construction of square root factor for solution of the lyapunov matrix equation abstract the lyapunov matrix equation is considered in this paper where the solution is a nonnegative definite matrix i e a matrix admitting decomposition in square root factors an algorithm for findings the square root factor without preliminary finding the solution itself is given

analysis of the solution of the sylvester equation using low rank adi with exact shifts abstract the solution to a general sylvester equation a x x b g f with a low rank right hand side is analyzed quantitatively through the low rank alternating directional implicit method lr adi with exact shifts new bounds and perturbation bounds on x are obtained a distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of a and b and the right hand side factors g and f numerical examples suggest that because of this inclusion of details new perturbation bounds are much sharper than the existing ones

notes on bipolar fuzzy graphs in this note we show by examples that definitions 3 1 5 6 and 5 7 example 5 8 propositions 5 9 and 5 10 and partial proof of propositions 5 11 and 5 12 in a previous paper by akram muhammad akram bipolar fuzzy graphs inform sci 181 2011 5548 5564 contain some flaws and then provide the correct versions finally we introduce a generalized bipolar fuzzy graph

bounds on the trace of a solution to the lyapunov equation with a general stable matrix abstract some new estimates for the eigenvalue decay rate of the lyapunov equation ax xa t b with a low rank right hand side b are derived the new bounds show that the right hand side b can greatly influence the eigenvalue decay rate of the solution this suggests a new choice of the adi parameters for the iterative solution the advantage of these new parameters is illustrated on second order damped systems with a low rank damping matrix

fuzzy chromatic number of union of fuzzy graphs an algorithm properties and its application abstract we focus on fuzzy graphs with crisp vertex and fuzzy edge sets a concept of the fuzzy chromatic number of these graphs based on fuzzy independent vertex set is used in this paper a modified algorithm called a fuzzy chromatic algorithm is developed to find the fuzzy chromatic number of union of fuzzy graphs running time and complexity of the algorithm are also analyzed furthermore we investigate some properties of the fuzzy chromatic number of union of fuzzy graphs finally an application of the fuzzy chromatic number to determine the number of phases of an integrated traffic light system is proposed we get different phases with different degrees of safety

fast frequency sweep of fem models via the balanced truncation proper orthogonal decomposition a fast frequency sweep method for wideband antennas and infinite arrays based on a singular value decomposition svd krylov model reduction method for frequency domain tangential vector finite elements tvfems is presented reduced models are constructed using balanced congruence transformations constructed from the dominant invariant subspace of the system s hankel matrix traditionally forming such matrix requires the intensive computation of gramians the proposed method only forms their low rank cholesky factors via a novel adaptive proper orthogonal decomposition pod sampling strategy leading to significant savings unlike some other model reduction methods balanced truncation pod bt pod is directly applicable to lossy and dispersive electromagnetic models numerical studies on large scale wideband antennas and infinite arrays show that the method is stable error controllable and without memory overheads capable of up to two orders of magnitude speed ups

trail to a lyapunov equation solver the lyapunov matrix equation ax xa b is n stable when all eigenvalues of the real nxn matrix a have positive real part when the real nxn matrix b is spd the solution x is spd it is of low rank when b cc where c is nxr with email protected n an efficient algorithm has been found for solving the low rank equation this algorithm is a result of over fifty years of research starting with seemingly unrelated development of alternating direction implicit adi iterative solution of elliptical systems the low rank algorithm may be applied to a full rank equation if one can approximate the right hand side by a sum of low rank matrices this may be attempted with the lanczos algorithm

the adi method for bounded real and positive real lur e equations we propose an algorithm for the numerical solution of the lur e equations in the bounded real and positive real lemma for stable systems the algorithm provides approximate solutions in low rank factored form we prove that the sequence of approximate solutions is monotonically increasing with respect to definiteness if the shift parameters are chosen appropriately the sequence is proven to be convergent to the minimal solution of the lur e equations the algorithm is based on the ideas of the recently developed adi iteration for algebraic riccati equations massoudi et al siam j matrix anal appl 2016 in particular the matrices obtained in our iteration express the optimal cost in a certain projected optimal control problem

discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

bounds in algebraic riccati and lyapunov equations a survey and some new results this paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic riccati and lyapunov equations that have been reported during the last two decades also presented are bounds for the unified riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete riccati equations can be unified by them

a multilevel technique for the approximate solution of operator lyapunov and algebraic riccati equations we consider multigrid or more appropriately multilevel techniques for the numerical solution of operator lyapunov and algebraic riccati equations the riccati equation which is quadratic plays an essential role in the solution of linear quadratic optimal control problems the linear lyapunov equation is important in the stability theory for linear systems and its solution is the primary step in the newton kleinman nk algorithm for the solution of algebraic riccati equations both equations are operator equations when the underlying linear system is infinite dimensional in this case finite dimensional discretization is required however as the level of discretization increases the convergence rate of the standard iterative techniques for solving high order matrix lyapunov and riccati equations decreases to deal with this multileveling is introduced into the iterative nk method for solving the algebraic riccati equation and smith s method for solving matrix lyapunov equations theoretical result

solution of lyapunov equations by alternating direction implicit iteration abstract in this report a new procedure is presented for solving the lyapunov matrix equation first the system is reduced to tridiagonal form with gaussian similarity transformations then the resulting system is solved with alternating direction implicit adi iteration a matrix commutation property essential for model problem convergence of adi iteration applied to elliptic difference equations is not needed for this application all stable lyapunov matrix equations are model adi problems

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

aspects of nonnormality for iterative methods abstract recently new optimal krylov subspace methods have been discovered for normal matrices in light of this novel ways to quantify nonnormality are considered in connection with various families of matrices we use as a criterion how for a given matrix these iterative methods introduced can be employed via e g inexpensive matrix factorizations the unitary orbit of the set of binormal matrices provides a natural extension of normal matrices its elements yield polynomially normal matrices of moderate degree in this context several matrix nearness problems arise

computing real low rank solutions of sylvester equations by the factored adi method abstract we investigate the factored alternating directions implicit adi iteration for large and sparse sylvester equations a novel low rank expression for the associated sylvester residual is established which enables cheap computations of the residual norm along the iteration and which yields a reformulated factored adi iteration the application to generalized sylvester equations is considered as well we also discuss the efficient handling of complex shift parameters and reveal interconnections between the adi iterates w r t those complex shifts this yields a further modification of the factored adi iteration which employs only an absolutely necessary amount of complex arithmetic operations and storage and which produces low rank solution factors consisting of entirely real data certain linear matrix equations such as e g cross gramian sylvester and stein equations are in fact special cases of generalized sylvester equations and we show how specially tailored low rank adi iterations can be deduced from the generalized factored adi iteration

lyapunov equations energy functionals and model order reduction of bilinear and stochastic systems we discuss the relation of a certain type of generalized lyapunov equations to gramians of stochastic and bilinear systems together with the corresponding energy functionals while gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems the relation of gramians and energy functionals for bilinear systems is less obvious we discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic gramians satisfying generalized lyapunov equations in any of the considered cases the definition of algebraic gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems we illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems

lur e equations and even matrix pencils abstract in this work we consider the so called lur e matrix equations that arise e g in model reduction and linear quadratic infinite time horizon optimal control we characterize the set of solutions in terms of deflating subspaces of even matrix pencils in particular it is shown that there exist solutions which are extremal in terms of definiteness it is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils

fuzzy intersection graphs abstract this paper extends the use of graphs to the modelling of systems under unceertainty when random assumptions do not adequately hold the paper includes the theoretical and computational aspects which are derived from the matrix representation of fuzzy intersection graphs their ability for applications is also discussed

pointwise characterizations of besov and triebel lizorkin spaces and quasiconformal mappings abstract in this paper the authors characterize in terms of pointwise inequalities the classical besov spaces b p q s and triebel lizorkin spaces f p q s for all s 0 1 and p q n n s both in r n and in the metric measure spaces enjoying the doubling and reverse doubling properties applying this characterization the authors prove that quasiconformal mappings preserve f n s q s on r n for all s 0 1 and q n n s a metric measure space version of the above morphism property is also established

bcr method for solving generalized coupled sylvester equations over centrosymmetric or anti centrosymmetric matrix abstract this paper introduces another version of biconjugate residual method bcr for solving the generalized coupled sylvester matrix equations over centrosymmetric or anti centrosymmetric matrix we prove this version of bcr algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round off errors furthermore a method is provided for choosing the initial matrices to obtain the least norm solution of the problem at last some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed

newton s method for solving cross coupled sign indefinite algebraic riccati equations for weakly coupled large scale systems in this paper a new algorithm for solving cross coupled sign indefinite algebraic riccati equations csares for weakly coupled large scale systems is proposed it is shown that since the proposed algorithm is based on the newton s method the quadratic convergence is attained moreover the local uniqueness of the convergence solutions for the csares is investigated finally in order to overcome the computation of large and sparse matrix related to the newton s method the fixed point algorithm and the alternating direction implicit adi method are combined

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

alternating direction implicit iteration for systems with complex spectra the alternating direction implicit adi iteration model problem has hitherto been the discretized dirichlet problem with an spd matrix splitting for which optimum iteration parameters are obtained as the solution due to w b jordan of a rational chebyshev minimax problem over the reals recently disclosed application to a class of sylvester equations bf ax bf xb bf c with bf a and bf b positive real matrices required generalization to complex spectra in the positive real half plane theory is developed here for applying jordan s parameters to complex domains which remain close to the real line and for predicting associated error reduction numerical studies are presented in support of this theory this analysis provided a basis for analysis of more general complex spectra which is reported in another paper

a generalized adi iterative method the adi iterative method for the solution of sylvester s equationax xb c proceeds by strictly alternating between the solution of the two equations begin gathered left a delta k 1 i right x 2k 1 x 2k left b delta k 1 i right c hfill x 2k 2 left b tau k 1 i right left a tau k 1 i right x 2k 1 c hfill end gathered fork 0 1 2 herex o is a given initial approximate solution and the k and k are real or complex parameters chosen so that the computed approximate solutionsx k converge rapidly to the solutionx of the sylvester equation ask increases this paper discusses the possibility of solving one of the equations in the adi iterative method more often than the other one i e relaxing the strict alternation requirement in order to achieve a higher rate of convergence our analysis based on potential theory shows that this generalization of the adi iterative method can give faster convergence than when strict alternation is required

a shift splitting hierarchical identification method for solving lyapunov matrix equations in the present paper we propose a hierarchical identification method sshi for solving lyapunov matrix equations which is based on the symmetry and skew symmetry splitting of the coefficient matrix we prove that the iterative algorithm consistently converges to the true solution for any initial values with some conditions and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factors appropriately furthermore we show that the method adopted can be easily extended to study iterative solutions of other matrix equations such as sylvester matrix equations finally we test the algorithms and show their effectiveness using numerical examples

mst mems model order reduction requirements and benchmarks the needs for model reduction in microsystem technology mst are described from an engineering perspective two representative mst model reduction benchmarks are presented in order to facilitate further development in this area the first benchmark application is from the area of electro thermal simulation the second one considers an electrostatically actuated beam as found in radio frequency applications model reduction is contrasted with compact modeling which currently enjoys widespread use among engineers and important challenges to be addressed are listed

adaptive rational krylov subspaces for large scale dynamical systems the rational krylov space is recognized as a powerful tool within model order reduction techniques for linear dynamical systems however its success has been hindered by the lack of a parameter free procedure which would effectively generate the sequence of shifts used to build the space in this paper we propose an adaptive computation of these shifts the whole procedure only requires us to inject some initial rough estimate of the spectral region of the matrix while further information is automatically generated during the process the approach is a full generalization to the nonsymmetric case of the idea first proposed in druskin et al 2010 18 and it is used for two important problems in control the approximation of the transfer function and the numerical solution of large lyapunov equations the procedure can be naturally extended to other related problems such as the solution of the sylvester equation and parametric or higher order systems several numerical experiments are proposed to assess the quality of the rational projection space over its most natural competitors

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

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

optimal alternating direction implicit parameters for nonsymmetric systems of linear equations the determination of optimal parameters for the well known alternating direction implicit adi method leads to a minimization problem for rational functions of given degree l on compact sets e and f that contain the eigenvalues of the matrices involved if e and f are real intervals the solution of this problem was given by zolotarev more than 100 years ago and by wachspress in the 1960s however in the complex case e g if the linear system arises from the discretization of non self adjoint elliptic differential equations little is known about this problem in this paper two approaches to the zolotarev problem in the complex plane are considered first generalized leja points are considered as an example for rational functions which are asymptotically minimal in addition for rectangular eigenvalue domains e and f the exact minimal solutions for l 2 are constructed and it is shown by numerical examples that the results obtained using the optimal parameters of degree 2 are comparable to tho

low memory and low complexity iterative schemes for a nonsymmetric algebraic riccati equation arising from transport theory abstract we reconsider newton s method and two fixed point methods for finding the minimal positive solution of a nonsymmetric algebraic riccati equation arising from transport theory we rewrite the subproblem of the newton and fixed point iterative schemes into an equivalent form with some special structure by the use of the particular structure of the subproblem we then present low memory and low complexity versions of these iterative methods with a factored alternating direction implicit iteration some properties of eigenvalues for iterative coefficient matrices in solving the subproblem are derived and the convergence of the proposed methods is established numerical experiments show that the new iterative schemes are highly efficient to obtain the minimal positive solution the proposed low memory and low complexity newton s method is particularly efficient for solving large scale riccati equation arising from transport theory

computationally enhanced projection methods for symmetric sylvester and lyapunov matrix equations abstract in the numerical treatment of large scale sylvester and lyapunov equations projection methods require solving a reduced problem to check convergence as the approximation space expands this solution takes an increasing portion of the overall computational effort when data are symmetric we show that the frobenius norm of the residual matrix can be computed at significantly lower cost than with available methods without explicitly solving the reduced problem for certain classes of problems the new residual norm expression combined with a memory reducing device make classical krylov strategies competitive with respect to more recent projection methods numerical experiments illustrate the effectiveness of the new implementation for standard and extended krylov subspace methods

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

lowest rank solutions of continuous and discrete lyapunov equations over symmetric cone abstract the low rank solutions of continuous and discrete lyapunov equations are of great importance but generally difficult to compute in control system analysis and design fortunately mesbahi and papavassilopoulos 1997 30 showed that with the semidefinite cone constraint the lowest rank solutions of the discrete lyapunov inequality can be efficiently solved by a linear semidefinite programming in this paper we further show that the lowest rank solutions of both the continuous and discrete lyapunov equations over symmetric cone are unique and can be exactly solved by their convex relaxations the symmetric cone linear programming problems therefore they are polynomial time solvable since the underlying symmetric cone is a more general algebraic setting which contains the semidefinite cone as a special case our results also answer an open question proposed by recht fazel and parrilo 2010 36

feje r walsh points for rational functions and their use in the adi iterative method abstract a new approach to the rational zolotarev problem in the complex plane is presented this minimization problem for rational functions arises from the determination of optimal parameters for the adi iterative method applied to nonsymmetric systems of linear equations based on a generalization due to walsh of the fejer points which are well known in connection to polynomial interpolation in the complex plane we are able to construct rational functions which are almost optimal for the rational zolotarev problem in an asymptotic sense for the construction of these fejer walsh points we make use of the schwarz christoffel mapping for doubly connected polygonal regions due to akhiezer and its recent implementation by dappen a numerical example illustrates the usefulness of the fejer walsh points as adi parameters

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

gramian based model reduction for data sparse systems model reduction is a common task within the simulation control and optimization of complex dynamical systems for instance in control and optimization problems for instationary partial differential equations pdes the associated large scale linear systems resulting from a semi discretization have to be solved repeatedly model reduction is a more and more accepted technique to attack these problems in reasonable time here we investigate model reduction for parabolic pdes based on balanced truncation bt the system theoretic properties of bt and the related method of singular perturbation approximation spa provide desirable features for the reduced order system the major computational task in bt is the solution of large scale lyapunov or stein equations depending on whether the semidiscretized or fully discretized system is considered therefore the method is often considered to be of limited use for really large scale applications i e problems that require gg1 gbyte of storage so that memory access becomes an issue here we develop effective implementations of bt and spa by exploiting the structure of the underlying pde problem this is achieved by a data sparse approximation of the semi discretized differential operator using the hierarchical matrix format furthermore we review how to integrate the corresponding formatted arithmetic in the sign function method for computing approximate solution factors of the lyapunov equations and suggest an iterative scheme for solving stein equations employing formatted arithmetic these approaches are well suited for a class of practically relevant problems and allow the application of bt and spa to systems coming from two dimensional 2d and three dimensional 3d finite element method fem and boundary element method bem discretizations the resulting methods are most beneficial in the context of bem discretizations for 3d problems where large scale dense matrices define the semi discretized control system we also analyze the additional approximation error in the reduced order model introduced by the hierarchical matrix approximation the efficiency of the methods is demonstrated for linear scalar convection diffusion reaction equations the numerical results confirm that the model reduction bt error and the hierarchical matrix approximation error can be balanced by choosing the tolerances and parameters involved appropriately

structure preserving tangential interpolation for model reduction of port hamiltonian systems port hamiltonian systems result from port based network modeling of physical systems and are an important example of passive state space systems in this paper we develop a framework for model reduction of large scale multi input multi output port hamiltonian systems via tangential rational interpolation the resulting reduced model is a rational tangential interpolant that retains the port hamiltonian structure hence it remains passive we introduce an h 2 inspired algorithm for effective choice of interpolation points and tangent directions and present several numerical examples illustrating its effectiveness

on the convergence of inexact newton methods for discrete time algebraic riccati equations abstract in this paper we present a convergence analysis of the inexact newton method for solving discrete time algebraic riccati equations dares for large and sparse systems the inexact newton method requires at each iteration the solution of a symmetric stein matrix equation these linear matrix equations are solved approximatively by the alternating directions implicit adi or smith s methods we give some new matrix identities that will allow us to derive new theoretical convergence results for the obtained inexact newton sequences we show that under some necessary conditions the approximate solutions satisfy some desired properties such as the d stability the theoretical results developed in this paper are an extension to the discrete case of the analysis performed by feitzinger et al 2009 8 for the continuous time algebraic riccati equations in the last section we give some numerical experiments

characterizations of fuzzy interval graphs abstract we use the min operator to define the fuzzy intersection graph of a family of fuzzy sets and explore relationships between properties of this fuzzy graph and its family of cut level graphs we show that each fuzzy graph without loops is the intersection graph of some family of fuzzy sets we show that the characterization of interval graphs by gilmore and hoffman naturally extends to fuzzy interval graphs while that of fulkerson and gross does not

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

on the benefits of the ldlt factorization for large scale differential matrix equation solvers abstract we propose efficient algorithms for solving large scale matrix differential equations in particular we deal with the differential riccati equations dre and state the applicability to the differential lyapunov equations dle we focus on methods based on standard versions of ordinary differential equations in the matrix setting the application of these methods yields algebraic lyapunov equations ales with a certain structure to be solved in every step the alternating direction implicit adi algorithm and krylov subspace based methods allow to exploit this special structure however a direct application of classic low rank formulations requires the use of complex arithmetic using an l d l t type decomposition of both the right hand side and the solution of the equation we avoid this problem thus the proposed methods are a more practical alternative for large scale problems arising in applications also they make the application of higher order methods feasible the numerical results show the better performance of the proposed methods compared to earlier formulations

near optimal frequency weighted interpolatory model reduction abstract this paper develops an interpolatory framework for weighted h 2 model reduction of mimo dynamical systems a new representation of the weighted h 2 inner products in mimo settings is introduced and used to derive associated first order necessary conditions satisfied by optimal weighted h 2 reduced order models equivalence of these new interpolatory conditions with earlier riccati based conditions given by halevi is also shown an examination of realizations for equivalent weighted h 2 systems leads then to an algorithm that remains tractable for large state space dimension several numerical examples illustrate the effectiveness of this approach and its competitiveness with frequency weighted balanced truncation and an earlier interpolatory approach the weighted iterative rational krylov algorithm

the sylvester equation and approximate balanced reduction abstract the purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems the resulting approach is completely automatic once an error tolerance is specified and also yields an error bound this is to be contrasted with existing projection methods namely pade via lanczos pvl and rational krylov which do not satisfy these properties our approach is based on the computation and approximation of the cross gramian of the system the cross gramian is the solution of a sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights our method produces a low rank approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the original system it is well suited to large scale problems because there are no matrix factorizations of the large sparse system matrix only matrix vector products are required

the accelerated gradient based iterative algorithm for solving a class of generalized sylvester transpose matrix equation in this paper we present an accelerated gradient based algorithm by minimizing certain criterion quadratic function for solving the generalized sylvester transpose matrix equation a x b c x t d f the idea is from ding and chen 2005 niu et al 2011 wang et al 2012 in which some efficient algorithms were developed for solving the sylvester matrix equation and the lyapunov matrix equation on the basis of the information generated in the previous half step we further introduce a relaxation factor to obtain the solution of the generalized sylvester transpose matrix equation we show that the iterative solution converges to the exact solution for any initial value provided that some appropriate assumptions finally some numerical examples are given to illustrate that the introduced iterative algorithm is efficient

finite rank adi iteration for operator lyapunov equations we give an algorithmic approach to the approximative solution of operator lyapunov equations for controllability motivated by the successfully applied alternating direction implicit adi iteration for matrix lyapunov equations we consider this method for the determination of gramian operators of infinite dimensional control systems in the case where the input space is finite dimensional this method provides approximative solutions of finite rank under the assumption of infinite time admissibility and boundedness of the semigroup we analyze convergence in several operator norms we show that under a mild assumption on the shift parameters convergence to the gramian is obtained particular emphasis is placed on systems governed by a heat equation with boundary control we present that adi iteration for the heat equation consists of solving a sequence of helmholtz equations two numerical examples are presented the first showing the benefit of adaptive finite elements and the second illustrating con

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

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

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function

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

on positive definite and skew hermitian splitting iteration methods for continuous sylvester equation ax xb c in this paper we present a positive definite and skew hermitian splitting pss iteration method for continuous sylvester equations ax xb c with positive definite semi definite matrices the theoretical analysis shows that the pss iteration method will converge unconditionally and the optimal parameter of the new method is presented moreover to reduce the computing cost an inexact variant of the pss iteration method ipss and the analysis of its convergence property in detail have been established numerical results show that this new method and its inexact invariant are efficient and robust solvers for this class of continuous sylvester equations

adi preconditioned krylov methods for large lyapunov matrix equations abstract in the present paper we propose preconditioned krylov methods for solving large lyapunov matrix equations ax xa t bb t 0 such problems appear in control theory model reduction circuit simulation and others using the alternating direction implicit adi iteration method we transform the original lyapunov equation to an equivalent symmetric stein equation depending on some adi parameters we then define the smith and the low rank adi preconditioners to solve the obtained stein matrix equation we apply the global arnoldi method and get low rank approximate solutions we give some theoretical results and report numerical tests to show the effectiveness of the proposed approaches

the modified conjugate gradient methods for solving a class of generalized coupled sylvester transpose matrix equations in this paper we consider the iteration solutions of generalized coupled sylvester transpose matrix equations a1xb1 c1ytd1 f1 a2yb2 c2xtd2 f2 when the coupled matrix equations are consistent we propose a modified conjugate gradient method to solve the equations and prove that a solution x y can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore we show that the minimum norm solution can be got by choosing a special kind of initial matrices when the coupled matrix equations are inconsistent we present another modified conjugate gradient method to find the least squares solution with the minimum norm finally some numerical examples are given to show the behavior of the considered algorithms

beltrami operators in the plane we determine optimal lp properties for the solutions of the general nonlinear elliptic system in the plane of the form fz h z fz h l c where h is a measurable function satisfying h z w1 h z w2 k w1 w2 and k is a constant k 1 we will also establish the precise invertibility and spectral properties in lp c for the operators i t i t and t where t is the beurling transform these operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations in two dimensions in particular we prove invertibility in lp c whenever 1 p 1 1 we also prove related results with applications to the regularity of weakly quasiconformal mappings

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

positive real and bounded real balancing for model reduction of descriptor systems we present an extension of the positive real and bounded real balanced truncation model reduction methods to large scale descriptor systems these methods are based on balancing the solutions of the projected lur e matrix equations important properties of these methods are that respectively passivity and contractivity are preserved in the reduced order models and that there exist approximation error bounds we also discuss the numerical solution of the projected lur e equations numerical examples are given

error bounds in the gap metric for dissipative balanced approximations abstract we derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation we prove these results by working in the context of dissipative driving variable systems as in behavioral and state signal systems theory in such a framework no prior distinction is made between inputs and outputs dissipativity preserving balanced truncation of dissipative driving variable systems is addressed and a gap metric error bound is obtained bounded real and positive real input state output systems are manifestations of a dissipative driving variable system through particular decompositions of the signal space under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow

weighted sobolev spaces and embedding theorems in the present paper we study embedding operators for weighted sobolev spaces whose weights satisfy the well known muckenhoupt a p condition sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities the proposed method is based on the concept of generalized quasiconformal homeomorphisms homeomorphisms with bounded mean distortion the choice of the homeomorphism type depends on the choice of the corresponding weighted sobolev space such classes of homeomorphisms induce bounded composition operators for weighted sobolev spaces with the help of these homeomorphism classes the embedding problem for non smooth domains is reduced to the corresponding classical embedding problem for smooth domains examples of domains with anisotropic holder singularities demonstrate the sharpness of our machinery comparatively with known results

regularity of the inverse of a planar sobolev homeomorphism let be a domain suppose that f w1 1loc r2 is a homeomorphism such that df x vanishes almost everywhere in the zero set of j f we show that f 1 w1 1loc f r2 and that df 1 y vanishes almost everywhere in the zero set of sharp conditions to quarantee that f 1 w1 q f r2 for some 1 q 2 are also given

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

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

matrix sign function methods for solving projected generalized continuous time sylvester equations in this technical note we investigate the numerical solution of the projected generalized sylvester equations via a matrix sign function method such equations arise in stability analysis and control problems for descriptor systems including model reduction based on balanced truncation unlike the classical matrix sign function iteration we propose a modification of the matrix sign function method that converges quadratically for pencils of arbitrary index numerical experiments report the effectiveness of the modified method

sylvester equations and projection based model reduction in this paper we establish a connection between krylov subspace techniques for multipoint pade interpolation and the use of sylvester equations for constructing reduced order models we also briefly point out that this connection partly extends to adi type techniques and to the smith iteration for computing approximate solutions of lyapunov equations

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

quasiconformal mappings and sharp estimates for the distance to l in some function spaces abstract we provide several estimates which involve the distance to l in some function spaces the composition operator induced by a quasiconformal mapping and the logarithm of the jacobian of a quasiconformal mapping our results are sharp in the two dimensional case

the adi iteration for lyapunov equations implicitly performs h2 pseudo optimal model order reduction abstracttwo approaches for approximating the solution of large scale lyapunov equations are considered the alternating direction implicit adi iteration and projective methods by krylov subspaces we show that they are linked in the way that the adi iteration can always be identified by a petrov galerkin projection with rational block krylov subspaces therefore a unique krylov projected dynamical system can be associated with the adi iteration which is proven to be an h2 pseudo optimal approximation this includes the generalisation of previous results on h2 pseudo optimality to the multivariable case additionally a low rank formulation of the residual in the lyapunov equation is presented which is well suited for implementation and which yields a measure of the obliqueness that the adi iteration is associated with

controllability observability and the solution of ax xb c a closed form finite series representation of the unique solution x of the matrix equation ax xb c is developed using this representation the image kernel and rank of x are related to the controllable and unobservable subspaces of the a c and c b pairs respectively bounds on the rank of x are obtainedin terms of the dimensions of these subspaces in the case that c has unitary rank an exact calculation of rank x is made the generic rank of x with a b fixed and c generic is evaluated

frequency limited balanced truncation with low rank approximations in this article we investigate model order reduction of large scale systems using frequency limited balanced truncation which restricts the well known balanced truncation framework to prescribed frequency regions the main emphasis is put on the efficient numerical realization of this model reduction approach we discuss numerical methods to take care of the involved matrix valued functions the occurring large scale lyapunov equations are solved for low rank approximations for which we also establish results regarding the eigenvalues of their solutions these results and also numerical experiments will show that the eigenvalues of the lyapunov solutions in frequency limited balanced truncation often decay faster than those in standard balanced truncation moreover we show in further numerical examples that frequency limited balanced truncation generates reduced order models which are significantly more accurate in the considered frequency region

on the adi method for sylvester equations this paper is concerned with the numerical solution of large scale sylvester equations ax xb c lyapunov equations as a special case in particular included with c having very small rank for stable lyapunov equations penzl 2000 22 and li and white 2002 20 demonstrated that the so called cholesky factor adi method with decent shift parameters can be very effective in this paper we present a generalization of the cholesky factor adi method for sylvester equations an easily implementable extension of penz s shift strategy for the lyapunov equation is presented for the current case it is demonstrated that galerkin projection via adi subspaces often produces much more accurate solutions than adi solutions

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