a global rational arnoldi method for model reduction abstract in this paper we propose two new approaches for model order reduction of large scale multi input multi output mimo linear time invariant dynamical systems lti these methods are based on a generalization of the global arnoldi algorithm which is used to generate projection subspaces an adaptive procedure for the selection of shift parameters is proposed and some simple arnoldi relations are established in order to compute error bounds on the transfer function numerical experiments showing the effectiveness of these approaches are provided

adaptive surface related multiple elimination the major amount of multiple energy in seismic data is related to the large reflectivity of the surface a method is proposed for the elimination of all surface related multiples by means of a process that removes the influence of the surface reflectivity from the data an important property of the proposed multiple elimination process is that no knowledge of the subsurface is required on the other hand the source signature and the surface reflectivity do need to be provided as a consequence the proposed process has been implemented adaptively meaning that multiple elimination is designed as an inversion process where the source and surface reflectivity properties are estimated and where the multiple free data equals the inversion residue results on simulated data and field data show that the proposed multiple elimination process should be considered as one of the key inversion steps in stepwise seismic inversion

model reduction of nonstationary lpv systems this paper focuses on the model reduction of nonstationary linear parameter varying nslpv systems we provide a generalization of the balanced truncation procedure for the model reduction of stable nslpv systems along with a priori error bounds then for illustration purposes this method is applied to reduce the model of a two mass translational system furthermore we give an approach for the model reduction of stabilizable and detectable systems which requires the development and use of coprime factorizations for nslpv models for the general class of eventually periodic lpv systems which includes periodic and finite horizon systems as special cases our results can be explicitly computed using semidefinite programming

identification of continuous time models the paper considers the problem of estimation of the transfer function of a continuous time dynamic system in the presence of colored noise the author introduces an operator transformation that allows for keeping a continuous time parametrization whereas the parameter estimation can be made by means of a discrete time maximum likelihood algorithm a comparison is made between the performance of the new method in comparison with a standard identification of an armax model the method is useful in cases where it is important to estimate the coefficients of a continuous time transfer function and where it is important to maintain a physical interpretation of the transfer function results

an inverse scattering series method for attenuating multiples in seismic reflection data we present a multidimensional multiple attenuation method that does not require any subsurface information for either surface or internal multiples to derive these algorithms we start with a scattering theory description of seismic data we then introduce and develop several new theoretical concepts concerning the fundamental nature of and the relationship between forward and inverse scattering these include 1 the idea that the inversion process can be viewed as a series of steps each with a specific task 2 the realization that the inverse scattering series provides an opportunity for separating out subseries with specific and useful tasks 3 the recognition that these task specific subseries can have different and more favorable data requirements convergence and stability conditions than does the original complete inverse series and most importantly 4 the development of the first method for physically interpreting the contribution that individual terms and pieces of terms in the inv

characterization of the solution to the optimal h model reduction problem abstract in this paper the h norm approximation of a given stable proper rational transfer function by a lower order stable proper rational transfer function is studied the solution method is based on the authors observation that the h model reduction problem can be converted into a hankel norm model reduction problem a simple characterization of the solution to the optimal h model reduction problem is developed

novel gramians for linear semistable systems in this paper the notions of pseudo gramians are introduced for linear time invariant semistable systems which allow multiple semisimple poles at the origin the proposed gramians are the generalizations of standard gramian matrices defined for asymptotically stable systems and they can be computed by a set of lyapunov equations furthermore it is shown that the controllability and observability of a semistable system are indicated by the ranks of the pseudo gramians and the controllability and observability energy functions are also characterized using the pseudo gramians additionally the h2 norm and h norm of a semistable system are analyzed and then the results are used for the model reduction of semistable systems finally the effectiveness of the methods is illustrated by an example of a gene regulation network

frequency weighted spl lscr sub spl infin norm and optimal hankel norm model reduction a new relative error model reduction method is proposed using frequency weighted balanced realization and explicit spl lscr sub spl infin norm error bounds are also derived for the relative error and multiplicative error the method only needs to solve two lyapunov equations it is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase this paper also gives a complete solution to the frequency weighted hankel norm approximation with antistable weighting these results are then applied to spl lscr sub spl infin norm model reduction and several numerically effective algorithms are proposed it is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions

relative performance of moveout based multiple suppression methods for amplitude variation with offset avo analysis and common midpoint cmp stacking we apply three moveout based methods for multiple suppression to simple synthetic common midpoint cmp gathers and compare their performance in terms of level of multiple rejection and more importantly in terms of 1 suitability of processed output for use in analysis of amplitude variations with offset avo 2 primary to multiple amplitude p m ratio of the data after cmp stacking and 3 preservation of primary wavelet shape and amplitude on the stack the three approaches are filtering in the frequency wavenumber f k domain the hampson method of filtering in the parabolic radon transform domain and a hybrid method that improves upon hampson s approach by using a variation of harlan s statistical pattern recognition approach to separate primaries from multiples in the parabolic radon transform domain the f k approach is unsuitable prior to avo analysis moreover it is little better in suppressing the multiples than the cmp stack itself hampson s method performs considerably better and is su

formulas for hankel singular values and vectors for a class of input delay systems this paper gives explicit formulas for hankel singular values and vectors for a class of multiple input delay systems the class consists of systems whose gramian is reducible to an integral operator with semi separable kernel this structure is exploited in deriving a transcendental equation for singular values

an optimal controller reduction by covariance equivalent realizations based upon the theory of covariance equivalent realizations 1 this note presents a technique to obtain reduced order controllers via an optimal projection of full order lqg controllers this projection is optimal in the sense that it is a weighted least square approximation to a covariance equivalent projection of the full order optimal closed loop system the technique which is capable of extracting optimal minimal lqg controllers whenever they exist is also shown to yield controllers that are equivalent to those obtained by component cost analysis 12

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

frequency weighted model reduction via quasi kalman decomposition a frequency weighted model reduction method based on quasi kalman decomposition is proposed for discrete systems the method is essentially an extension of the technique presented by lu and lee 1985 to the frequency weighted case and can handle both single sided as well as double sided weights two illustrative examples show the effectiveness of the method and results are compared with the method developed by enns 1984

reduced order compensators via balancing and central control design for a structural control problem real time control of a physical system necessitates controllers that are low order in this article we compare two balanced truncation methods as a means of designing low order controllers for a nonlinear cable mass system the first is the standard technique of balanced truncation the second linear quadratic gaussian lqg balanced truncation can be thought of as balancing based on the controller and states that are important from the perspective of control and filter design are retained the control design applied to each reduced order model is the central controller we provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing also described in this article is a method for reducing computational time in solving algebraic riccati equations for the design of low order lqg balanced controllers

a jacobi type method for computing balanced realizations a new numerical scheme for computing balancing coordinate transformations in linear systems theory is presented the method is closely related to the jacobi method for diagonalizing symmetric matrices here the minimization of the sum of traces of the gramians by orthogonal and nonorthogonal jacobi type rotations is considered the algorithm is shown to be globally convergent to a balancing transformation that arranges the hankel singular values in a prescribed ordering local quadratic convergence of the algorithm is shown

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

model reduction by balanced truncation for systems with nuclear hankel operators we prove the h infinity error bounds for lyapunov balanced truncation and for optimal hankel norm approximation under the assumption that the hankel operator is nuclear this is an improvement of the result from glover curtain and partington siam j control optim 26 1998 pp 863 898 where additional assumptions were made the proof is based on convergence of the schmidt pairs of the hankel operator in a sobolev space we also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation

consistent impulse response estimation and system realization from noisy data a novel method for the estimation of a finite part of the impulse response is presented the method is consistent in case the measured output data are prone to colored noise the combination of kung s realization algorithm yields a consistent system identification algorithm that is particularly useful for the identification of systems with uncoupled deterministic and stochastic dynamics this algorithm is called irear a detailed linear sensitivity analysis of the impulse response estimation and kung s realization algorithms yields novel error and covariance formulae for the estimated impulse response and system matrices and the poles and frequency response functions that result from the identified system description

order reduction of large scale second order systems using krylov subspace methods in order reduction of large scale linear time invariant systems krylov subspace methods based on moment matching are among the best choices today however in many technical fields models typically consist of sets of second order differential equations and krylov subspace methods cannot directly be applied two methods for solving this problem are presented in this paper 1 an approach by su and craig is generalized and the number of matching moments is increased 2 a new approach via first order models is presented resulting in an even higher number of matching moments both solutions preserve the specific structure of the second order type model

frequency dependent error bounds for uncertain linear models we study frequency dependent error bounds for approximation and truncation of linear dynamic models with uncertainty a possible application is in model simplification relevant for control design the uncertainty is described by quadratic constraints and the error bounds are calculated based on solutions to linear matrix inequalities

bundle reduction and the alignment distance on spaces of state space lti systems this paper introduces a large class of differential geometric distances between finite dimensional linear dynamical systems collectively called the alignment distance contrary to the existing distances the alignment distance is based on the state space description of dynamical systems and is defined on the manifolds of systems of fixed order and fixed input output dimension under a matrix rank constraint e g minimality controllability or observability while the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory distances natural to this structure have not been studied the starting point for defining such a distance is to identify a linear system of order n with its equivalence class of state space realizations all related by the so called similarity action i e state space change of basis under gl n the lie group of nonsingular n times n matrices the main idea of the alignment distance is to first find the best state space change of basis that brings a realization of a system as close as possible to a realization of another system the alignment step and then compare the aligned realizations a direct implementation of this idea due to noncompactness of gl n is complicated however using the notion of reduction of the structure group of a principal bundle we show that the change of basis can be restricted to an orthogonal change of basis provided one uses realizations in a reduced subbundle this key observation brings about significant computational benefits as a technical contribution possibly of independent interest we show that several forms of realization balancing available in the control literature have differential geometric significance and are indeed examples of reducing the structure group from gl n to its subgroup of orthogonal matrices o n the alignment distance can be defined for stable and unstable systems discrete or continuous time and stochastic systems

identification and model reduction of time varying discrete time systems abstract linear discrete time variable systems are considered and an important class of uniform realizations is defined the necessary and sufficient conditions for a pulse response to be uniformly realizable are obtained two model reduction schemes one via balanced realizations and the other via hankel matrix are proposed the latter approach can be considered as an identification algorithm for the two dimensional pulse response sequence h k i it is shown that the two approaches yield an identical reduced model which is always asymptotically stable

on rotational invariance of lattice boltzmann schemes we propose the derivation of acoustic type isotropic partial differential equations that are equivalent to linear lattice boltzmann schemes with a density scalar field and a momentum vector field as conserved moments the corresponding linear equivalent partial differential equations are generated with a new berliner version of the taylor expansion method the details of the implementation are presented these ideas are applied for the d2q9 d2q13 d3q19 and d3q27 lattice boltzmann schemes some limitations associated with necessary stability conditions are also presented

a subspace algorithm for balanced state space system identification in an earlier paper by moonen et at 1989 an algorithm has been introduced for identifying multivariable linear systems directly from input output data this new algorithm falls in the class of so called subspace methods and resembles impulse response based realization algorithms in this paper we extend this algorithm by incorporating a balancing step such that the identified model is always in balanced coordinates with this modification one obtains a data driven counterpart for kung s realization algorithm

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

an hp proper orthogonal decomposition moving least squares approach for molecular dynamics simulation abstract the application of model order reduction for molecular dynamic systems exhibits intrinsic complexities due to the highly nonlinear and nonlocal properties of such systems the most costly computational part of such work is the calculation of potential energy and inter atomic force fields we introduce the so called hp proper orthogonal decomposition moving least squares hp pod mls method not only to tackle this force computation but also to improve the accuracy of reduced displacement approximations in an adaptive and controllable manner the approach combines the proper orthogonal decomposition pod method with the moving least squares mls one to interpolate the inter atomic force field at an arbitrary time instance this approximated force field is then fed into the reduced molecular dynamic md governing equation to compute the reduced order displacement field in fact this pod mls approach is already able to handle efficiently the force computation in the online computational stage i e the online force computational cost is independent of the dimension of the full md space however its corresponding approximated displacement error will be very large if the considered total time interval is long the proposed hp pod mls algorithm will divide adaptively and automatically the global temporal interval into many smaller local subintervals h refinement and then build the sets of local reduced basis functions over each of these subintervals p refinement the goal is to reduce the corresponding displacement errors to a desired tolerance over these subintervals the effectiveness of the algorithm is demonstrated by performing reduced order simulations of several molecular dynamic systems

suppression of multiple reflections using the radon transform multiple suppression using a variant of the radon transform is discussed this transform differs from the classical radon transform in that the integration surfaces are hyperbolic rather than planar this specific hyperbolic surface is equivalent to parabolae in terms of computational expense but more accurately distinguishes multiples from primary reflections the forward transform separates seismic arrivals by their differences in traveltime moveout multiples can be suppressed by an inverse transform of only part of the data examples show that multiples are effectively attenuated in prestack and stacked seismograms

approximation of fir by iir digital filters an algorithm based on balanced model reduction an algorithm for the approximation of finite impulse response fir filters by infinite impulse response iir filters is presented the algorithm is based on a concept of balanced model reduction the matrix inversions normally associated with this procedure are notoriously error prone due to ill conditioning of the special matrix forms required this difficulty is circumvented here by directly formulating a reduced state space system description which is input output equivalent to the system that would more conventionally be obtained following the explicit step of constructing an interim balanced realization examples of fir by iir filter approximations are included

an implicit deflation method for ill conditioned sylvester and lyapunov equations this paper describes an effective algorithm for solving ill conditioned sylvester or lyapunov equations these equations can be solved by conventional methods for ill conditioned linear systems however such methods are not efficient since they require on the order of n 6 arithmetic operations on order n 2 data for these ill conditioned matrix equations an implicit deflation algorithm is proposed to implement a certain svd based minimum norm least squares approximate solution the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical details are also discussed

subspace algorithms for the identification of multivarible dynamic errors in variables models abstract we consider the problem of identifying multivariable finite dimensional linear time invariant systems from noisy input output measurements apart from the fact that both the measured input and output are corrupted by additive white noise the output may also be contaminated by a term which is caused by a white input process noise furthermore all these noise processes are allowed to be correlated with each other we shall develop a solution to this problem in the framework of subspace identification and we shall show that our algorithms give consistent estimates when the system is operating in open or closed loop two realistic simulation studies are presented to demonstrate the practical applicability of the proposed algorithms 1997 elsevier science ltd

some results on energy conserving lattice boltzmann models we consider the problem of energy conserving lattice boltzmann models a major difficulty observed in previous studies is the coupling between the viscous and thermal waves even at moderate wave numbers we propose a theoretical framework based on the knowledge of the partial equivalent equations of the lattice boltzmann scheme at several orders of precision with the help of linearized models inviscid and dissipative advective acoustics and classical acoustics we suggest natural sets of relations for the parameters of lattice boltzmann schemes the application is proposed for three two dimensional schemes numerical test cases for simple linear and nonlinear waves establish that the main difficulty in the previous contributions can now be overcome

h optimal control part 1 model matching our aim in this paper is to develop a new approach for solving the h optimal control problem where the feedback arrangement takes the form of a linear fractional transformation the paper is in two parts in part 1 a basic kind of model matching problem is considered given rational matrices m s and n s the h norm of an error function defined as e s m s n s q s is minimized or bounded subject to e s and q s being stable closed form state space characterizations are obtained for both e s and q s the results established here will be used in part 2 of the paper hung 1989 to solve the h optimal control problem

an algorithmic method for the simplification of linear dynamic scalar systems a comprehensive method is proposed for obtaining reduced order models of a scalar linear time invariant system described by the transfer function g s the method is a viable alternative to the pade approximation technique for model order reduction it approximates the differential operations d i g s dsi proportional to the time moments by the divided difference equivalents and finally leads to linear equations in the unknown coefficients of the reduced model r s it is shown that a parameter governs the stability and the low frequency matching quality of the model heuristic criteria are given for selecting a proper the method is illustrated by some examples from the literature

high resolution velocity gathers and offset space reconstruction we present a high resolution procedure to reconstruct common midpoint cmp gathers first we describe the forward and inverse transformations between offset and velocity space then we formulate an underdetermined linear inverse problem in which the target is the artifacts free aperture compensated velocity gather we show that a sparse inversion leads to a solution that resembles the infinite aperture velocity gather the latter is the velocity gather that should have been estimated with a simple conjugate operator designed from an infinite aperture seismic array this high resolution velocity gather is then used to reconstruct the offset space the algorithm is formally derived using two basic principles first we use the principle of maximum entropy to translate prior information about the unknown parameters into a probabilistic framework in other words to assign a probability density function to our model second we apply bayes s rule to relate the a priori probability density function pdf with the pdf corresponding to the experimental uncertainties likelihood function to construct the a posteriori distribution of the unknown parameters finally the model is evaluated by maximizing the a posteriori distribution when the problem is correctly regularized the algorithm converges to a solution characterized by different degrees of sparseness depending on the required resolution the solutions exhibit minimum entropy when the entropy is measured in terms of burg s definition we emphasize two crucial differences in our approach with the familiar burg method of maximum entropy spectral analysis first burg s entropy is minimized rather than maximized which is equivalent to inferring as much as possible about the model from the data second our approach uses the data as constraints in contrast with the classic maximum entropy spectral analysis approach where the autocorrelation function is the constraint this implies that we recover not only amplitude information but also phase information which serves to extrapolate the data outside the original aperture of the array the tradeoff is controlled by a single parameter that under asymptotic conditions reduces the method to a damped least squares solution finally the high resolution or aperture compensated velocity gather is used to extrapolate near and far offset traces

topology of the set of asymptotically stable minimal systems it is shown that the set of asymptotically stable and minimal single input single output systems of order n has n 1 connected components in the case of asymptotically stable and minimal multiple input multiple output systems however there is only one connected component

linear lattice boltzmann schemes for acoustic parameter choices and isotropy properties in this paper we investigate the numerous possible parameter choices for linear lattice boltzmann schemes according to the definition of the order of isotropy given in augier et al in press 4 this property written in a general framework that includes all of the d dq q schemes can be understood through a group operation the resulting relationships between the parameters of the scheme defining the equilibrium states and relaxation times yield a rigorous methodology that should be followed if one is to ensure isotropy at a given order for acoustic applications in two spacial dimensions namely d 2q 9 and d 2q 1 3 schemes this methodology is used to propose a full description of the sets of parameters that are isotropic at order m email protected 1 2 3 4 5 for d 2q 9 and email protected 1 2 for d 2q 1 3 we include numerical illustrations for the d 2q 9 scheme obtained with the code lbmpy developed in the laboratory of mathematics of the university paris sud

on the use of modal derivatives for nonlinear model order reduction summary modal derivative is an approach to compute a reduced basis for model order reduction of large scale nonlinear systems that typically stem from the discretization of partial differential equations in this way a complex nonlinear simulation model can be integrated into an optimization problem or the design of a controller based on the resulting small scale state space model we investigate the approximation properties of modal derivatives analytically and thus lay a theoretical foundation of their use in model order reduction which has been missing so far concentrating on the application field of structural mechanics and structural dynamics we show that the concept of modal derivatives can also be applied as nonlinear extension of the craig bampton family of methods for substructuring we furthermore generalize the approach from a pure projection scheme to a novel reduced order modeling method that replaces all nonlinear terms by quadratic expressions in the reduced state variables this complexity reduction leads to a frequency preserving nonlinear quadratic state space model numerical examples with carefully chosen nonlinear model problems and three dimensional nonlinear elasticity confirm the analytical properties of the modal derivative reduction and show the potential of the proposed novel complexity reduction methods along with the current limitations copyright 2016 john wiley sons ltd

balanced realizations for linear systems a variational approach this paper develops a new geometric approach to balanced realizations which enables balanced realizations for arbitrary linear systems to be defined and studied for an arbitrary unitarily invariant strictly plurisubharmonic function on the set of realizations of a given transfer function the class of realizations is considered that minimizes the function based on results from invariant theory and complex analysis a general theorem on the existence and uniqueness properties of such function minimizing realizations is derived if the function is the sum of the traces of the controllability and observability gramians the usual class of balanced realizations is obtained other choices of functions yield different new classes of function minimizing realizations

the cumulant lattice boltzmann equation in three dimensions theory and validation we propose analyze and validate a lattice boltzmann model with a cumulant collision operator the new model is analytically and numerically shown to poses smaller errors than a moment based multiple relaxation time lattice boltzmann model we demonstrate the usability of the cumulant lattice boltzmann model by simulations of flow around a sphere for reynolds numbers from 200 to 105

model reduction of large scale systems by least squares abstract in this paper we introduce an approximation method for model reduction of large scale dynamical systems this is a projection which combines aspects of the svd and krylov based reduction methods this projection can be efficiently computed using tools from numerical analysis namely the rational krylov method for the krylov side of the projection and a low rank smith type iteration to solve a lyapunov equation for the svd side of the projection for discrete time systems the proposed approach is based on the least squares fit of the r 1 th column of a hankel matrix to the preceding r columns where r is the order of the reduced system the reduced system is asymptotically stable matches the first r markov parameters of the full order model and minimizes a weighted h 2 error the method is also generalized for moment matching at arbitrary interpolation points application to continuous time systems is achieved via the bilinear transformation numerical examples prove the effectiveness of the approach the proposed method is significant because it combines guaranteed stability and moment matching together with an optimization criterion

arma spectral estimation of narrow band processes via model reduction the problem of estimating autoregressive moving average arma models for narrowband processes is considered the following approach is proposed estimate a high order autoregressive ar approximation of the process by model reduction based on a truncated internally balanced realization or optimal hankel norm model reduction reduce the order of this high order ar estimate to find a lower order arma model this algorithm gives arma spectral estimates with excellent resolution properties without using iterative numerical minimization methods as for the maximum likelihood method how to take the narrowband assumption into account in the model reduction step is discussed in detail

parametrization and optimization of reduced order filters this paper deals with the design of reduced order filters for linear signal models to ensure that the transfer function from the noise to the filtering error is not only stable but also has a minimum h sub 2 norm one of the major results in the paper is the discovery and parametrization of a set of filters of fixed order which lead to stable filtering error transfer functions such a parametrization is given in terms of an arbitrary orthogonal projection matrix and a given full order filter as a consequence the problem of minimizing the h sub 2 norm of the filtering error transfer function over the set of reduced order filters can be formulated as an equivalent unconstrained parametric optimization problem over a compact manifold a gradient flow based algorithm is proposed to find an optimal solution to the problem nice properties of the algorithm including the convergence property are established theoretically as well as demonstrated numerically with an example

discrete radon transform this paper describes the discrete radon transform drt and the exact inversion algorithm for it similar to the discrete fourier transform dft the drt is defined for periodic vector sequences and studied as a transform in its own right casting the forward transform as a matrix vector multiplication the key observation is that the matrix although very large has a block circulant structure this observation allows construction of fast direct and inverse transforms moreover we show that the drt can be used to compute various generalizations of the classical radon transform rt and in particular the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves in fact we describe not a single transform but a class of transforms representatives of which correspond in one way or another to discrete versions of the rt and its generalizations an interesting observation is that the exact inversion algorithm cannot be obtained directly from radon s inversion formula given the fact that the rt has no nontrivial one dimensional analog exact invertibility makes the drt a useful tool geared specifically for multidimensional digital signal processing exact invertibility of the drt flexibility in its definition and fast computational algorithm affect present applications and open possibilities for new ones some of these applications are discussed in the paper

stable model order reduction for time domain exterior vibro acoustic finite element simulations abstract this paper presents a novel method that enables model order reduction of a fully coupled exterior vibro acoustic finite element model for time domain simulations the method preserves the stability of the full model and reduces the amount of degrees of freedom significantly with only a moderate amount of calculation complexity infinite elements are used on the finite element boundary to satisfy the sommerfeld radiation condition two different strategies to calculate the reduced order model are compared the first strategy works with a split reduced basis and can be applied on any fully stable model the second strategy starts from a modified everstine formulation and directly builds a reduced basis from the full model leading to more compact reduced order models furthermore a method is derived to perform explicit time integration on the reduced system while avoiding the inversion of the mass matrix which might not be possible due to the presence of the infinite elements also this method is shown to preserve the stability of the model and a computationally efficient way for implementation of the method is discussed the effectiveness of the novel methodology is demonstrated with two numerical models

a lyapunov theory for linear time delay systems some relationships between a lyapunov based stability test and linear control system models with time delays are explored the main result is a generalization of the result that if a b is a reachable pair of matrices then square a is a stability matrix if and only if there is a positive definite matrix k such that ak ka bb

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

optimum weighted stack for multiple suppression in water deeper than about 300 m common midpoint stack is moderately effective at suppressing multiple reflections the stacking process acts as a low pass multiple filter rejecting to some extent frequencies greater than the reciprocal of the multiple s residual moveout on the farthest offset trace a weighted stack can often be even more effective at attenuating multiples if the weights are chosen such that the multiple filter response has low amplitudes at those frequencies where the multiple is strong a least squares technique is developed to determine optimum weights based on the multiple s frequency content the water depth the primary velocity and on a stabilization factor equivalent to white noise in deconvolution filter design to be truly optimum the weights must vary both spatially to account for water depth changes and temporally to account for sediment velocity changes and mute patterns but the filter responses are not very sensitive to the input parameters and need to be redesigned only infrequently the superiority of weighted stacks to uniform stacks is demonstrated by application to three deep water examples in all cases weighted stacks attenuate multiples more effectively than do conventional uniform stacks

balanced pod for model reduction of linear pde systems convergence theory we consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems the algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations the algorithm is related to the proper orthogonal decomposition and it was first proposed for systems of ordinary differential equations by rowley int j bifurc chaos appl sci eng 15 3 997 1013 2005 for the convergence analysis we consider the algorithm in terms of the hankel operator of the system rather than the product of the system gramians as originally proposed by rowley for exponentially stable systems with bounded finite rank input and output operators we prove that the balanced realization can be expressed in terms of balancing modes which are related to the hankel operator the balancing modes are required to be smooth and this can cause computational difficulties for pde systems we show how this smoothness requirement can be lessened for parabolic systems and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems we prove entrywise convergence of the matrices in the approximate reduced order models in both cases and present numerical results for two example pde systems

brief paper on the stability of a reduced order filter based on dominant singular value decomposition of the system dynamics the stability of a reduced order filter rof the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics is examined a definition of s detectability is introduced it is found that the observability of all unstable and neutral singular vectors s detectability is a sufficient condition for the existence of a stable filter

another balanced controller reduction algorithm the problem of controller reduction has been addressed by a number of authors 1 4 using the concept of balanced coordinates originally proposed by moore 5 this note establishes a more direct link between the riccati balancing methods in 2 4 and moore s work and proposes yet another candidate method as a logical extension of 2 4

model order reduction by balanced proper orthogonal decomposition and by rational interpolation we show that model order reduction by rational interpolation also known as moment matching or rational krylov can be seen as the special case of balanced proper orthogonal decomposition where all the snapshots are retained and particular numerical procedures are used to obtain the snapshots

on the structure of balanced and other principal representations of siso systems a new matrix w co which can be considered as a cross gramian matrix which contains information about both controllability and observability is defined for single input single output linear systems using this matrix the structural properties of linear systems are studied in the context of principal component analysis the matrix w co can be used in obtaining balanced and other principal representations without computation of the controllability and the observability gramians the importance of this matrix in model order reduction is highlighted

error bounds for balanced model reduction of linear time varying systems error bounds are developed for balanced truncation of linear time varying systems leading to an extension of the twice the sum of the tail formulas well known in the time invariant case the approach relies on an operator theoretic framework for analysis of linear time varying systems this provides a multivariable notion of frequency for such systems which are thus characterized by rational functions of many complex variables allowing the problem to be formulated in the linear fractional framework using a time varying version of standard necessary conditions for reduced order modeling based on convex operator inequalities we show that these error bounds for balanced truncation are related to the closest possible reduced order modeling error in a sense which parallels the time invariant case

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

computation of approximate null vectors of sylvester and lyapunov operators this paper describes an effective algorithm for computing approximate null vectors of certain matrix operators associated with sylvester or lyapunov equations the singular value decomposition and rank revealing qr methods are two widely used stable algorithms for numerical determination of the rank and nullity of a matrix a these methods however are not readily applicable to sylvester and lyapunov operators since they require on the order of n sup 6 arithmetic operations on order n sup 2 data for these problems a variant of inverse power iteration is employed to compute orthonormal bases for singular subspaces associated with the small singular values the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical aspects are considered and a direct refinement technique is proposed to enhance the convergence of the algorithm

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

on balanced realizations of bounded real and positive real functions based on normalized coprime factorizations with respect to indefinite metrics and the construction of suitable characteristic functions the ober balanced canonical forms for the classes of bounded real and positive real functions are derived this uses a matrix representation of the shift realization with respect to a basis related to sets of orthogonal polynomials

equivalent partial differential equations of a lattice boltzmann scheme we show that when we formulate the lattice boltzmann equation with a small time step dt and an associated space scale dx a taylor expansion joined with the so called equivalent equation methodology leads to establishing macroscopic fluid equations as a formal limit we recover the euler equations of gas dynamics at the first order and the compressible navier stokes equations at the second order

partial fraction expansion based frequency weighted model reduction technique with error bounds in this technical note we present a new frequency weighted balancing related technique which is based on a parametrized combination of the unweighted balanced truncation technique 9 8 and the partial fraction expansion technique 11 the reduced order models which are guaranteed to be stable in the case of double sided weighting are obtained by either direct truncation or singular perturbation approximation simple elegant and easily computable a priori error bounds are also derived numerical examples and comparison with other well known techniques show the effectiveness of the proposed technique

stable lattice boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves we follow the mathematical framework proposed by bouchut 2003 22 and present in this contribution a dual entropy approach for determining equilibrium states of a lattice boltzmann scheme this method is expressed in terms of the dual of the mathematical entropy relative to the underlying conservation law it appears as a good mathematical framework for establishing a h theorem for the system of equations with discrete velocities the dual entropy approach is used with d1q3 lattice boltzmann schemes for the burgers equation it conducts to the explicitation of three different equilibrium distributions of particles and induces naturally a nonlinear stability condition satisfactory numerical results for strong nonlinear shocks and rarefactions are presented we prove also that the dual entropy approach can be applied with a d1q3 lattice boltzmann scheme for systems of linear and nonlinear acoustics and we present a numerical result with strong nonlinear waves for nonlinear acoustics we establish also a negative result with the present framework the dual entropy approach cannot be used for the shallow water equations

brief paper balanced truncation model reduction for systems with inhomogeneous initial conditions we present a rigorous approach to extend balanced truncation model reduction btmr to systems with inhomogeneous initial conditions we provide an estimate for the error between the input output maps of the original and of the reduced initial value system and we illustrate numerically the superiority of our approach over the naive application of btmr when btmr is applied to linear time invariant systems with inhomogeneous initial conditions it is crucial that the initial data are well represented by the subspaces generated by btmr this requirement is often ignored or it is avoided by making the restrictive assumption that the initial data are zero to ensure that the initial data are well represented by the btmr subspaces we add auxiliary inputs determined by the initial data

quartic parameters for acoustic applications of lattice boltzmann scheme using the taylor expansion method we show that it is possible to improve the lattice boltzmann method for acoustic applications we derive a formal expansion of the eigenvalues of the discrete approximation and fit the parameters of the scheme to enforce fourth order accuracy the corresponding discrete equations are solved with the help of symbolic manipulation the solutions are obtained in the case of d3q27 lattice boltzmann scheme various numerical tests support the coherence of this approach

parabolic radon transform sampling and efficiency a good choice of the sampling in the transform domain is essential for a successful application of the parabolic radon transform the parabolic radon transform is computed for each temporal frequency and is essentially equivalent to the nonuniform fourier transform this leads to new and useful insights in the parabolic radon transform using nonuniform fourier theory we derive a minimum sampling interval for the curvature parameter and a maximum curvature range for which stability is guaranteed for general irregular sampling a significantly smaller sampling interval requires stabilization if diagonal stabilization is used no gain in resolution is obtained in contrast to conventional implementations the curvature sampling interval is proposed to be inversely proportional to the temporal frequency this results in improved quality of the transform and yields significant savings in computation time

implementation of the compact interpolation within the octree based lattice boltzmann solver musubi abstract a sparse octree based parallel implementation of the lattice boltzmann method for non uniform meshes is presented in this paper to couple grids of different resolutions a second order accurate compact interpolation is employed and further extended into three dimensions the compact interpolation requires only four source elements from the adjacent level for both two and three dimensions thus improves the computational efficiency moreover a weight based domain decomposition algorithm and the level wise arrangement of elements is explained in detail the second order convergence for both velocity and strain rate are validated numerically in the taylor green vortex test case additionally the 3d flow around a sphere in the reynolds number range of 100 1000 is investigated good agreement between simulated results and those from literature is observed which provides further evidence for the accuracy of our method

theory of the lattice boltzmann equation symmetry properties of discrete velocity sets the lattice boltzmann equation replaces continuous particle velocity space by a finite set the velocity distribution function then varies over a finite dimensional vector space instead of over an infinite dimensional function space the number of linearly independent moments of the distribution function in a lattice boltzmann model cannot exceed the number of velocities finite dimensionality therefore necessarily induces linear dependences among the moments that do not exist in a continuous theory given a finite velocity set it is important to know which moments are free of these dependences elementary group theory is applied to the solution of this problem it is found that decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group makes it straightforward to uncover linear dependences among the moments the construction of some standard two and three dimensional models is reviewed from this viewpoint and procedures for constructing higher dimensional models are suggested