optimal controllers for finite wordlength implementation when a controller is implemented by a digital computer with a d and d a conversion numerical errors can severely affect the performance of the control system there exist realizations of a given controller transfer function exhibiting arbitrarily large effects from computational errors assuming sufficient excitation of the system the problem of designing an optimal controller in the presence of both external disturbances and internal roundoff errors is solved the results reduce to the standard lqg controller when infinite precision computation is used for finite precision however the separation principle does not hold a penalty is also added to the cost function to penalize the sum of the wordlengths used to compute the fractional part of each state variable of the controller this sum can be used to represent the lower bound on computer memory needed for controller synthesis it measures controller complexity and is minimized penalized here

robust stability of interval matrices a computational approach a simple computational procedure is given to determine the robust stability of matrices containing interval parameter uncertainty the computation consists of determining the maximal phase difference of the characteristic polynomial evaluated over the vertices of the uncertain parameter set at each point of the stability boundary and verifying that this maximal phase difference is less than radians this follows as a direct application of the mapping theorem which allows us effectively to treat rank one perturbations of matrices we discuss the conservatism of the method and show how the solution may be improved to any desired level of accuracy we give several numerical examples of systems with state space perturbations and compare our results with those obtained by other methods

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

comments on robust fragile or optimal with reply the issue of fragile controllers produced by using popular robust and optimal control synthesis methods was raised by keel bhattacharyya in the above mentioned paper ibid vol 42 1997 the commenter points out that the controller fragility problem described in the paper is 1 well known in the literature 2 due to the poor choice of examples 3 can easily be fixed by changing the optimization criteria and 4 can be fixed by changing the implementation in reply keel bhattacharyya disagree with him on all counts and explain their argument

generalized linear array models with applications to multidimensional smoothing summary data with an array structure are common in statistics and the design or regression matrix for analysis of such data can often be written as a kronecker product factorial designs contingency tables and smoothing of data on multidimensional grids are three such general classes of data and models in such a setting we develop an arithmetic of arrays which allows us to define the expectation of the data array as a sequence of nested matrix operations on a coefficient array we show how this arithmetic leads to low storage high speed computation in the scoring algorithm of the generalized linear model we refer to a generalized linear array model and apply the methodology to the smoothing of multidimensional arrays we illustrate our procedure with the analysis of three data sets mortality data indexed by age at death and year of death spatially varying microarray background data and disease incidence data indexed by age at death year of death and month of death

eigencurves for two parameter sturm liouville equations this paper concerns two parameter sturm liouville problems of the form p x y q x y lambda r x mu y quad a leqslant x leqslant b with self adjoint boundary conditions at a and b the set of lambda mu in bf r 2 for which there exists a nontrivial y satisfying the differential equation and the boundary conditions turns out to be a countable union of graphs of analytic functions our focus is on these graphs which are termed eigencurves in the literature although eigencurves have been used in a variety of ways for about a century they seem comparatively underdeveloped in their own right our plan is to give motivation for the topic elementary properties of eigencurves illustrations on a simple example first studied by richardson in 1918 and since then by several authors and some natural questions which may whet the reader s appetite some of these questions lead to new types of inverse sturm liouville problems

balanced realizations via gradient flow techniques abstract the task of finding a class of balanced minimal realizations is shown to be equivalent to finding limiting solutions of certain gradient flow differential equations by viewing such algebraic tasks in the context of calculus they are amenable to analog computational solutions or parallel processing machines perhaps even neural networks the convergence rates of the differential equations is exponential and consequentially convergence is rapid and numerical stability properties are attractive

guarantees for the kronecker fast johnson lindenstrauss transform using a coherence and sampling argument abstract in the recent paper jin kolda ward arxiv 1909 04801 it is proved that the kronecker fast johnson lindenstrauss transform kfjlt is in fact a johnson lindenstrauss transform which had previously only been conjectured in this paper we provide an alternative proof of this for when the kfjlt is applied to kronecker vectors using a coherence and sampling argument our proof yields a different bound on the embedding dimension which can be combined with the bound in the paper by jin et al to get a better bound overall as a stepping stone to proving our result we also show that the kfjlt is a subspace embedding for matrices with columns that have kronecker product structure lastly we compare the kfjlt to four other sketch techniques in numerical experiments on both synthetic and real world data

recursive fast algorithm and the role of the tensor product the use of the tensor product for modeling and designing fft algorithms is addressed the benefit of the tensor product approach lies in the strong connection between certain tensor product constructs and important computer architectures the scope of the tensor product approach is generalized to include a much larger class of fast recursive algorithms this greatly enhances the versatility of the tensor product technique and brings many different algorithms to the level of understanding and flexibility enjoyed by the fft

optimal realizations of floating point implemented digital controllers with finite word length considerations the closed loop stability issue of finite precision realizations is investigated for digital controllers implemented in floating point arithmetic unlike the existing methods which only address the effect of the mantissa bits in floating point implementation to the sensitivity of closed loop stability the sensitivity of closed loop stability is analyzed with respect to both the mantissa and exponent bits of floating point implementation a computationally tractable fwl closed loop stability measure is then defined and the method of computing the value of this measure is given the optimal controller realization problem is posed as searching for a floating point realization that maximizes the proposed fwl closed loop stability measure and a numerical optimization technique is adopted to solve for the resulting optimization problem simulation results show that the proposed design procedure yields computationally efficient controller realizations with enhanced fwl closed loop stability performance

solving an infinite horizon adverse selection model through finite policy graphs this paper studies an infinite horizon adverse selection model with an underlying markov information process it introduces a graphic representation of continuation contracts and continuation payoff frontiers namely finite policy graph and provides an algorithm to approximate the optimal policy graph through iterations the algorithm performs an additional step after each value iteration replacing dominated points on the previous continuation payoff frontier by points on the new frontier and reevaluating the new frontier this dominance free reevaluation step accelerates the convergence of the continuation payoff frontiers numerical examples demonstrate the effectiveness of this algorithm and properties of the optimal contracts

fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains a time dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered by applying the energy method well posed boundary conditions for the continuous problem are derived summation by parts sbp operators for the space and time discretization together with a weak imposition of boundary and initial conditions using simultaneously approximation terms sats lead to a provable fully discrete energy stable conservative finite difference scheme we show how to construct a time dependent sat formulation that automatically imposes boundary conditions when and where they are required we also prove that a uniform flow field is preserved i e the numerical geometric conservation law ngcl holds automatically by using sbp sat in time and space the developed technique is illustrated by considering an application using the linearized euler equations the sound generated by moving boundaries numerical calculations corroborate the stability and accuracy of the new fully discrete approximations

a formula for computation of the real stability radius this paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane

partially observable markov decision processes a geometric technique and analysis this paper presents a novel framework for studying partially observable markov decision processes pomdps with finite state action observation sets and discounted rewards the new framework is solely based on future reward vectors associated with future policies which is more parsimonious than the traditional framework based on belief vectors it reveals the connection between the pomdp problem and two computational geometry problems i e finding the vertices of a convex hull and finding the minkowski sum of convex polytopes which can help solve the pomdp problem more efficiently the new framework can clarify some existing algorithms over both finite and infinite horizons and shed new light on them it also facilitates the comparison of pomdps with respect to their degree of observability as a useful structural result

conic geometric optimization on the manifold of positive definite matrices we develop geometric optimization on the manifold of hermitian positive definite hpd matrices in particular we consider optimizing two types of cost functions i geodesically convex g convex and ii log nonexpansive ln g convex functions are nonconvex in the usual euclidean sense but convex along the manifold and thus allow global optimization ln functions may fail to be even g convex but still remain globally optimizable due to their special structure we develop theoretical tools to recognize and generate g convex functions as well as cone theoretic fixed point optimization algorithms we illustrate our techniques by applying them to maximum likelihood parameter estimation for elliptically contoured distributions a rich class that substantially generalizes the multivariate normal distribution we compare our fixed point algorithms with sophisticated manifold optimization methods and obtain notable speedups

multi output separable gaussian process towards an efficient fully bayesian paradigm for uncertainty quantification computer codes simulating physical systems usually have responses that consist of a set of distinct outputs e g velocity and pressure that evolve also in space and time and depend on many unknown input parameters e g physical constants initial boundary conditions etc furthermore essential engineering procedures such as uncertainty quantification inverse problems or design are notoriously difficult to carry out mostly due to the limited simulations available the aim of this work is to introduce a fully bayesian approach for treating these problems which accounts for the uncertainty induced by the finite number of observations our model is built on a multi dimensional gaussian process that explicitly treats correlations between distinct output variables as well as space and or time the proper use of a separable covariance function enables us to describe the huge covariance matrix as a kronecker product of smaller matrices leading to efficient algorithms for carrying out inference and predictions the novelty of this work is the recognition that the gaussian process model defines a posterior probability measure on the function space of possible surrogates for the computer code and the derivation of an algorithmic procedure that allows us to sample it efficiently we demonstrate how the scheme can be used in uncertainty quantification tasks in order to obtain error bars for the statistics of interest that account for the finite number of observations

finite wordlength controller realisations using the specialised implicit form a specialised implicit state space representation is introduced to deal with finite wordlength effects in controller implementations this specialised implicit form provides a macroscopic description of the algorithm to be implemented so it constitutes a unifying framework allowing to encompass various implementation forms such as the operator the direct form ii transposed observer based and many other realisations usually considered separately in the literature different measures quantifying the finite wordlength effects on the overall closed loop behaviour are defined in this new context they concern both stability and performance the gap with the infinite precision case is evaluated classically through the coefficient sensitivity and roundoff noise analysis the problem of determining a realisation with minimum finite wordlength effects can subsequently be solved using appropriate numerical methods the approach is illustrated with an example

reduction of controller fragility by pole sensitivity minimization presents a method for the reduction of controller fragility the method is based on the sensitivity of closed loop poles to perturbations in the controller parameters by means of a state space parameterization of the controller the closed loop pole sensitivity can be reduced a controller fragility measure based on the closed loop pole sensitivity is proposed conditions for the optimal state space realization of the controller are presented along with a numerical method for obtaining the solution

a three domain covariance framework for eeg meg data in this paper we introduce a covariance framework for the analysis of single subject eeg and meg data that takes into account observed temporal stationarity on small time scales and trial to trial variations we formulate a model for the covariance matrix which is a kronecker product of three components that correspond to space time and epochs trials and consider maximum likelihood estimation of the unknown parameter values an iterative algorithm that finds approximations of the maximum likelihood estimates is proposed our covariance model is applicable in a variety of cases where spontaneous eeg or meg acts as source of noise and realistic noise covariance estimates are needed such as in evoked activity studies or where the properties of spontaneous eeg or meg are themselves the topic of interest like in combined eeg fmri experiments in which the correlation between eeg and fmri signals is investigated we use a simulation study to assess the performance of the estimator and investigate the influence of different assumptions about the covariance factors on the estimated covariance matrix and on its components we apply our method to real eeg and meg data sets

optimizing stability bounds of finite precision pid controller structures this paper investigates a recently derived lower bound stability measure for sampled data controller structures subject to finite wordlength fwl constraints the optimal realization of the digital pid controller with fwl considerations is formulated as a nonlinear optimization problem and an efficient strategy based on adaptive simulated annealing asa is adopted to solve this complex optimization problem a numerical example of optimizing the finite precision pid controller structure for a simulated steel rolling mill system is presented to illustrate the effectiveness of the proposed strategy

improved parallel qr method for large least squares problems involving kronecker products abstract a new algorithm is presented for the efficient solution of large least squares problems in which the coefficient matrix of the linear system is a kronecker product of two smaller dimension matrices the solution algorithm is based on qr factorizations of the smaller dimension matrices near perfect load balancing is achieved by exploiting a commutativity property of the kronecker product and communication requirements are minimized by employing a binary exchange algorithm for matrix transposition the parallel algorithm is presented and timing results are shown from test runs on an intel i 860 computer

penalized least squares estimation of volterra filters and higher order statistics volterra filters vfs and higher order statistics hos are important tools for nonlinear analysis processing and modeling despite their highly desirable properties the transfer of vfs and hos to real world signal processing problems has been hindered by the requirement of very large data records needed to obtain reliable estimates the identification of vfs and the estimation of hos both fall into the category of ill posed estimation problems we develop penalized least squares pls estimation methods for vfs and hos it is shown that pls is a very effective way to incorporate prior information of the problem at hand without directly constraining the estimation procedure hence pls produces much more reliable estimates the main contributions of this paper are the development of appropriate penalizing functionals and cross validation procedures for pls based vf identification and hos estimation

tensor product basis approximations for volterra filters the paper studies approximations for a class of nonlinear filters known as volterra filters although the volterra filter provides a relatively simple and general representation for nonlinear filtering it is often highly overparameterized due to the large number of parameters the utility of the volterra filter is limited the overparameterization problem is addressed in the paper using a tensor product basis approximation tpba in many cases a volterra filter may be well approximated using the tpba with far fewer parameters hence the tpba offers considerable advantages over the original volterra filter in terms of both implementation and estimation complexity furthermore the tpba provides useful insight into the filter response the paper studies the crucial issue of choosing the approximation basis several methods for designing an appropriate approximation basis and error bounds on the resulting mean square output approximation error are derived certain methods are known to be nearly optimal

rank equalities and inequalities for kronecker products of matrices with applications some rank equalities and inequalities are established for kronecker products of matrices as applications upper and lower bounds are presented for the dimensions of the ranges of the two well known linear transformations t 1 x x axb and t 2 x ax xb

optimal realisations of fixed point implemented digital controller with the smallest dynamic range a novel approach is proposed to design optimal finite word length fwl realizations of digital controllers implemented in fixed point arithmetic a minimax based search procedure is first formulated to obtain an optimal controller realization that optimizes an fwl closed loop stability measure since this fwl closed loop stability measure is solely linked to the fractional part or precision of fixed point format the resulting realization may not have the smallest dynamic range a measure is then derived to indicate the dynamic range of fixed point implemented realization by choosing an appropriate orthogonal transformation of this dynamic range measure of the optimal precision controller realization a numerical optimization method is developed to make the controller realization having the smallest dynamic range without sacrificing fwl closed loop stability robustness the proposed approach is more efficient than a direct optimization of some combined fwl closed loop stability and dynamic range measure

computing supergame equilibria we present a general method for computing the set of supergame equilibria in infinitely repeated games with perfect monitoring and public randomization we present a three stage algorithm that constructs a convex set containing the set of equilibrium values constructs another convex set contained in the set of equilibrium values and produces strategies that support them we explore the properties of this algorithm by applying it to familiar games

restoring images degraded by spatially variant blur restoration of images that have been blurred by the effects of a gaussian blurring function is an ill posed but well studied problem any blur that is spatially invariant can be expressed as a convolution kernel in an integral equation fast and effective algorithms then exist for determining the original image by preconditioned iterative methods if the blurring function is spatially variant however then the problem is more difficult in this work we develop fast algorithms for forming the convolution and for recovering the original image when the convolution functions are spatially variant but have a small domain of support this assumption leads to a discrete problem involving a banded matrix we devise an effective preconditioner and prove that the preconditioned matrix differs from the identity by a matrix of small rank plus a matrix of small norm numerical examples are given related to the hubble space telescope hst wide field planetary camera the algorithms that we develop are applicable to other ill posed integral equations as well

analysis of a dynamic adverse selection model with asymptotic efficiency this paper studies an infinite horizon adverse selection model with an underlying two state markov decision process it introduces a novel approach that constructs the continuation payoff frontier exactly as the fixed point of a functional operator if the model supports an incentive compatible first best icfb contract the continuation payoff frontier can be efficiently constructed and the principal s optimal contracts converge to icfb contracts over time the existence of an icfb contract is implied by the common assumption of private values and is a fairly general scenario the paper generalizes some key findings in the dynamic adverse selection literature to this scenario

on perturbation bounds of kronecker product linear systems and their level 2 condition numbers the classical perturbation theory of linear systems ax b is extended to kronecker product linear systems email protected b x d upper bounds are derived for the normwise and componentwise condition number the nearness to singularity and the sensitivity of the condition numbers are analyzed

quarks identification of large scale kronecker vector autoregressive models in this paper we address the identification of two dimensional 2 d spatial temporal dynamical systems described by the vector autoregressive var form the coefficient matrices of the var model are parametrized as sums of kronecker products when the number of terms in the sum is small compared to the size of the matrices such a kronecker representation efficiently models large scale var models estimating the coefficient matrices in least squares sense gives rise to a bilinear estimation problem that is tackled using an alternating least squares als algorithm regularization or parameter constraints on the coefficient matrices allows to induce temporal network properties such as stability as well as spatial properties such as sparsity or toeplitz structure convergence of the regularized als is proved using fixed point theory a numerical example demonstrates the advantages of the new modeling paradigm it leads to comparable variance of the prediction error with the unstructured least squares estimation of var models however the number of parameters grows only linearly with respect to the number of nodes in the 2 d sensor network instead of quadratically in the case of fully unstructured coefficient matrices

comments on on the structure of digital controllers with finite word length consideration in the above mentioned paper by li ibid vol 43 1998 a condition is presented as necessary and sufficient to give the optimal realization of state space controllers for minimal sensitivity of all closed loop eigenvalues subject to finite word length effects the condition is shown here not to be sufficient in addition the condition for minimal sensitivity of a single closed loop eigenvalue presented in the above mentioned paper is not complete the complete conditions are presented in this paper

monotonically convergent algorithms for symmetric tensor approximation abstract reduced rank approximations to symmetric tensors find use in data compaction and in multi user blind source separation we derive iterative algorithms which feature monotonic convergence to a minimum of a frobenius norm approximation criterion for a certain rank r tucker product version of the approximation problem the approach exploits the gradient inequality for convex functions to establish monotonic convergence while sparing the cumbersome step size analysis required from a manifold gradient approach it likewise overcomes some limitations of symmetric versions of alternating least squares the computational load per iteration amounts to computing an unfolded matrix and a qr decomposition

brief paper insensitive h filter design for continuous time systems with respect to filter coefficient variations this paper is concerned with the problem of designing insensitive h filters for linear continuous time systems coefficient sensitivity functions of transfer functions with respect to filter additive multiplicative coefficient variations are defined and the h norms of the sensitivity functions are used to measure the sensitivity of the transfer functions with respect to additive filter coefficient variations in addition in order to deal with the filter design problem for the multiplicative filter coefficient variation case new measures based on the average of the sensitivity functions are also defined consequently the insensitive h filter design problem is reduced to a multi objective filter design problem which minimizes the coefficient s sensitivity and meets the prescribed h norm constraint simultaneously first a novel method for designing insensitive h filters subjected to additive filter coefficient variations is given in terms of the linear matrix inequality lmi optimization techniques furthermore based on the new sensitivity measures the obtained results are extended to the multiplicative coefficient variation case in addition an indirect method for solving the multiplicative variations is also proposed finally two numerical examples are provided to demonstrate the effectiveness of the proposed method

the spectra of multiplicative attribute graphs abstract a multiplicative attribute graph is a random graph in which vertices are represented by random words of length t in a finite alphabet and the probability of adjacency is a symmetric function t t 0 1 these graphs are a generalization of stochastic kronecker graphs and both classes have been shown to exhibit several useful real world properties we establish asymptotic bounds on the spectra of the adjacency matrix and the normalized laplacian matrix for these two families of graphs under certain mild conditions as an application we examine various properties of the stochastic kronecker graph and the multiplicative attribute graph including the diameter clustering coefficient chromatic number and bounds on low congestion routing

controller approximation approaches for preserving h sub spl infin performance investigates the design of reduced order controllers using an h sub spl infin framework given a stabilizing controller which satisfies a prespecified level of closed loop h sub spl infin performance sufficient conditions are derived for another controller to be stabilizing and satisfy the same level of h sub spl infin performance such controllers are said to be p spl gamma admissible where p is the model of the plant under consideration and spl gamma is the required level of prespecified h sub spl infin performance the conditions are expressed as norm bounds on particular frequency weighted errors where the weights are selected to make a specific transfer function a contraction the design of reduced order p spl gamma admissible controllers is then formulated as a frequency weighted model reduction problem it is advantageous for the required weights to be large in some sense solutions which minimize either the trace or the determinant of the inverse weights are characterized we show that the procedure for minimizing the determinant of the inverse weights always gives a direction where the weights are the best possible to conclude we demonstrate by way of a numerical example that when used in conjunction with a combined model reduction convex optimization scheme the proposed design procedures are effective in substantially reducing controller complexity

optimal finite wordlength linear quadratic regulation the authors consider the design of an lqg regulator implemented in fixed point arithmetic which is optimized with respect to its state wordlength and structure the approach extends previous design methods so that arithmetic errors are directly included the optimal finite controller gains are computed as a function of the arithmetic wordlength subject to an l sub 2 scaling constant on the controller states the design method used permits the cost for the near optimal structure to be developed so that this can then be used as a basis for comparison to other less complex structures

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

the matrix equation xa bx r and its applications abstract we study the well known sylvester equation xa bx r in the case when a and b are given and r is known up to its first n 1 rows we prove new results on the existence and uniqueness of x our results essentially state that in case a is a nonderogatory matrix there always exists a solution to this equation a solution is uniquely determined by its first row x 1 and there is an interesting relationship between x 1 and the rows of r we also give a complete characterization of the nonsingularity of x in this case as applications of our results we develop direct methods for constructing symmetrizers and commuting matrices computing the characteristic polynomial of a matrix and finding the numbers of common eigenvalues between a and b some well known important results on symmetrizers bezoutians and inertia are recovered as special cases

on the structure of digital controllers with finite word length consideration it is well known that a stable discrete time control system may become unstable when the digital controller is actually implemented with a digital control processor due to the finite word length effects and that the stability behavior of the system depends on the structure of the digital controller due to the difficulties in adopting the existing stability robustness measures a new measure is derived which is tractable and hence enables us to handle the digital controller structure problem easily the optimal controller structure problem is formulated by maximizing this measure an analytical solution is presented an algorithm is developed for finding sparser structures

kronecker products and shuffle algebra relates three classical concepts viz mixed radix number system kronecker product of matrices and perfect shuffle it presents an algebra which describes the hardware organization of the computation of a product mv where m is a matrix in kronecker product form and v is a vector the algebraic formalism describes both the blockwise structure of the computation and the various possible connection patterns

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

low rank estimation of higher order statistics low rank estimators for higher order statistics are considered in this paper the bias variance tradeoff is analyzed for low rank estimators of higher order statistics using a tensor product formulation for the moments and cumulants in general the low rank estimators have a larger bias and smaller variance than the corresponding full rank estimator and the mean squared error can be significantly smaller this makes the low rank estimators extremely useful for signal processing algorithms based on sample estimates of the higher order statistics the low rank estimators also offer considerable reductions in the computational complexity of such algorithms the design of subspaces to optimize the tradeoffs between bias variance and computation is discussed and a noisy input noisy output system identification problem is used to illustrate the results

time and lag recursive computation of cumulants from a state space model time and lag recursive algorithms for the computation of the cumulants of the state vector and the output process of a multiple input multiple output time varying state space model are derived using a kronecker product representation for the cumulants of vector processes the noise processes are not assumed to be stationary symmetry relations for the cumulants of vector processes are discussed computational aspects are examined in detail it is conjectured that the recursive equations derived will be useful in developing state estimators and even optimal controllers based on higher order statistics

a second order exponential time differencing scheme for non linear reaction diffusion systems with dimensional splitting abstract a second order l stable exponential time differencing etd method is developed by combining an etd scheme with approximating the matrix exponentials by rational functions having real distinct poles rdp together with a dimensional splitting integrating factor technique a variety of non linear reaction diffusion equations in two and three dimensions with either dirichlet neumann or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second order implicit explicit schemes an additional performance boost is gained through further use of basic parallelization techniques

on condition numbers for least squares with quadric inequality constraint abstract in this paper we will study normwise mixed and componentwise condition numbers for the linear mapping of the solution for general least squares with quadric inequality constraint glsqi and its standard form lsqi we will introduce the mappings from the data space to the interested data space and the frechet derivative of the introduced mapping can deduced through matrix differential techniques based on condition number theory we derive the explicit expressions of normwise mixed and componentwise condition numbers for the linear function of the solution for glsqi and lsqi also easier computable upper bounds for mixed and componentwise condition numbers are given numerical example shows that the mixed and componentwise condition numbers can tell us the true conditioning of the problem when its data is sparse or badly scaled compared with normwise condition numbers the mixed and componentwise condition number can give sharp perturbation bounds

sampling of stochastic operators we develop sampling methodology aimed at identifying stochastic operators that satisfy a support size restriction on the autocorrelation of the operators stochastic spreading function the data that we use to reconstruct the operator or in some cases only the autocorrelation of the spreading function is based on the response of the unknown operator to a known deterministic test signal

affinity learning with diffusion on tensor product graph in many applications we are given a finite set of data points sampled from a data manifold and represented as a graph with edge weights determined by pairwise similarities of the samples often the pairwise similarities which are also called affinities are unreliable due to noise or due to intrinsic difficulties in estimating similarity values of the samples as observed in several recent approaches more reliable similarities can be obtained if the original similarities are diffused in the context of other data points where the context of each point is a set of points most similar to it compared to the existing methods our approach differs in two main aspects first instead of diffusing the similarity information on the original graph we propose to utilize the tensor product graph tpg obtained by the tensor product of the original graph with itself since tpg takes into account higher order information it is not a surprise that we obtain more reliable similarities however it comes at the price of higher order computational complexity and storage requirement the key contribution of the proposed approach is that the information propagation on tpg can be computed with the same computational complexity and the same amount of storage as the propagation on the original graph we prove that a graph diffusion process on tpg is equivalent to a novel iterative algorithm on the original graph which is guaranteed to converge after its convergence we obtain new edge weights that can be interpreted as new learned affinities we stress that the affinities are learned in an unsupervised setting we illustrate the benefits of the proposed approach for data manifolds composed of shapes images and image patches on two very different tasks of image retrieval and image segmentation with learned affinities we achieve the bull s eye retrieval score of 99 99 percent on the mpeg 7 shape dataset which is much higher than the state of the art algorithms when the data points are image patches the ncut with the learned affinities not only significantly outperforms the ncut with the original affinities but it also outperforms state of the art image segmentation methods

a unified closed loop stability measure for finite precision digital controller realizations implemented in different representation schemes a computationally tractable unified finite word length closed loop stability measure is derived which is applicable to fixed point floating point and block floating point representation schemes both the dynamic range and precision of an arithmetic scheme are considered in this new unified measure for each arithmetic scheme the optimal controller realization problem is defined and a numerical optimization approach is adopted to solve it numerical examples are used to illustrate the design procedure and to compare the optimal controller realizations in different representation schemes

structured data sparse approximation to high order tensors arising from the deterministic boltzmann equation we develop efficient data sparse representations to a class of high order tensors via a block many fold kronecker product decomposition such a decomposition is based on low separation rank approximations of the corresponding multivariate generating function we combine the sinc interpolation and a quadrature based approximation with hierarchically organised block tensor product formats different matrix and tensor operations in the generalised kronecker tensor product format including the hadamard type product can be implemented with the low cost an application to the collision integral from the deterministic boltzmann equation leads to an asymptotical cost o n 4 log n o n 5 log n in the one dimensional problem size n depending on the model kernel function which noticeably improves the complexity o n 6 log n of the full matrix representation

structured mixed and componentwise condition numbers of some structured matrices structured matrices such as cauchy vandermonde toeplitz hankel and circulant matrices are considered in this paper we apply a kronecker product based technique to deduce the structured mixed and componentwise condition numbers for the matrix inversion and for the corresponding linear systems