a parallel element by element solution scheme a parallel element by element scheme is developed for finite element solution of elliptic boundary value problems it is shown that for a broad class of computational grids the solution algorithm is totally parallelizable moreover the conversion of an existing serial ebe code to a parallel code is seen to be quite simple the method is implemented on the alliant fx 8 and sequent balance parallel computers and speedup performance studies are conducted for a representative elliptic pde in two dimensions the present procedure can be applied quite generally to other finite element applications and the associated programs can be modified in a straightforward way to implement the method

lattice filter based multivariate autoregressive spectral estimation with joint model order and estimation bandwidth adaptation the problem of parametric autoregressive model based estimation of a time varying spectral density function of a multivariate nonstationary process is considered it is shown that estimation results can be considerably improved if identification of the autoregressive model is carried out using the two sided doubly exponentially weighted lattice algorithm which combines results yielded by two one sided lattice algorithms running forward in time and backward in time respectively it is also shown that the model order and the most appropriate estimation bandwidth can be efficiently selected using the suitably modified akaike s final prediction error criterion

relative entropy and the multivariable multidimensional moment problem entropy like functionals on operator algebras have been studied since the pioneering work of von neumann umegaki lindblad and lieb the best known are the von neumann entropy rho trace rhologrho and a generalization of the kullback leibler distance s rhoparsigma trace rhologrho rhologsigma referred to as quantum relative entropy and used to quantify distance between states of a quantum system the purpose of this paper is to explore and s as regularizing functionals in seeking solutions to multivariable and multidimensional moment problems it will be shown that extrema can be effectively constructed via a suitable homotopy the homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems this is accomplished by a renormalization of a riemannian metric induced by entropy functionals as an application we discuss the inverse problem of describing power spectra which are consistent with second order statistics which has been the main motivation behind the present work

multichannel relative entropy spectrum analysis a new relative entropy method is presented for estimating the power spectral density matrix for multichannel data given correlation values for linear combinations of the channels and given an initial estimate of the spectral density matrix a derivation of the method from the relative entropy principle is given the basic approach is similar in spirit to the multisignal relative entropy spectrum analysis of johnson and shore but the results differ significantly because the present method does not arbitrarily require the final distributions of the various channels to be independent for the special case of separately estimating the spectra of a signal and noise given the correlations of their sum multichannel relative entropy spectrum analysis turns into a two stage procedure first a smooth power spectrum model is fitted to the correlations of the signal plus noise then final estimates of the spectra and cross spectra are obtained through linear filtering for the special case where p uniformly spaced correlations are known and where the initial estimate of the signal plus noise spectrum is all pole with order p or less this method fits a standard maximum entropy autoregressive spectrum to the noisy correlations then linearly filters to calculate the signal and noise spectra and cross spectra consideration is given to the case where only an initial estimate of the noise power spectrum is available an illustrative example is given

on a class of preconditioned iterative methods on parallel computers the standard implementations of iterative solvers for finite element and finite difference methods frequently use a diagonal jacobi preconditioner particularly for element by element schemes however for such methods the actual order of the condition number with respect to mesh size is not reduced by the preconditioner in the present paper we describe an iterative method where in addition the condition number is reduced by an order of magnitude moreover the scheme may also be implemented as an element by element method the method uses a generalized ssor preconditioner and a wave front or multi frontal ordering of the mesh nodes for a general irregular finite element mesh a striped irregular wave front ordering may be used the performance of the method as well as various iterative acceleration techniques for a parallel computer are examined in the numerical studies

stochastic bridges of linear systems we consider particles obeying langevin dynamics while being at known positions and having known velocities at the two end points of a given interval their motion in phase space can be modeled as an ornstein uhlenbeck process conditioned at the two end points a generalization of the brownian bridge using standard ideas from stochastic optimal control we construct a stochastic differential equation sde that generates such a bridge that agrees with the statistics of the conditioned process as a degenerate diffusion higher order linear diffusions are also considered in general a time varying drift is sufficient to modify the prior sde and meet the end point conditions when the drift is obtained by solving a suitable differential lyapunov equation the sde models correctly the statistics of the bridge these types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions

distances and riemannian metrics for multivariate spectral densities the paper is concerned with the practical problem of how to compare power spectral densities of multivariable time series for scalar time series several notions of distance divergences have been proposed and studied starting from the early 1970s while multivariable ones have only recently began to receive any attention in the paper two classes of divergence measures inspired by classical prediction theory are introduced these divergences naturally induce riemannian metrics on the cone of multivariable densities the metrics amount to the quadratic term in the divergence between infinitesimally close to each other power spectra for one of the two we provide explicit formulae for the corresponding geodesics and geodesic distance a close connection between the geometry of power spectra and the geometry of the fisher rao metric is noted

sample mean versus sample frechet mean for combining complex wishart matrices a statistical study the space of covariance matrices is a non euclidean space the matrices form a manifold which if equipped with a riemannian metric becomes a riemannian manifold and recently this idea has been used for comparison and clustering of complex valued spectral matrices which at a given frequency are typically modeled as complex wishart distributed random matrices identically distributed sample complex wishart matrices can be combined via a standard sample mean to derive a more stable overall estimator however using the riemannian geometry their so called sample frechet mean can also be found we derive the expected value of the determinant of the sample frechet mean and the expected value of the sample frechet mean itself the population frechet mean is shown to be a scaled version of the true covariance matrix the risk under convex loss functions for the standard sample mean is never larger than for the frechet mean in simulations the sample mean also performs better for the estimation of an important functional derived from the estimated covariance matrix namely partial coherence

the caratheodory fejer pisarenko decomposition and its multivariable counterpart when a covariance matrix with a toeplitz structure is written as the sum of a singular one and a positive scalar multiple of the identity the singular summand corresponds to the covariance of a purely deterministic component of a time series whereas the identity corresponds to white noise this is the caratheacuteodory fejeacuter pisarenko cfp decomposition in the present paper we study multivariable analogs for block toeplitz matrices as well as for matrices with the structure of state covariances of finite dimensional linear systems which include block toeplitz ones to this end we develop theory which addresses questions of existence uniqueness and realization of multivariable power spectra possibly having deterministic components we characterize state covariances which admit only a deterministic input power spectrum and we explain how to realize multivariable power spectra which are consistent with singular state covariances via decomposing the contribution of the singular part we then show that multivariable decomposition of a state covariance in accordance with a deterministic component white noise hypothesis for the input does not exist in general we finally reinterpret the cfp dictum and consider replacing the scalar multiple of the identity by a covariance of maximal trace which is admissible as a summand the summand can be either block diagonal corresponding to white noise or have a short range correlation structure corresponding to a moving average component the trace represents the maximal variance energy that can be accounted for by a process at the input e g noise with the aforementioned structure and this maximal solution can be computed via convex optimization the decomposition of covariances and spectra according to the range of their time domain correlations is an alternative to the cfp dictum with potentially great practical significance

geometric methods for spectral analysis this paper explores a geometric framework for modeling nonstationary but slowly varying time series based on the assumption that short windowed power spectra capture their spectral character and that energy transference in the frequency domain has a physical significance the framework relies on certain notions of transportation distance and their respective geodesics to model possible nonparametric changes in the power spectral density with respect to time we discuss the relevance of this framework to applications in spectral tracking spectral averaging and speech morphing

on the geometry of maximum entropy problems we show that a simple geometric result suffices to derive the form of the optimal solution in a large class of finite and infinite dimensional maximum entropy problems concerning probability distributions spectral densities and covariance matrices these include burg s spectral estimation method and dempster s covariance completion as well as various recent generalizations of the above we then apply this orthogonality principle to the new problem of completing a block circulant covariance matrix when an a priori estimate is available

fixed point algorithms for estimating power means of positive definite matrices estimating means of data points lying on the riemannian manifold of symmetric positive definite spd matrices has proved of great utility in applications requiring interpolation extrapolation smoothing signal detection and classification the power means of spd matrices with exponent p in the interval 1 1 interpolate in between the harmonic mean p 1 and the arithmetic mean p 1 while the geometric cartan karcher mean which is the one currently employed in most applications corresponds to their limit evaluated at 0 in this paper we treat the problem of estimating power means along the continuum p 1 1 given noisy observed measurement we provide a general fixed point algorithm mpm and we show that its convergence rate for p 0 5 deteriorates very little with the number and dimension of points given as input along the whole continuum mpm is also robust with respect to the dispersion of the points on the manifold noise much more than the gradient descent algorithm usually employed to estimate the geometric mean thus mpm is an efficient algorithm for the whole family of power means including the geometric mean which by mpm can be approximated with a desired precision by interpolating two solutions obtained with a small p value we also present an approximated version of the mpm algorithm with very low computational complexity for the special case p finally we show the appeal of power means through the classification of brain computer interface event related potentials data

conal distances between rational spectral densities this paper generalizes thompson and hilbert metrics to the space of spectral densities the resulting complete metric space has the differentiable structure of a finsler manifold with explicit geodesics the corresponding distances are filtering invariant can be computed efficiently and admit geodesic paths that preserve rationality these are properties of fundamental importance in many engineering applications

multivariate spectral estimation based on the concept of optimal prediction in this technical note we deal with a spectrum approximation problem arising in three like multivariate spectral estimation approaches the solution to the problem minimizes a suitable divergence index with respect to an a priori spectral density we derive a new divergence family between multivariate spectral densities which takes root in the prediction theory under mild assumptions on the a priori spectral density the approximation problem based on this new divergence family admits a family of solutions moreover an upper bound on the complexity degree of these solutions is provided

a new family of high resolution multivariate spectral estimators in this paper we extend the beta divergence family to multivariate power spectral densities similarly to the scalar case we show that it smoothly connects the multivariate kullback leibler divergence with the multivariate itakura saito distance we successively study a spectrum approximation problem based on the beta divergence family which is related to a multivariate extension of the three spectral estimation technique it is then possible to characterize a family of solutions to the problem an upper bound on the complexity of these solutions will also be provided finally we will show that the most suitable solution of this family depends on the specific features required from the estimation problem

distances and riemannian metrics for spectral density functions we introduce a differential geometric structure for spectral density functions of discrete time random processes this is quite analogous to the riemannian structure of information geometry which is used to study perturbations of probability density functions and which is based on the fisher information metric herein we introduce an analogous riemannian metric which we motivate with a problem in prediction theory it turns out that this problem also provides a prediction theoretic interpretation to the itakura distortion measure which relates to our metric geodesies and geodesic distances are characterized in closed form and hence the geodesic distance between two spectral density functions provides an explicit intrinsic pseudo metric on the cone of density functions certain other distortion measures that involve generalized means of spectral density functions are shown to lead to the same riemannian metric finally an alternative riemannian metric is introduced which is motivated by an analogous problem involving smoothing instead of prediction

robust preconditioners for linear elasticity fem analyses summary this paper deals with two forms of preconditioner which can be easily used with a conjugate gradient solver to replace a direct solution subroutine in a traditional engineering nite element package they are tested in such a package final over a range of 2 d and 3 d elasticity problems from geotechnical engineering quadratic basis functions are used a number of modications to the basic incomplete choleski ic 0 factorization preconditioner are considered an algorithm to reduce positive o diagonal entries is shown in numerical experiments to ensure stability but at the expense of slow convergence an alternative algorithm of jennings and malik is more successful and a relaxation parameter is introduced which can make a further signicant improvement in performance while maintaining stability a heuristic for determining a near optimal value of is proposed a second form of preconditioning symmetrically scaled element by element due to bartelt is also shown to perform robustly over a range of problems it does not require assembly of the global stiness matrix and has great potential for parallelization 1997 by john wiley sons ltd

decay rates of solutions of an anisotropic inhomogeneous n dimensional viscoelastic equation with polynomially decaying kernels we consider the anisotropic and inhomogeneous viscoelastic equation and we prove that the first and second order energy decay polynomially as time goes to infinity when the relaxation function also decays polynomially to zero that is if the kernelg ijkl satisfies dot g ijkl leqq c 0 g ijkl 1 frac 1 p and g ijkl g ijkl 1 frac 1 p in l 1 mathbb r for p 2such that 2 m 1 p then the first and second order energy decay as frac 1 1 t q withq 2 m 1

a three dimensional explicit preconditioned solver a new class of generalized approximate inverse matrix gaim techniques based on the concept of lu sparse factorization procedures is introduced for computing explicitly generalized approximate inverses of large sparse unsymmetric matrices of regular structure without inverting the decomposition factors explicit preconditioned iterative methods in conjunction with modified forms of the gaim techniques are presented for solving numerically boundary value problems in three dimensions the numerical implementation of these algorithms is presented and fortran subroutines are given

smooth interpolation of covariance matrices and brain network estimation part ii we propose an approach to use the state covariance of autonomous linear systems to track time varying covariance matrices of nonstationary time series following concepts from the riemannian geometry we investigate three types of covariance paths obtained by using different quadratic regularizations of system matrices the first quadratic form induces the geodesics based on the hellinger bures metric related to optimal mass transport omt theory and quantum mechanics the second type of quadratic form leads to the geodesics based on the fisher rao metric from information geometry in the process we introduce a weighted omt interpretation of the fisher rao metric for multivariate gaussian distributions a main contribution of this work is the introduction of the third type of covariance paths which are steered by system matrices with rotating eigenspaces the three types of covariance paths are compared using two examples with synthetic data and real data from resting state functional magnetic resonance imaging respectively

an interpretation of the dual problem of the three like approaches spectral estimation can be performed using the so called three like approach such method leads to a convex optimization problem whose solution is characterized through its dual problem in this paper we show that the dual problem can be seen as a new parametric spectral estimation problem this interpretation implies that the three like solution is optimal in terms of closeness to the correlogram over a certain parametric class of spectral densities describing arma models enriching in this way its meaningfulness

the thermoelastic and viscoelastic contact of two rods we consider the thermoelastic and viscoelastic contact problem of two rods and prove the existence of a weak solution using a penalization method and compensated compactness moreover for the thermoelastic contact we show that the weak solution converges to zero exponentially as time goes to infinity and for the viscoelastic contact we prove that the weak solution decays to zero with the same rates as the relaxation functions do

time and spectral domain relative entropy a new approach to multivariate spectral estimation the concept of spectral relative entropy rate is introduced for jointly stationary gaussian processes using classical information theoretic results we establish a remarkable connection between time and spectral domain relative entropy rates this naturally leads to a new spectral estimation technique where a multivariate version of the itakura saito distance is employed it may be viewed as an extension of the approach called three introduced by byrnes georgiou and lindquist in 2000 which in turn followed in the footsteps of the burg jaynes maximum entropy method spectral estimation is here recast in the form of a constrained spectrum approximation problem where the distance is equal to the processes relative entropy rate the corresponding solution entails a complexity upper bound which improves on the one so far available in the multichannel framework indeed it is equal to the one featured by three in the scalar case the solution is computed via a globally convergent matricial newton type algorithm simulations suggest the effectiveness of the new technique in tackling multivariate spectral estimation tasks especially in the case of short data records

mean and median of psd matrices on a riemannian manifold application to detection of narrow band sonar signals the rich information in the power spectral density psd matrix of a received signal can be extracted in different ways for various purposes in signal processing care however must be taken when distances between psd matrices are considered because these matrices form a riemannian manifold in the signal space in this paper two riemannian distances rd are introduced for the measurement of distances between psd matrices on the manifold the principle by which the geodesics on the manifold can be lifted to a euclidean subspace isometric with the tangent space of the manifold is also explained and emphasized this leads to the concept that any optimization involving the rd on the manifold can be equivalently performed in this euclidean subspace the application of this principle is illustrated by the development of iterative algorithms to find the riemannian means and riemannian medians according to the two rd these distance measures are then applied to the detection of narrow band sonar signals exploration of the translation of measure reference as well as the application of optimum weighting show substantial improvement in detection performance over classical detection method