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

new convergence results on the global gmres method for diagonalizable matrices in the present paper we give some new convergence results of the global gmres method for multiple linear systems in the case where the coefficient matrix a is diagonalizable we derive new upper bounds for the frobenius norm of the residual we also consider the case of normal matrices and we propose new expressions for the norm of the residual

approximate solution of large sparse lyapunov equations describes a simple method for efficiently estimating the dominant eigenvalues and eigenvectors of the solution to a lyapunov equation without first solving the equation explicitly the method is based on the power method and matrix vector multiplications and is particularly suitable for problems where those multiplications can be done efficiently such as where the coefficient matrices are large and sparse or low rank the same idea is directly applicable to balanced truncation order reduction of linear systems

an adaptive block tangential method for multi input multi output dynamical systems abstract in this paper we present a new approach for model order reduction in large scale dynamical systems with multiple inputs and multiple outputs mimo this approach will be named adaptive block tangential arnoldi algorithm abtaa and is based on interpolation via block tangential krylov subspaces requiring the selection of shifts and tangent directions via an adaptive procedure we give some algebraic properties and present some numerical examples to show the effectiveness of the proposed method

parallel algorithms for algebraic riccati equations the matrix sign function is the basis of a parallel algorithm for solving the generalized algebraic riccati equation three forms of the algorithm were implemented and tested on a distributed memory hypercube multiprocessor performance results indicate that the method is an excellent means of solving large scale problems on a parallel computer

on the linear independence measure of logarithms of rational numbers in this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and in particular the linear independence measure of 1 log 2 log 3 log 5 and of 1 log 2 log 3 log 5 log 7 we also give a method to search for polynomials of smallest norm on a real interval a b which may be suitable for computing or improving the linear independence measure of logarithms of rational numbers

a global harmonic arnoldi method for large non hermitian eigenproblems with an application to multiple eigenvalue problems the global arnoldi method can be used to compute exterior eigenpairs of a large non hermitian matrix a but it does not work well for interior eigenvalue problems based on the global arnoldi process that generates an f orthonormal basis of a matrix krylov subspace we propose a global harmonic arnoldi method for computing certain harmonic f ritz pairs that are used to approximate some interior eigenpairs we propose computing the f rayleigh quotients of the large non hermitian matrix with respect to harmonic f ritz vectors and taking them as new approximate eigenvalues they are better and more reliable than the harmonic f ritz values the global harmonic arnoldi method inherits convergence properties of the harmonic arnoldi method applied to a larger matrix whose distinct eigenvalues are the same as those of the original given matrix some properties of the harmonic f ritz vectors are presented as an application assuming that a is diagonalizable we show that the global harmonic arnoldi method is able to solve multiple eigenvalue problems both in theory and in practice to be practical we develop an implicitly restarted global harmonic arnoldi algorithm with certain harmonic f shifts suggested in particular this algorithm can be adaptively used to solve multiple eigenvalue problems numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill conditioned multiple eigenproblems

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

data driven parametrized model reduction in the loewner framework we introduce a new method for constructing reduced order models that accurately approximate the frequency domain behavior of large scale parametrized linear dynamical systems using multivariate rational interpolation we compute reduced order models that match transfer function measurements of the large scale system the main tools are a new barycentric formula and loewner matrices formed directly from measurements this data driven approach introduces a new degree of freedom for parametrized model reduction the ability to choose separate reduced orders for each parameter more precisely each reduced order is determined by computing the rank of appropriate loewner matrices moreover we show how to control the pointwise approximation error through a new formula involving the barycentric form and the smallest singular value of a loewner matrix finally we also give new state space realizations for multivariate rational functions and conclude with several numerical examples showcasing the effectivene

on global hessenberg based methods for solving sylvester matrix equations abstract in the first part of this paper we investigate the use of hessenberg based methods for solving the sylvester matrix equation a x x b c to achieve this goal the sylvester form of the global generalized hessenberg process is presented using this process different methods based on a petrov galerkin or on a minimal norm condition are derived in the second part we focus on the sgl cmrh method which is based on the sylvester form of the hessenberg process with pivoting strategy combined with a minimal norm condition in order to accelerate the sgl cmrh method a preconditioned framework of this method is also considered it includes both fixed and flexible variants of the sgl cmrh method moreover the connection between the flexible preconditioned sgl cmrh method and the fixed one is studied and some upper bounds for the residual norm are obtained in particular application of the obtained theoretical results is investigated for the special case of solving linear systems of equations with several right hand sides finally some numerical experiments are given in order to evaluate the effectiveness of the proposed methods

algorithms for model reduction of large dynamical systems three algorithms for the model reduction of large scale continuous time time invariant linear dynamical systems with a sparse or structured transition matrix and a small number of inputs and outputs are described they rely on low rank approximations to the controllability and observability gramians which can efficiently be computed by adi based iterative low rank methods the first two model reduction methods are closely related to the well known square root method and schur method which are balanced truncation techniques the third method is a heuristic balancing free technique the performance of the model reduction algorithms is studied in numerical experiments

hankel transform domain analysis of open circular microstrip radiating structures a full wave analysis of a circular disk conductor printed on a dielectric substrate backed by a ground plane is presented the method is based on the galerkin s method applied in the hankel transform domain and is numerically quite efficient several numerical examples are compared with available experimental data the agreement is found to be quite good and the validity of the present method is confirmed

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

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

the smallest perron numbers a perron number is a real algebraic integer of degree d 2 whose conjugates are i such that max 2 i d i in this paper we compute the smallest perron numbers of degree d 24 and verify that they all satisfy the lind boyd conjecture moreover the smallest perron numbers of degree 17 and 23 give the smallest house for these degrees the computations use a family of explicit auxiliary functions these functions depend on generalizations of the integer transfinite diameter of some compact sets in c

entropies and factorizations of topological markov shifts on presente une caracterisation des valeurs possibles pour l entropie topologique de decalages de markov si est un nombre de penon alors il y a une matrice non negative aperiodique entiere dont le rayon spectral est

on the generating sequences of regular languages on k symbols the main result is a characterization of the generating sequences of the length of words in a regular language on k symbols we say that a sequence s of integers is regular if there is a finite graph g with two vertices i t such that sn is the number of paths of length n from i to t in g thus the generating sequence of a regular language is regular we prove that a sequence s is the generating sequence of a regular language on k symbols if and only if both sequences s sn n 0 and t kn sn n 0 are regular

the integer chebyshev problem the main problem of this chapter is to find a polynomial in z n of minimal supremum norm on an interval this is p1 and it is of a slightly different flavour than most of the other problems in this book in that there is no restriction on the size of the coefficients we now state p1 with greater precision

numerical solution of the lyapunov equation by approximate power iteration abstract we present the approximate power iteration api algorithm for the computation of the dominant invariant subspace of the solution x of large order lyapunov equations ax xa t q 0 without first computing the matrix x itself the api algorithm is an iterative procedure that uses krylov subspace bases in computing estimates of matrix vector products x v in a power iteration sequence application of the api algorithm requires that a a t x admits a good low rank solution then api provides an orthogonal basis of a subspace that closely approximates the dominant x invariant subspace of corresponding dimension analytical convergence results are also presented

finite time balanced truncation for linear systems via shifted legendre polynomials abstract in this paper we present a finite time model order reduction method for linear systems via shifted legendre polynomials the main idea of the approach is to use finite time empirical gramians which are constructed from impulse responses by solving block tridiagonal linear systems to generate approximate balanced system for the large scale system the balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted legendre polynomials without individual reduction of the gramians and a separate eigenvector solve then the reduced order model is constructed by truncating the states corresponding to the small approximate hankel singular values hsvs the stability preservation of the reduced model is briefly discussed and in combination with the dominant subspace projection method we modify the reduction procedure to alleviate the shortcomings of the above method which may unexpectedly lead to unstable systems even though the original one is stable furthermore the properties of the resulting reduced models are considered finally numerical experiments are given to demonstrate the effectiveness of the proposed methods

strong shift equivalence and shear adjacency of nonnegative square integer matrices abstract for two square matrices a b of possibly different sizes with nonnegative integer entries write a 1 b if a rs and b sr for some two nonnegative integer matrices r s the transitive closure of this relation is called strong shift equivalence and is important in symbolic dynamics where it has been shown by r f williams to characterize the isomorphism of two topological markov chains with transition matrices a and b one invariant is the characteristic polynomial up to factors of however no procedure for deciding strong shift equivalence is known even for 2 2 matrices a b in fact for n n matrices with n 2 no nontrivial sufficient condition is known this paper presents such a sufficient condition that a and b are in the same component of a directed graph whose vertices are all n n nonnegative integer matrices sharing a fixed characteristic polynomial and whose edges correspond to certain elementary similarities for n 2 this result gives confirmation of strong shift equivalences that previously could not be verified for n 2 previous results are strengthened and the structure of the directed graph is determined

algebraic integers whose conjugates all lie in an ellipse we find all 15909 algebraic integers whose conjugates all lie in an ellipse with two of them nonreal while the others lie in the real interval 1 2 this problem has applications to finding certain subgroups of sl 2 c we use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of c this gives good bounds for the coefficients of the minimal polynomial of a

truncated balanced realization of a stable non minimal state space system in this paper we present a numerically reliable algorithm to compute the balanced realization of a stable state space system that may be arbitrarily close to being unobservable and or uncontrollable the resulting realization which is known to be a good approximation of the original system must be minimal and therefore may contain a reduced number of states depending on the choice of partitioning of the hankel singular values this algorithm can be used either as a form of minimal realization or of model reduction this illustrates that in finite precision arithmetic these two procedures are closely related in addition to real matrix multiplication the algorithm only requires the solution of two lyapunov equations and one singular value decomposition of an upper triangular matrix

the maximal modulus of an algebraic integer the maximal modulus of an algebraic integer is the absolute value of its largest conjugate we compute the minimum of the maximal modulus of all algebraic integers of degree d which are not roots of unity for d at most 12 the computations suggest that the minimum is never attained for a reciprocal algebraic integer the truth of this conjecture would show that the conjecture of schinzel and zassenhaus follows from a theorem of smyth we further test our conjecture by computing the minimum of the maximal modulus of all reciprocal algebraic integers of degree d which are not roots of unity for d at most 16 our computations strongly suggest that the best constant in the conjecture of schinzel and zassenhaus is 1 5 log 3 where 0 is the smallest p v number they also shed some light on a recent conjecture of lind concerning the perron numbers

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

model reduction of a class of discrete time nonlinear systems abstract this paper considers the problem of model reduction of a class of discrete time systems subject to lipschitzian nonlinearities it is shown that under some conditions the nonlinear system can be either approximated by a discrete time linear time invariant system or a nonlinear system of reduced order the computation of the matrices of the reduced order system is carried out through the solutions of a set of linear matrix inequalities the proposed design is approved by the simulation of reduced order dynamics of a mass spring system subject to a nonlinear friction and a linear electric circuit with uncertain parameters

multilayered media green s functions in integral equation formulations a compact representation is given of the electric and magnetic type dyadic green s functions for plane stratified multilayered uniaxial media based on the transmission line network analog along the aids normal to the stratification furthermore mixed potential integral equations are derived within the framework of this transmission line formalism for arbitrarily shaped conducting or penetrable objects embedded in the multilayered medium the development emphasizes laterally unbounded environments but an extension to the case of a medium enclosed by a rectangular shield is also included

fast and accurate mom analysis of periodic arrays of multilayered stacked rectangular patches with application to the design of reflectarray antennas the scattering of plane waves by periodic arrays of stacked rectangular patches in multilayered substrates is a problem that has to be solved many times when designing reflectarray antennas made of those patches under the local periodicity assumption the solution to the periodic multilayered problem has been traditionally carried out by means of the galerkin s version of the method of moments mom in the spectral domain this approach involves the computation of double infinite summations and whereas some of these summations converge very fast some other converge very slowly in this paper the slowly convergent summations are computed by making use of an enhanced mixed potential integral equation mpie formulation of the mom in the spatial domain this enhanced formulation is based on the interpolation of the multilayered periodic green s functions and on the efficient computation of the four dimensional 4 d integrals leading to the mom matrix entries both the novel hybrid spectral spatial mom code and the standard spectral domain mom code have been used for the design of a contoured beam reflectarray antenna it has been verified that the spectral spatial mom code requires cpu times that are typically 30 times smaller than those required by the pure spectral domain mom code

an optimal projection method for the reduced order modeling of incompressible flows a new method to build reduced order models roms for incompressible flows is addressed in this paper proper orthogonal decompositions pods are used to efficiently represent both the fluid velocity and pressure fields with a small number of spatial modes a minimum residual projection is then developed for building the reduced models in which the velocity and pressure temporal coefficients are fully coupled comparisons of numerical calculations based on the proposed approach with results arising from the classical galerkin projection are exposed on three fluid flows a transient ventilated cavity a periodic lid driven cavity and a transient mixed convection flow it is shown that both stability and accuracy of the roms are strongly enhance when the minimum residual projection is used

on optimality of approximate low rank solutions of large scale matrix equations abstract in this paper we discuss some optimality results for the approximation of large scale matrix equations in particular this includes the special case of lyapunov and sylvester equations respectively we show a relation between the iterative rational krylov algorithm and a riemannian optimization method which recently has been shown to locally minimize a certain energy norm of the underlying lyapunov operator moreover we extend the results for a more general setting leading to a slight modification of irka by means of some numerical test examples we show the efficiency of the proposed methods

hessian based model reduction for large scale systems with initial condition inputs reduced order models that are able to approximate output quantities of interest of high fidelity computational models over a wide range of input parameters play an important role in making tractable large scale optimal design optimal control and inverse problem applications we consider the problem of determining a reduced model of an initial value problem that spans all important initial conditions and pose the task of determining appropriate training sets for reduced basis construction as a sequence of optimization problems we show that under certain assumptions these optimization problems have an explicit solution in the form of an eigenvalue problem yielding an efficient model reduction algorithm that scales well to systems with states of high dimension furthermore tight upper bounds are given for the error in the outputs of the reduced models the reduction methodology is demonstrated for a large scale contaminant transport problem

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

integral equation formulation of microstrip antennas the problem of a microstrip antenna covered by a dielectric is formulated in terms of coupled integro differential equations with the current distribution on the microstrip patch as an unknown variable galerkin s method is used to solve for the unknown patch current using the present formulation the resonant frequency and bandwidth of a rectangular microstrip antenna are determined and compared with experimental results

trace of totally positive algebraic integers and integer transfinite diameter the diameter of a totally real algebraic integer of degree d with conjugates 1 2 d is diam d 1 for all positive integers k and n diam 2 cos 2k n is less than 4 r m robinson has computed modulo integer translations all the other totally real algebraic integers with diam 4 for d 8 we have done the computations for all d 15 we use a large family of explicit auxiliary functions related to generalized integer transfinite diameter of real intervals they give good bounds for the coefficients of the minimal polynomial of for d 15 we prove a recent conjecture of capparelli et al mathematics subject classification 2000 primary 11c08 11y40

the use of vector transforms in solving some electromagnetic scattering problems it is shown that there exists a class of vector transforms and vector series expansions that are independent of their scalar counterparts they can be used to simplify the solutions of a class of vector electromagnetic scattering problems the applications are shown of vector fourier hankel and mathieu transforms and series expansions to the formal solutions of scattering by rectangular circular and elliptical disks and open ended waveguides the problems solved are canonical problems however more complex problems for applications in microwaves microstrip integrated circuits geophysical probing etc can be solved in a similar fashion

infinite to one codes and markov measures we study the structure of infinite to one continuous codes between subshifts of finite type and the behaviour of markov measures under such codes we show that if an infinite to one code lifts one markov measure to a markov measure then it lifts each markov measure to uncountably many markov measures and the fibre over each markov measure is isomorphic to any other fibre calling such a code markovian we characterize markovian codes through pressure we show that a simple condition on periodic points necessary for the existence of a code between two subshifts of finite type is sufficient to construct a markovian code several classes of markovian codes are studied in the process of proving illustrating and providing contrast to the main results a number of examples and counterexamples are given in particular we give a continuous code between two bernoulli shifts such that the defining vector of the image is not a clustering of the defining vector of the domain

model reduction of neutral linear and nonlinear time invariant time delay systems with discrete and distributed delays the problem of model reduction by moment matching for linear and nonlinear differential time delay systems is studied the class of models considered includes neutral differential time delay systems with discrete delays and distributed delays the description of moment is revisited by means of a sylvester like equation for linear time delay systems and by means of the center manifold theory for nonlinear time delay systems in addition the moments at infinity are characterized for both linear and nonlinear time delay systems parameterized families of models achieving moment matching are given the parameters can be exploited to derive delay free reduced order models or time delay reduced order models with additional properties e g interpolation at an arbitrary large number of points finally the problem of obtaining a reduced order model of an unstable system is discussed and solved

hybrid reduced order modeling applied to nonlinear models summary reduced order modeling plays an indispensible role for most real world complex models the objective of this manuscript is to hybridize local and global sensitivity analysis methods to enable the application of reduced order modeling to complex nonlinear models often encountered in real system design and analysis calculations for example nuclear reactors this is achieved by first employing local variational methods to identify important nonlinear features of the original model that are required to reach a user defined accuracy for the reduced model this information is obtained by sampling local first order derivatives of a pseudoresponse utilizing a modified representation of an infinite series expansion around some reference point the resulting derivative information is aggregated in a subspace of dimension much less than the dimension of the input parameter space the accuracy of the reduced model can be mathematically quantified using a bounding norm next global sensitivity methods are employed to exhaustively search the reduced subspace for sensitivity information the theory and implementation details of the proposed method are exposed in this manuscript numerical tests based on prototype nonlinear functions and radiation transport models with many input parameters and many responses are conducted as proof of principle copyright 2012 john wiley sons ltd

integer transfinite diameter and polynomials with small mahler measure in this work we show how suitable generalizations of the integer transfinite diameter of some compact sets in c give very good bounds for coefficients of polynomials with small mahler measure by this way we give the list of all monic irreducible primitive polynomials of z x of degree at most 36 with mahler measure less than 1 324 and of degree 38 and 40 with mahler measure less than 1 31

polynomials with small mahler measure we describe several searches for polynomials with integer coefficients and small mahler measure we describe the algorithm used to test mahler measures we determine all polynomials with degree at most 24 and mahler measure less than 1 3 test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40 and check certain sparse polynomials with height 1 and degree as large as 181 we find a new limit point of mahler measures near 1 309 four new salem numbers less than 1 3 and many new polynomials with small mahler measure none has measure smaller than that of lehmer s degree 10 polynomial

exploiting structure in large scale electrical circuit and power system problems abstract the rapid increase in complexity of systems such as electrical circuits and power systems calls for the development of efficient numerical methods in many cases direct application of standardized methods for numerical problems is computationally not feasible or inefficient however the performance of such methods can be improved considerably by taking into account the structure of the underlying problem in this paper we describe when and how this mathematical and or physical structure can be exploited to arrive at efficient algorithms that also suffer less from other numerical issues such as round off errors eigenvalue and stability problems are considered in particular but applications to other problems are shown as well

matrix krylov subspace methods for large scale model reduction problems the present paper considers matrix krylov subspace methods for solving large coupled lyapunov matrix equations of the form ap pa t bb t 0 and a tq qa c tc 0 where a is a real nxn matrix b and c t are real nxs and nxr matrices respectively with email protected n and email protected n such problems appear in many areas of control theory such as the computation of the controllability and observability gramians of a stable linear time invariant lti system the proposed methods are based on the global arnoldi and lanczos processes in the second part we show how to use matrix krylov subspace techniques to obtain a reduced order model for lti systems this will be done by approximating the corresponding transfer functions finally numerical experiments are reported to illustrate the behavior and the effectiveness of the global arnoldi and lanczos processes when applied to solve some large coupled lyapunov equations and to approximate some transfer functions

a priori space time separated representation for the reduced order modeling of low reynolds number flows abstract reduced order models roms in fluid dynamics are nowadays mostly developed by performing a projection of the navier stokes equations onto a low dimensional space basis this basis is usually obtained through proper orthogonal decomposition pod which remains one the most efficient techniques to compress precomputed data the main drawback of a posteriori pod based roms is however their lacks of reliability as their parameters are varied preventing their direct uses within optimization algorithms the goal of the present article is to obtain an a priori low dimensional space time separated representation of the fluid fields without precomputed data the approach is based on the use of space time proper generalized decomposition pgd definitions which are successfully applied in several fields but whose uses in fluid dynamics remain scarce their applications to the navier stokes equations are indeed not straightforward due to the pressure velocity coupling the divergence free constraint and the non linear convective term the roms are built here from a space time weak formulation of the chorin temam prediction correction scheme more particularly a priori space time separated representations are obtained by applying the galerkin based progressive pgd definition the minimax pgd definition is moreover experimented on a linear stokes simplified case the related algorithms are explicitly given and illustrated on a transient lid driven cavity flow it is shown that the a priori space time separated representations converge to the full model solution as the decomposition order increases the ability of the resulting rom to learn iteratively from its own error is highlighted the progressive pgd algorithm can be used to effectively enrich incomplete pod and pgd precomputed space bases such as those obtained with different parameter values this may allow to ensure the accuracy of a rom as the parameter is varied which is of crucial interest for optimization problems

a mathematical biography of danny c sorensen abstract on the occasion of his 65th birthday we briefly recount dan sorensen s profound contributions to optimization numerical linear algebra and model order reduction for dynamical systems

theoretical and experimental studies of the resonant frequencies of the equilateral triangular microstrip antenna a critical study of the resonant frequency of the equilateral triangular patch antenna is presented the contributions include 1 comparison of the previous experimental data of j s dahele and k f lee see ibid vol ap 35 no 1 p 100 1 1987 with moment method results 2 new measurements and moment method results for epsilon sub r 10 5 and comparison with gang s hypothesis 3 verification of the relationship among the various modes by measurements and by the moment method and 4 a curve fitting formula yielding the resonant frequency of the lowest mode which is within 1 of the value obtained from moment method analysis

an iterative svd krylov based method for model reduction of large scale dynamical systems abstract in this paper we propose a model reduction algorithm for approximation of large scale linear time invariant dynamical systems the method is a two sided projection combining features of the singular value decomposition svd based and the krylov based model reduction techniques while the svd side of the projection depends on the observability gramian the krylov side is obtained via iterative rational krylov steps the reduced model is asymptotically stable matches certain moments and solves a restricted h 2 minimization problem we present modifications to the proposed approach for employing low rank gramians in the reduction step and also for reducing discrete time systems several numerical examples from various disciplines verify the effectiveness of the proposed approach it performs significantly better than the q cover a yousouff r e skelton covariance equivalent realizations with applications to model reduction of large scale systems in c t leondes ed control and dynamic systems vol 22 academic press 1985 pp 273 348 a yousouff d a wagie r e skelton linear system approximation via covariance equivalent realizations journal of math anal and appl 196 1985 91 115 and the least squares s gugercin a c antoulas model reduction of large scale systems by least squares linear algebra appl 415 2 3 2006 290 321 methods that have a similar projection structure to the proposed method also in terms of both the h 2 and h error measures its performance is comparable to or sometimes better than that of balanced truncation moreover the method proves to be robust with respect to the perturbations due to usage of approximate gramians

projection methods for large lyapunov matrix equations abstract in the present paper we propose krylov subspace methods for solving large lyapunov matrix equations of the form ax xat bbt 0 where a and b are real n n and n s matrices respectively with s n such problems appear in many areas of control theory such as the computation of hankel singular values model reduction algorithms and others the proposed methods are based on the arnoldi process we show how to extract low rank approximate solutions to lyapunov matrix equations and give some theoretical results finally some numerical tests will be reported to illustrate the effectiveness of the proposed method

error and convergence in numerical implementations of the conjugate gradient method em problems the conjugate gradient method has previously been applied in electromagnetics in two ways to moment method matrices and directly to continuous operator equations numerical implementations of these two methods are shown here to be equivalent it is concluded that the advantage of the conjugate gradient method is therefore its potential computational efficiency as a solution procedure not its ability to achieve a more exact solution than the moment method

time domain model order reduction of general orthogonal polynomials for linear input output systems for a class of large linear input output systems we present a new model order reduction algorithm based on general orthogonal polynomials in the time domain the main idea of the algorithm is first to expand the unknown state variables in the space spanned by orthogonal polynomials then the coefficient terms of polynomial expansion are calculated by a recurrence formula the basic procedure is to use the coefficient terms to generate a projection matrix many classic methods with orthogonal polynomials are special cases of the general approach the proposed approach has a good computational efficiency and preserves the stability and passivity under certain condition numerical experiments are reported to verify the theoretical analysis

a general minimal residual krylov subspace method for large scale model reduction this paper considers approximating a given nth order stable transfer matrix g s by an rth order stable transfer matrix g sub r s in which n spl gt r and where n is large the arnoldi process is used to generate a basis to a part of the controllability subspace associated with the realization of g s and a residual error is defined for any approximation in this subspace we establish that minimizing the l sub spl infin norm of this residual error over the set of stable approximations leads to a 2 block distance problem finally the solution of this distance problem is used to construct reduced order approximate models the behavior of the algorithms is illustrated with a simple example

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

singular perturbational model reduction of balanced systems the balanced representations of linear systems due to moore are shown to be natural and convenient mediums for singular perturbational model reduction

krylov subspace methods for the sylvester equation abstract we describe galerkin and minimal residual algorithms for the solution of sylvester s equation ax xb c the algorithms use krylov subspaces forwhich orthogonal bases are generated by the arnoldi process for certain choices of krylov subspaces the computation of the solution splits into the solution of many independent subproblems this makes the algorithms suitable for implementation on parallel computers

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 the smallest value of the maximal modulus of an algebraic integer the house of an algebraic integer of degree d is the largest modulus of its conjugates for d 1 of degree d say m d as a consequence we improve matveev s theorem on the lower bound of m d we show that in this range the conjecture of schinzelzassenhaus is satisfied the minimal polynomial of any algebraic integer whose house is equal to m d is a factor of a bi tri or quadrinomial the computations use a family of explicit auxiliary functions these functions depend on generalizations of the integer transfinite diameter of some compact sets in c they give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer whose house is small

a schur method for balanced truncation model reduction it is shown that a not necessarily balanced state space realization of the moore reduced model can be computed directly without balancing via projections defined in terms of arbitrary bases for the left and right eigenspaces associated with the large eigenvalues of the product pq of the reachability and controllability grammians two specific methods for computing these bases are proposed one based on the ordered schur decomposition of pq and the other based on the cholesky factors of p and q the algorithms perform reliably even for nonminimal models

a fast adaptive model reduction method based on takenaka malmquist systems abstract an adaptive model reduction method is proposed for linear time invariant systems based on the continuous time rational orthogonal basis takenaka malmquist basis the method is to find an adaptive approximation in the energy sense by selecting optimal points for the rational orthogonal basis the stability of the reduced models holds and the steady state values of step responses are kept to be equal furthermore the method automatically ensures the reduced system to be in the hardy space h 2 the existence of the best approximation in the hardy space h 2 by n blaschke forms is proved in the proposed approach the effectivity of this method is illustrated through three well known examples

on normal and skew hermitian splitting iteration methods for large sparse continuous sylvester equations this paper is concerned with some generalizations of the hermitian and skew hermitian splitting hss iteration for solving continuous sylvester equations the main contents we will introduce are the normal and skew hermitian splitting nss iteration methods for the continuous sylvester equations it is shown that the new schemes can outperform the standard hss method in some situations theoretical analysis shows that the nss methods converge unconditionally to the exact solution of the continuous sylvester equations moreover we derive the upper bound of the contraction factor of the nss iterations numerical experiments further show the effectiveness of our new methods

resonance frequency of a rectangular microstrip patch the microstrip patch resonant frequency problem is formulated in terms of an integral equation using vector fourier transforms using galerkins s method in solving the integral equation the resonant frequency of the microstrip patch is studied with both chebyshev polynomials and sinusoidal functions as basis functions in the case of the chebyshev polynomials the edge singularity is included but it is not important for convergence furthermore the resonant frequency of the microstrip patch is ascertained with a perturbation calculation the results for galerkin s method and experiments are in good agreement the perturbation calculation agrees asymptotically with galerkin s method with the aim of developing a computer aided design formula the solutions obtained by galerkin s method are interpolated with a three variable polynomial

a characterization of all solutions to the four block general distance problem all solutions to the four block general distance problem which arises in h infty optimal control are characterized the procedure is to embed the original problem in an all pass matrix which is constructed it is then shown that part of this all pass matrix acts as a generator of all solutions special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all pass transfer functions as an application necessary and sufficient conditions are found for the existence of an h infty optimal controller following that a descriptor representation of all solutions is derived

balanced truncation model reduction for semidiscretized stokes equation we discuss model reduction of linear continuous time descriptor systems that arise in the control of semidiscretized stokes equations balanced truncation model reduction methods for descriptor systems are presented these methods are closely related to the proper and improper controllability and observability gramians and hankel singular values of descriptor systems the gramians can be computed by solving projected generalized lyapunov equations important properties of the balanced truncation approach are that the asymptotic stability is preserved in the reduced order system and there is an a priori bound on the approximation error we demonstrate the application of balanced truncation model reduction to the semidiscretized stokes equation

eigenvalue decay bounds for solutions of lyapunov equations the symmetric case abstract we present two new bounds for the eigenvalues of the solutions to a class of continuous and discrete time lyapunov equations these bounds hold for lyapunov equations with symmetric coefficient matrices and right hand side matrices of low rank they reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix