the numerical integration of odes based on linear multistep formulas are considered more in detail for initial value problems theoretical and practical arguments are given to show that the block circulant preconditioner can give fast preconditioned iterations for various classes of differential problems moreover the p circulants a recently introduced circulant approximation for unsymmetric toeplitz matrices are shown to be more suitable sometimes than other circulant matrices for the underlying block preconditioner

on three grid fourier analysis for multigrid in this paper we present three grid fourier analysis for multigrid methods due to the recursive structure of a multigrid iteration this analysis can be deduced from the well known two grid fourier analysis the coarse grid correction part of multigrid algorithms can be more accurately evaluated with the three grid analysis we apply the analysis to several scalar equations and discretizations with an emphasis on problems with a multigrid coarse grid correction difficulty like upwind discretizations of the convection diffusion equation the main focus lies on possible improvements by carefully chosen galerkin operators and or by an additional acceleration with restarted gmres gmres m numerical test calculations validate the theoretical predictions

particle markov chain monte carlo methods summary markov chain monte carlo and sequential monte carlo methods have emerged as the two main tools to sample from high dimensional probability distributions although asymptotic convergence of markov chain monte carlo algorithms is ensured under weak assumptions the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and or if highly correlated variables are updated independently we show here how it is possible to build efficient high dimensional proposal distributions by using sequential monte carlo methods this allows us not only to improve over standard markov chain monte carlo schemes but also to make bayesian inference feasible for a large class of statistical models where this was not previously so we demonstrate these algorithms on a non linear state space model and a levy driven stochastic volatility model

fourier analysis of gmres m preconditioned by multigrid this paper deals with convergence estimates of gmres m saad and schultz siam j sci statist comput 7 1986 pp 856 869 preconditioned by multigrid brandt math comp 31 1977 pp 333 390 hackbusch multi grid methods and applications springer berlin 1985 fourier analysis is a well known and useful tool in the multigrid community for the prediction of two grid convergence rates brandt math comp 31 1977 pp 333 390 stuben and trottenberg in multigrid methods lecture notes in math 960 k stuben and u trottenberg eds springer berlin pp 1 176 this analysis is generalized here to the situation in which multigrid is a preconditioner since it is possible to obtain the whole spectrum of the two grid iteration matrix a preconditioned krylov subspace acceleration method like gmres m implicitly builds up a minimal residual polynomial the determination of the polynomial coefficients is easily possible and can be done explicitly since from fourier analysis a simple block diagonal two grid iteration matrix results based on the gmres m polynomial sharp theoretical convergence estimates can be obtained which are compared with estimates based on the spectrum of the iteration matrix several numerical scalar test problems are computed in order to validate the theoretical predictions

efficient preconditioners for optimality systems arising in connection with inverse problems this paper is devoted to the numerical treatment of linear optimality systems os that arise in connection with inverse problems for partial differential equations if such inverse problems are regularized by tikhonov regularization then it follows from standard theory that the associated os is well posed provided that the regularization parameter alpha is positive and that the involved state equation satisfies suitable assumptions we explain and analyze how certain mapping properties of the operators appearing in the os can be employed to define efficient preconditioners for finite element fe approximations of such systems the key feature of the scheme is that the number of iterations needed to solve the preconditioned problem by the minimal residual method is bounded independently of the mesh parameter h used in the fe discretization and increases only moderately as alpha rightarrow0 more specifically if the stopping criterion for the iteration process is defined in terms of the associated energy norm then the number of iterations required in the severely ill posed case cannot grow faster than o ln alpha 2 our analysis is based on a careful study of the operators involved which yields the distribution of the eigenvalues of the preconditioned os finally the theoretical results are illuminated by a number of numerical experiments addressing both a model problem studied by borzi kunisch and kwak siam j control optim 41 2003 pp 1477 1497 and an inverse problem arising in connection with electrocardiography nielsen cai and lysaker math biosci 210 2007 pp 523 553

an optimal preconditioner for a class of saddle point problems with a penalty term iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems an optimal preconditioner which is independent of the discretization and the penalty parameter is constructed this approach is then used to design an iterative method with a convergence rate bounded independently of the lame parameters occurring in the equations of linear elasticity

multigrid methods for implicit runge kutta and boundary value method discretizations of parabolic pdes advanced time discretization schemes for stiff systems of ordinary differential equations odes such as implicit runge kutta and boundary value methods have many appealing properties however the resulting systems of equations can be quite large and expensive to solve many techniques exploiting the structure of these systems have been developed for general odes for spatial discretizations of time dependent partial differential equations pdes these techniques are in general not sufficient and also the structure arising from spatial discretization has to be taken into consideration we show here that for time dependent parabolic problems this can be done by multigrid methods as in the stationary elliptic case the key to this approach is the use of a smoother that updates several unknowns at a spatial grid point simultaneously the overall cost is essentially proportional to the cost of integrating a scalar ode for each grid point combination of the multigrid principle with both time stepping and waveform relaxation techniques is described together with a convergence analysis numerical results are presented for the isotropic heat equation and a general diffusion equation with variable coefficients

brief paper detection and estimation for abruptly changing systems the problem of state estimation and system structure detection for discrete stochastic dynamical systems with parameters which may switch among a finite set of values is considered the switchings are modelled by a markov chain with known transition probabilities a brief survey and a unified treatment of the existing suboptimal algorithms are provided the optimal algorithms require exponentially increasing memory and computations with time simulation results comparing the various suboptimal algorithms are presented

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

preconditioning techniques for a mixed stokes darcy model in porous media applications we study numerical methods for a mixed stokes darcy model in porous media applications the global model is composed of two different submodels in a fluid region and a porous media region coupled through a set of interface conditions the weak formulation of the coupled model is of a saddle point type the mixed finite element discretization applied to the saddle point problem leads to a coupled indefinite and nonsymmetric linear system of algebraic equations we apply the preconditioned gmres method to solve the discrete system and are particularly interested in efficient and effective decoupled preconditioning techniques several decoupled preconditioners are proposed theoretical analysis and numerical experiments show the effectiveness and efficiency of the preconditioners effects of physical parameters on the convergence performance are also investigated

order optimal preconditioners for implicit runge kutta schemes applied to parabolic pdes in this paper we show that standard preconditioners for parabolic pdes discretized by implicit euler or crank nicolson schemes can be reused for higher order fully implicit runge kutta time discretization schemes we prove that the suggested block diagonal preconditioners are order optimal for a stable and irreducible runge kutta schemes with invertible coefficient matrices the theoretical investigations are confirmed by numerical experiments

the shifted rayleigh mixture filter for bearings only tracking of maneuvering targets this paper introduces the shifted rayleigh mixture filter srmf which is based on jump markov linear systems the formulation permits the presence of clutter for bearings only tracking problems involving maneuvering targets the conditional density of the target state given the available measurements evolves as a growing mixture of probability density functions associated with a history of manoeuvre modes similar to other mixture algorithms the srmf approximates this conditional density by a gaussian mixture of fixed order unlike the extended or unscented kalman filters the shifted rayleigh filter incorporates an exact calculation of the posterior density when the prior is assumed to be gaussian given the latest bearings measurement computer simulations are provided to demonstrate the performance of the algorithm

analysis of iterative methods for the steady and unsteady stokes problem application to spectral element discretizations a new and detailed analysis of the basic uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady stokes operator is presented the paper focuses on the following new aspects explicit construction of the uzawa pressure operator spectrum for a semiperiodic model problem general relationship of the convergence rate of the uzawa procedure to classical inf sup discretization analysis and application of the method to high order variational discretization

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

a new algebraic multigrid approach for stokes problems standard discretizations of stokes problems lead to linear systems of equations in saddle point form making difficult the application of algebraic multigrid methods in this paper a new approach is proposed it consists in first transforming the system by pre and post multiplication with simple algebraic sparse block triangular matrices this is a form of pre conditioning in the literal sense designed to make sure that the transformed matrix is well adapted to multigrid in particular after transformation all the diagonal blocks are symmetric and positive definite and correspond to or resemble a discrete laplace operator then to each of these diagonal blocks is associated a prolongation that works well for it using any relevant algebraic or geometric multigrid method next a multigrid scheme for the global system is naturally set up by combining these partial prolongations with a galerkin coarse grid matrix for this approach combined with damped jacobi smoothing a uniform two grid convergence bound is derived for the global system under the assumption that the two grid schemes for the different diagonal blocks are themselves uniformly convergent this result is illustrated by a few examples showing further that time dependent problems and variable viscosity can be handled in a natural way without requiring parameter adjustment a numerical comparison also shows that the new approach can be more effective than state of the art block preconditioning techniques

multigrid waveform relaxation on spatial finite element meshes the discrete time case the efficiency of numerically solving time dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously the theoretical properties of one such method namely the discrete time multigrid waveform relaxation method are investigated for systems of ordinary differential equations obtained by spatial finite element discretisation of linear parabolic initial boundary value problems the results are compared to the corresponding continuous time results the theory is illustrated for a one dimensional and a two dimensional model problem and checked against results obtained by numerical experiments

analysis of the minimal residual method applied to ill posed optimality systems we analyze the performance of the minimal residual minres method applied to linear karush kuhn tucker systems arising in connection with inverse problems such optimality systems typically have a saddle point structure and have unique solutions for all alpha 0 where alpha is the parameter employed in the tikhonov regularization unfortunately the associated spectral condition number is very large for small values of alpha which strongly indicates that their numerical treatment is difficult our main result shows that a broad range of linear ill posed optimality systems can be solved efficiently with the minres method this result is obtained by carefully analyzing the spectrum of the associated saddle point operator except for a few isolated eigenvalues the spectrum consists of three bounded intervals krylov subspace methods handle such problems very well for severely ill posed cases techniques based on chebyshev polynomials are applied to prove that the number of iterations needed by

a multilevel algorithm for mixed problems we describe a multilevel algorithm for the numerical solution of symmetric indefinite problems which arise e g from mixed finite element approximations of the stokes equation the main difficulty besides the indefiniteness is the lack of regularity of the solution of the corresponding continuous problem this is overcome by introducing a scale of mesh dependent norms the convergence rate of the described algorithm is bounded independently of the meshsize for convenience we only discuss jacobi relaxation as smoothing operator in detail in the last section we comment on lanczos type smoothing procedures

multigrid analysis for the time dependent stokes problem certain implicit time stepping procedures for the incompressible stokes or navier stokes equations lead to a singular perturbed stokes type problem at each time step the paper presents a convergence analysis of a geometric multigrid solver for the system of linear algebraic equations resulting from the disretization of the problem using a finite element method several smoothing iterative methods are considered a smoother based on distributive iterations the braess sarazin and inexact uzawa smoother convergence analysis is based on smoothing and approximation properties in special norms a robust independent of time step and mesh parameter estimate is proved for the two grid and multigrid w cycle convergence factors

an ell _ 1 laplace robust kalman smoother robustness is a major problem in kalman filtering and smoothing that can be solved using heavy tailed distributions e g l1 laplace this paper describes an algorithm for finding the maximum a posteriori map estimate of the kalman smoother for a nonlinear model with gaussian process noise and l1 laplace observation noise the algorithm uses the convex composite extension of the gauss newton method this yields convex programming subproblems to which an interior point path following method is applied the number of arithmetic operations required by the algorithm grows linearly with the number of time points because the algorithm preserves the underlying block tridiagonal structure of the kalman smoother problem excellent fits are obtained with and without outliers even though the outliers are simulated from distributions that are not l1 laplace it is also tested on actual data with a nonlinear measurement model for an underwater tracking experiment the l1 laplace smoother is able to construct a smoothed fit without data removal from data with very large outliers

on line inference for hidden markov models via particle filters we consider the on line bayesian analysis of data by using a hidden markov model where inference is tractable conditional on the history of the state of the hidden component a new particle filter algorithm is introduced and shown to produce promising results when analysing data of this type the algorithm is similar to the mixture kalman filter but uses a different resampling algorithm we prove that this resampling algorithm is computationally efficient and optimal among unbiased resampling algorithms in terms of minimizing a squared error loss function in a practical example that of estimating break points from well log data our new particle filter outperforms two other particle filters one of which is the mixture kalman filter by between one and two orders of magnitude

sequential monte carlo samplers summary we propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space each distribution being known up to a normalizing constant these probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential monte carlo methods this methodology allows us to derive simple algorithms to make parallel markov chain monte carlo algorithms interact to perform global optimization and sequential bayesian estimation and to compute ratios of normalizing constants we illustrate these algorithms for various integration tasks arising in the context of bayesian inference

flexible multiple semicoarsening for three dimensional singularly perturbed problems we present robust parallel multigrid based solvers for 3d scalar partial differential equations the robustness is obtained by combining multiple semicoarsening strategies matrix dependent transfer operators and a krylov subspace acceleration the basis for the 3d preconditioner is a 2d method with multiple semicoarsened grids based on the mg s method from c w oosterlee appl numer math 19 1995 pp 115 128 and t washio and c w oosterlee gmd arbeitspapier 949 gmd st augustin germany 1995 the 2d method is generalized to three dimensions with a line smoother in the third dimension the method based on semicoarsening has been parallelized with the grid partitioning technique j linden b steckel and k stuben parallel comput 7 1988 pp 461 475 o a mcbryan et al impact comput sci engrg 3 1991 pp 1 75 and is evaluated as a solver and as a preconditioner on a mimd machine the robustness of the 3d method is shown for finite volume and finite difference discretizations of 3d anisotropic diffusion equations and convection dominated convection diffusion problems

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

a note on the uniform inf sup condition for the brinkman problem in highly heterogeneous media abstract in this short note we study the interpolation spaces based uniform inf sup condition for brinkman equation in highly heterogeneous domains the result is known to hold in the case of a uniform constant permeability our numerical construction and experiments show that it is not true for the taylor hood and the mini elements applied to problems with highly heterogeneous coefficients

multi grid methods for stokes and navier stokes equations in the present paper we introduce transforming iterations an approach to construct smoothers for indefinite systems this turns out to be a convenient tool to classify several well known smoothing iterations for stokes and navier stokes equations and to predict their convergence behaviour epecially in the case of high reynolds numbers using this approach we are able to construct a new smoother for the navier stokes equations based on incomplete lu decompositions yielding a highly effective and robust multi grid method besides some qualitative theoretical convergence results we give large numerical comparisons and tests for the stokes as well as for the navier stokes equations for a general convergence theory we refer to 29

multilevel iteration for mixed finite element systems with penalty in this paper the authors consider the solution of the discrete systems that arises when a mixed finite element approach is used to approximate the solution of second order elliptic boundary value problems by the introduction of a penalty parameter these equations can be approximated by the solution of a symmetric and positive definite penalty system on the velocity subspace iterative procedures are developed and analyzed for this penalty system based on the hierarchical basis approach as well as on the standard multigrid approach finally numerical experiments are presented that illustrate the convergence behavior suggested by the theory

multigrid waveform relaxation on spatial finite element meshes the continuous time case the waveform relaxation method and its multigrid acceleration are studied as solution procedures for the system of ordinary differential equations obtained by finite element discretisation of a linear parabolic initial boundary value problem the convergence properties of the continuous time algorithm are theoretically investigated on finite length and infinite length time intervals in addition quantitative convergence estimates and numerical results are presented for one dimensional and two dimensional model problems

interior penalty preconditioners for mixed finite element approximations of elliptic problems it is established that an interior penalty method applied to second order elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method this relation can be utilized in order to construct preconditioners for the discrete mixed system as an example a family of additive schwarz preconditioners for these systems is constructed numerical examples which confirm the theoretical results are also presented

association pattern discovery via theme dictionary models summary discovering patterns from a set of text or more generally categorical data is an important problem in many disciplines such as biomedical research linguistics artificial intelligence and sociology we consider here the well known market basket problem that is often discussed in the data mining community and is also quite ubiquitous in biomedical research the data under consideration are a set of baskets where each basket contains a list of items our goal is to discover themes which are defined as subsets of items that tend to co occur in a basket we describe a generative model i e the theme dictionary model for such data structures and describe two likelihood based methods to infer themes that are hidden in a collection of baskets we also propose a novel sequential monte carlo method to overcome computational challenges using both simulation studies and real applications we demonstrate that the new approach proposed is significantly more powerful than existing methods such as association rule mining and topic modelling in detecting weak and subtle interactions in the data

a particle filtering scheme for processing time series corrupted by outliers the literature in engineering and statistics is abounding in techniques for detecting and properly processing anomalous observations in the data most of these techniques have been developed in the framework of static models and it is only in recent years that we have seen attempts that address the presence of outliers in nonlinear time series for a target tracking problem described by a nonlinear state space model we propose the online detection of outliers by including an outlier detection step within the standard particle filtering algorithm the outlier detection step is implemented by a test involving a statistic of the predictive distribution of the observations such as a concentration measure or an extreme upper quantile we also provide asymptotic results about the convergence of the particle approximations of the predictive distribution and its statistics and assess the performance of the resulting algorithms by computer simulations of target tracking problems with signal power observations

notes on a saddle point reformulation of mixed variational problems abstract we summarize some general ideas regarding approximation of mixed variational problems using saddle point reformulation we consider the concepts of optimal and almost optimal or test norm and provide estimates for the continuity and stability constants a preconditioning strategy for solving the discrete mixed formulations is used in combination with the special test norms we further provide a choice for a discrete trial space that depends on the choice of a standard test space and leads to discrete stability when using the appropriate test norm examples to illustrate how the stability of the saddle point discretization can be improved using special test norms are included

detecting change points for shifts in mean and variance using fuzzy classification maximum likelihood change point algorithms knowing the time of changes called change point cp in a process is crucial for engineers to recognize the root cause fast and accurately since special causes may induce simultaneous changes in mean and variance detecting changes in both at once is required many methodologies in quality control were developed for detecting changes in either mean or variance only and process parameters were assumed known often however they are rarely known exactly and a small estimation error may lead to unfavorable cp estimates fuzzy partitioning is better suited to cases of vague boundaries between two segments which appear very often in reality a new mechanism called fuzzy classification maximum likelihood change point fcml cp algorithm is proposed to detect shifts in mean and variance simultaneously a cp framework is transferred into a mixture model and then a fcml cp algorithm is created through fuzzy classification maximum likelihood procedures the proposed fcml cp can be applied to phase i and ii processes without knowledge of in control process parameters it can estimate multiple cps of process mean or and variance simultaneously the effectiveness and superiority of fcml cp are shown by extensive experiments with numerical and real data sets specifically the proposed fcml cp is superior to the commonly used statistical mixture likelihood approach using expectation maximization em algorithm it is much more time saving especially the remarkable performance of fcml cp in detecting cps for small changes is particularly important and helpful for engineers to recognize the special cause fast and correctly since an out of control signal resulted from small changes is usually delayed long we propose a new algorithm fcml cp to detect change points in a statistical process the mixture likelihood embedded fuzzy c partition is utilized to estimate the change points the method outperforms statistical likelihood approaches various experiments show the effectiveness and practicability of the proposed method excellent performance in detecting small changes is helpful for root cause analyses

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 robust finite element method for darcy stokes flow finite element methods for a family of systems of singular perturbation problems of a saddle point structure are discussed the system is approximately a linear stokes problem when the perturbation parameter is large while it degenerates to a mixed formulation of poisson s equation as the perturbation parameter tends to zero it is established basically by numerical experiments that most of the proposed finite element methods for stokes problem or the mixed poisson s system are not well behaved uniformly in the perturbation parameter this is used as the motivation for introducing a new robust finite element which exhibits this property

evaluation of st preconditioners for saddle point problems the general block st decomposition of the saddle point problem is used as a preconditioner to transform the saddle point problem into an equivalent symmetric and positive definite system such a decomposition is called a block st preconditioner two general block st preconditioners are proposed for saddle point problems with symmetric and positive definite 1 1 block some estimations of the condition number of the preconditioned system are given the same study is done for singular 1 1 block

a new gaussian mixture algorithm for gmti tracking under a minimum detectable velocity constraint this paper introduces a new methodology to account for doppler blind zone constraints arising for example in ground moving target indicator gmti tracking applications in such problems target measurements are suppressed when the range rate doppler of the target drops below a specified threshold in magnitude the minimum detectable velocity the proposed method employing gaussian mixture approximations to the filtering density differs from earlier gaussian mixture approaches in the way missed measurements are modelled the distinctive feature of the algorithm as compared with other gaussian mixture filters is that it is based on an exact calculation of the filtering density when a measurement is not recorded algorithms that result from applying this methodology are simple to implement and computationally undemanding simulation results indicate a uniform improvement in estimation accuracy over that of earlier proposed analytic techniques and a tracking performance comparable to that of state of the art particle filters

a domain embedding preconditioner for the lagrange multiplier systems finite element approximations for the dirichlet problem associated to a second order elliptic differential equation are studied the purpose of this paper is to discuss domain embedding preconditioners for discrete systems the essential boundary condition on the interior interface is removed by introducing lagrange multipliers the associated discrete system with a saddle point structure is preconditioned by a block diagonal preconditioner the main contribution of this paper is to propose a new operator constructed from the h div inner product for the block of the preconditioner corresponding to the multipliers

a multigrid method for a parameter dependent problem in solid mechanics in the finite element calculations of problems in solid mechanics the method of selected reduced integration sri is frequently used to eliminate locking phenomena often sri is equivalent to the application of a mixed method when multigrid methods are applied the formulation as a mixed method is by far superior this is shown by an analysis of the timoshenko beam

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

on the eigenvalues of a class of saddle point matrices we study spectral properties of a class of block 2 2 matrices that arise in the solution of saddle point problems these matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system we give conditions for having a positive real spectrum and for ensuring diagonalizability of the matrix in particular we show that these properties hold for the discrete stokes operator and we discuss the implications of our characterization for augmented lagrangian formulations for krylov subspace solvers and for certain types of preconditioners

a parallelizable preconditioner for the iterative solution of implicit runge kutta type methods the main diculty in the implementation of most standard implicit runge kutta irk methods applied to sti ordinary dierential equations odes is to eciently solve the nonlinear system of equations in this article we propose the use of a preconditioner whose decomposition cost for a parallel implementation is equivalent to the cost for the implicit euler method the preconditioner is based on the w transformation of the rk coecient matrices discovered by hairer and wanner for sti odes the preconditioner is by construction asymptotically exact for methods with an invertible rk coecient matrix the methodology is particularly useful when applied to super partitioned additive runge kutta spark methods the nonlinear system can be solved by inexact simplied newton iterations at each simplied newton step the linear system can be approximately solved by an iterative method applied to the preconditioned linear system c 1999 published by elsevier science b v all rights reserved

blind deconvolution via sequential imputations abstract the sequential imputation procedure is applied to adaptively and sequentially reconstruct discrete input signals that are blurred by an unknown linear moving average channel and contaminated by additive gaussian noises a problem known as blind deconvolution in digital communication a rejuvenation procedure for improving the efficiency of sequential imputation is introduced and theoretically justified the proposed method does not require the channel to be nonminimum phase and can be used in real time signal restoration two simulated systems are studied to illustrate the proposed method our result shows that the ideas of multiple imputations and flexible simulation techniques are as powerful in engineering as in survey sampling

multilevel iterative methods for mixed finite element discretizations of elliptic problems for solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed the methods are within the general framework of the product multiplicative scheme for operators in a hilbert space proposed recently by bramble pasciak wang and xu 5 6 26 27 and make use of certain multilevel decomposition of the corresponding spaces for the flux variable

stochastic analysis of scattered field by building facades using polynomial chaos this paper presents a statistical assessment of scattered field from a building facade having rando m physical and geometrical parameters a simple inhomogeneous model is considered for the building and the calculation met hod is based on green s functions the basis of polynomial chaos expansion method is explained and applied to estimate the sca ttered electric field from a building facade having 8 random parameters in specular and non specular scenarios uncertainty analysis and total output distribution are discussed in differen t diffraction zones of the building ireless communications particularly mobile networks experienced a great expansion worldwide these networks are very dense in urban areas and th e electromagnetic field distribution is highly depend ent on city structures it is essential to have predictive tool s to assess as accurately as possible the distribution of electrom agnetic fields in order to enable optimized implementation of base stations and respect the constraints regarding huma n exposure wave propagation simulators are based on different models of urban environment with different assumptions on the building architectural details given the adopted model the y incorporate appropriate calculation methods we can mention empirical statistical theoretical site specific models or a combination of them to generate a hybrid model 1 empirical models 2 are based on the extensive measurement campaigns and are highly related to the structure of the city statistical models 3 are also based on measurements and are u sed to give wide band multipath model of the environment theoretical physical 4 models are obtained by i mposing some general idealized conditions on the geometry they usually use physical optics with or without diffrac tion phenomena site specific models 5 depend on a lar ge number of parameters which are related to a given s ite and their computing time depending on the complexity o f the environment can be long these models usually employ ray

spectra and pseudospectra of waveform relaxation operators we describe the spectra and pseudospectra of continuous time and discrete time waveform relaxation operators since the spectrum of the finite interval waveform relaxation operator is only a singleton the spectrum of the infinite interval operator is typically used to describe the behavior of waveform relaxation algorithms here we show that the pseudospectrum is a more useful tool for analyzing convergence of waveform iterations on finite intervals and prove that the pseudospectrum of the infinite interval operator is in fact the limit of the pseudospectra of the finite interval operators

uniform preconditioners for the time dependent stokes problem implicit time stepping procedures for the time dependent stokes problem lead to stationary singular perturbation problems at each time step these singular perturbation problems are systems of saddle point type which formally approach a mixed formulation of the poisson equation as the time step tends to zero preconditioners for discrete analogous of these systems are discussed the preconditioners uses standard positive definite elliptic preconditioners as building blocks and lead to condition numbers which are bounded uniformly with respect to the time step and the spatial discretization the construction of the discrete preconditioners is related to the mapping properties of the corresponding continuous system

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

a uniformly stable fortin operator for the taylor hood element we construct a new fortin operator for the lowest order taylor hood element which is uniformly stable both in l 2 and h 1 the construction which is restricted to two space dimensions is based on a tight connection between a subspace of the taylor hood velocity space and the lowest order nedelec edge element general shape regular triangulations are allowed for the h 1 stability while some mesh restrictions are imposed to obtain the l 2 stability as a consequence of this construction a uniform inf sup condition associated the corresponding discretizations of a parameter dependent stokes problem is obtained and we are able to verify uniform bounds for a family of preconditioners for such problems without relying on any extra regularity ensured by convexity of the domain

fuzzy change point algorithms for regression models change point cp regression models have been widely applied in various fields where detecting cps is an important problem detecting the location of cps in regression models could be equivalent to partitioning data points into clusters of similar individuals in the literature fuzzy clustering has been widely applied in various fields but it is less used in locating cps in cp regression models in this paper a new method called fuzzy cp fcp algorithm is proposed to detect the cps and simultaneously estimate the parameters of regression models the fuzzy c partitions concept is first embedded into the cp regression models any possible collection of all cps is considered as a partitioning of data with a fuzzy membership we then transfer these memberships into the pseudomemberships of data points belonging to each individual cluster and therefore we can obtain the estimates for model parameters by the fuzzy c regressions method subsequently we use the fuzzy c means clustering to obtain the new iterates of the cp collection memberships by minimizing an objective function concerning the deviations between the predicted response values and data values we illustrate the new approach with several numerical examples and real datasets experimental results actually show that the proposed fcp is an effective and useful cp detection algorithm for cp regression models and can be applied to various fields such as econometrics medicine quality control and signal processing

navier stokes equations in rotation form a robust multigrid solver for the velocity problem the topic of this paper is motivated by the navier stokes equations in rotation form linearization and application of an implicit time stepping scheme results in a linear stationary problem of oseen type in well known solution techniques for this problem such as the uzawa or schur complement method a subproblem consisting of a coupled nonsymmetric system of linear equations of diffusion reaction type must be solved to update the velocity vector field in this paper we analyze a standard finite element method for the discretization of this coupled system and we introduce and analyze a multigrid solver for the discrete problem both for the discretization method and the multigrid solver the question of robustness with respect to the amount of diffusion and variation in the convection field is addressed we prove stability results and discretization error bounds for the galerkin finite element method we present a convergence analysis of the multigrid method which shows the robustness of the solver results of numerical experiments are presented which illustrate the stability of the discretization method and the robustness of the multigrid solver

a robust multigrid method for the time dependent stokes problem we propose a coupled multigrid method for generalized stokes flow problems such problems occur as subproblems in implicit time stepping approaches for time dependent stokes problems the discretized stokes system is a large scale linear system whose condition number depends on the grid size of the spatial discretization and of the length of the time step recently for this problem a coupled multigrid method has been proposed where in each smoothing step a poisson problem has to be solved approximately for the pressure field in the present paper we propose a coupled multigrid method where the solution of such subproblems is not needed we prove that the proposed method shows robust convergence behavior in the grid size of the spatial discretization and of the length of the time step