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

parallelization of the rational arnoldi algorithm rational krylov methods are applicable to a wide range of scientific computing problems and the rational arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational krylov space typically the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration we explore the option of solving several linear systems simultaneously thus constructing the rational krylov basis in parallel if this is not done carefully the basis being orthogonalized may become badly conditioned leading to numerical instabilities in the orthogonalization process we introduce the new concept of continuation pairs which gives rise to a near optimal parallelization strategy that allows to control the growth of the condition number of this nonorthogonal basis as a consequence we obtain a significantly more accurate and reliable parallel rational arnoldi algorithm the computational benefits are illustrated using several numerical examples from different application areas

automatic smoothness detection of the resolvent krylov subspace method for the approximation of c_0 semigroups the resolvent krylov subspace method builds approximations to operator functions f a times a vector v for the semigroup and related operator functions this method is proved to possess the favorable property that the convergence is automatically faster when the vector v is smoother the user of the method does not need to know the presented theory and alterations of the method are not necessary in order to adapt to the possibly unknown smoothness of v the findings are illustrated by numerical experiments

a projection method for generalized eigenvalue problems using numerical integration in this paper we propose a method for finding certain eigenvalues of a generalized eigenvalue problem that lie in a given domain of the complex plane the proposed method projects the matrix pencil onto a subspace associated with the eigenvalues that are located in the domain via numerical integration

on the commutative factorization of n n matrix wiener hopf kernels with distinct eigenvalues in this article we present a method for factorizing n x n matrix wiener hopf kernels where n 2 and the factors commute we are motivated by a method posed by jones jones 1984a proc r soc a 393 185 192 to tackle a narrower class of matrix kernels however no matrix of jones form has yet been found to arise in physical wiener hopf models in contrast the technique proposed herein should find broad application to illustrate the approach we consider a 3 x 3 matrix kernel arising in a problem from elastostatics while this kernel is not of jones form we shall show how it can be factorized commutatively we discuss the essential difference between our method and that of jones and explain why our method is a generalization the majority of wiener hopf kernels that occur in canonical diffraction problems are however strictly non commutative for 2x2 matrices abrahams has shown that one can overcome this difficulty using pade approximants to rearrange a non commutative kernel into a partial commutative form an approximate factorization can then be derived by considering the dynamic analogue of antipov s model we show for the first time that abrahams pade approximant method can also be employed within a 3x3 commutative matrix form

convergence analysis of an extended krylov subspace method for the approximation of operator functions in exponential integrators we analyze the convergence of an extended krylov subspace method for the approximation of operator functions that appear in exponential integrators for operators the size of the polynomial part of the extended krylov subspace is restricted according to the smoothness of the initial data this restriction for the continuous operator has a significant influence on the approximation of matrix functions evaluated for matrices stemming from space discretizations of the continuous operator we prove convergence of the method for the continuous operator and in the discrete case this leads to a convergence independent of the norm of the discretized operator uniformly over all possible grids the analysis is illustrated by numerical experiments

chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems abstract the present paper deals with the problem of computing a few of the eigenvalues with largest or smallest real parts of a large sparse nonsymmetric matrix we present a general acceleration technique based on chebyshev polynomials and discuss its practical application to arnoldi s method and the subspace iteration method the resulting algorithms are compared with the classical ones in a few experiments which exhibit a sharp superiority of the arnoldi chebyshev approach

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

a lanczos model order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media in this paper we present a krylov subspace model order reduction technique for time and frequency domain electromagnetic wave fields in linear dispersive media starting point is a self consistent first order form of maxwell s equations and the constitutive relation this form is discretized on a standard staggered yee grid while the extension to infinity is modeled via a recently developed global complex scaling method by applying this scaling method the time or frequency domain electromagnetic wave field can be computed via a so called stability corrected wave function since this function cannot be computed directly due to the large order of the discretized maxwell system matrix krylov subspace reduced order models are constructed that approximate this wave function we show that the system matrix exhibits a particular physics based symmetry relation that allows us to efficiently construct the time and frequency domain reduced order models via a lanczos type reduction algorithm the frequency domain models allow for frequency sweeps meaning that a single model provides field approximations for all frequencies of interest and dominant field modes can easily be determined as well numerical experiments for two and three dimensional configurations illustrate the performance of the proposed reduction method

on investigating gmres convergence using unitary matrices for a given matrix a and right hand side b this paper investigates unitary matrices generating with some right hand sides c the same gmres residual norms as the pair a b we give characterizations of this class of unitary matrices and point out the relationship with krylov subspaces and krylov residual subspaces for the pair a b we investigate the eigenvalues of these unitary matrices in relation to the convergence behavior of gmres for the pair a b and describe the indispensable role of the eigenvector information we conclude with a formula for the gmres residual norms generated by a normal matrix b in terms of its eigenvalues and components of the right hand side c in the eigenvector basis

an implicit filter for rational krylov using core transformations abstract the rational krylov method is a powerful tool for computing a selected subset of eigenvalues in large scale eigenvalue problems in this paper we study a method to implicitly apply a filter in a rational krylov iteration by directly acting on a qr factorized representation of the hessenberg pair from the rational krylov method this filter is used to restart the iteration which is generally required to limit the orthogonalization and storage costs the contribution in this paper is threefold we reformulate existing procedures in terms of operations on core transformations this has the advantage of improved convergence monitoring secondly we demonstrate that the extended qz method is a special case of this more general method finally numerical experiments show the validity and the increased accuracy of the new approach compared with existing methods

a fast algorithm for parabolic pde based inverse problems based on laplace transforms and flexible krylov solvers we consider the problem of estimating parameters in large scale weakly nonlinear inverse problems for which the underlying governing equations is a linear time dependent parabolic partial differential equation a major challenge in solving these inverse problems using newton type methods is the computational cost associated with solving the forward problem and with repeated construction of the jacobian which represents the sensitivity of the measurements to the unknown parameters forming the jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time dependent parabolic partial differential equations corresponding to multiple sources and receivers we propose an efficient method based on a laplace transform based exponential time integrator combined with a flexible krylov subspace approach to solve the resulting shifted systems of equations efficiently our proposed solver speeds up the computation of the forward and adjoint problems thus yielding significant speedup in total inversion time we consider an application from transient hydraulic tomography tht which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests the algorithms discussed are applied to a synthetic example taken from tht to demonstrate the resulting computational gains of this proposed method

lyapunov inverse iteration for identifying hopf bifurcations in models of incompressible flow the identification of instability in large scale dynamical systems caused by hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems a new method developed in k meerbergen and a spence siam j matrix anal appl 31 2010 pp 1982 1999 avoids this computation instead performing an inverse iteration for a certain set of real eigenvalues that requires the solution of a large scale lyapunov equation at each iteration in this study we refine the lyapunov inverse iteration method to make it more robust and efficient and we examine its performance on challenging test problems arising from fluid dynamics various implementation issues are discussed including the use of inexact inner iterations and the impact of the choice of iterative solution for the lyapunov equations and the effect of eigenvalue distribution on performance numerical experiments demonstrate the robustness of the algorithm

on the computation of a very large number of eigenvalues for selfadjoint elliptic operators by means of multigrid methods recent results in the study of quantum manifestations in classical chaos raise the problem of computing a very large number of eigenvalues of selfadjoint elliptic operators the standard numerical methods for large eigenvalue problems cover the range of applications where a few of the leading eigenvalues are needed they are not appropriate and generally fail to solve problems involving a number of eigenvalues exceeding a few hundreds further the accurate computation of a large number of eigenvalues leads to much larger problem dimension in comparison with the usual case dealing with only a few eigenvalues a new method is presented which combines multigrid techniques with the lanczos process the resulting scheme requires o mn arithmetic operations and o n storage requirement where n is the number of unknowns and m the number of needed eigenvalues the discretization of the considered differential operators is realized by means of p finite elements and is applicable on general geometries numerical experiments validate the proposed approach and demonstrate that it allows to tackle problems considered to be beyond the range of standard iterative methods at least on current workstations the ability to compute more than 9000 eigenvalues of an operator of dimension exceeding 8 million on a pc shows the potential of this method practical applications are found e g in the numerical simulation of quantum billiards

a refined shift and invert arnoldi algorithm for large unsymmetric generalized eigenproblems abstract the shift and invert arnoldi method has been popularly used for computing a number of eigenvalues close to a given shift and or the associated eigenvectors of a large unsymmetric matrix pair but there is no guarantee for the approximate eigenvectors ritz vectors obtained by this method to converge even though the subspace is good enough in order to correct this problem a refined shift and invert arnoldi method is proposed that uses certain refined ritz vectors to approximate the desired eigenvectors the refined ritz vectors can be computed cheaply and reliably by small sized singular value decompositions it is shown that the refined method converges a refined shift and invert arnoldi algorithm is developed and several numerical examples are reported comparisons are drawn on the refined algorithm and the shift and invert arnoldi algorithm indicating that the former is considerably more efficient than the latter

generalized block lanczos methods for large unsymmetric eigenproblems generalized block lanczos methods for large unsymmetric eigenproblems are presented which contain the block arnoldi method and the block arnoldi algorithms are developed the convergence of this class of methods is analyzed when the matrix a is diagonalizable upper bounds for the distances between normalized eigenvectors and a block krylov subspace are derived and a priori theoretical error bounds for ritz elements are established compared with generalized lanczos methods which contain arnoldi s method the convergence analysis shows that the block versions have two advantages first they may be efficient for computing clustered eigenvalues second they are able to solve multiple eigenproblems however a deep analysis exposes that the approximate eigenvectors or ritz vectors obtained by general orthogonal projection methods including generalized block methods may fail to converge theoretically for a general unsymmetric matrix a even if corresponding approximate eigenvalues or ritz values do since the convergence of ritz vectors needs more sufficient conditions which may be impossible to satisfy theoretically than that of ritz values does the issues of how to restart and to solve multiple eigenproblems are addressed and some numerical examples are reported to confirm the theoretical analysis

lanczos versus subspace iteration for solution of eigenvalue problems in this paper we consider solution of the eigen problem in structural analysis using a recent version of the lanczos method and the subspace method the two methods are applied to examples and we conclude that the lanczos method has advantages which are too good to overlook

paraexp a parallel integrator for linear initial value problems a novel parallel algorithm for the integration of linear initial value problems is proposed this algorithm is based on the simple observation that homogeneous problems can typically be integrated much faster than inhomogeneous problems an overlapping time domain decomposition is utilized to obtain decoupled inhomogeneous and homogeneous subproblems and a near optimal krylov method is used for the fast exponential integration of the homogeneous subproblems we present an error analysis and discuss the parallel scaling of our algorithm the efficiency of this approach is demonstrated with numerical examples

the rkfit algorithm for nonlinear rational approximation the rkfit algorithm outlined in m berljafa and s ga ttel generalized rational krylov decompositions with an application to rational approximation siam j matrix anal appl 2015 is a krylov based approach for solving nonlinear rational least squares problems this paper puts rkfit into a general framework allowing for its extension to nondiagonal rational approximants and a family of approximants sharing a common denominator furthermore we derive a strategy for the degree reduction of the approximants as well as methods for their conversion to partial fraction form for the efficient evaluation and root finding we also discuss commons and differences of rkfit and the popular vector fitting algorithm a matlab implementation of rkfit is provided and numerical experiments including the fitting of a mimo dynamical system and an optimization problem related to exponential integration demonstrate its applicability

error estimates and evaluation of matrix functions via the faber transform the need to evaluate expressions of the form f a or f a b where f is a nonlinear function a is a large sparse n times n matrix and b is an n vector arises in many applications this paper describes how the faber transform applied to the field of values of a can be used to determine improved error bounds for popular polynomial approximation methods based on the arnoldi process applications of the faber transform to rational approximation methods and in particular to the rational arnoldi process also are discussed

rational qr iteration without inversion in this manuscript a new method will be presented for performing a qr iteration with a i a i 1 qr without explicit inversion of the factor a i 1 a qr method driven by a rational function is attractive since convergence can occur at both sides of the matrix each step of this new iteration consists of two substeps in the explicit version first an rq factorization of the initial matrix a i rq will be computed followed by a qr factorization of the matrix a i q h the factorization of a i q h can be computed in an intelligent manner exploiting properties of the already known rq factorization of a i the similarity transformation yielding the qr step is defined by the unitary factor q in the qr factorization of the transformed matrix a i q h examples will be given illustrating how to efficiently compute the factorization for some specific classes of matrices the novelties of this approach with respect to these matrix classes will be discussed

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

a characterization for solutions of the monge kantorovich mass transport problem a measure theoretical approach is presented to study the solutions of the monge kantorovich optimal mass transport problems this approach together with kantorovich duality provide an effective tool to answer a long standing question about the support of optimal plans for the mass transport problem involving general cost functions we also establish a criterion for the uniqueness

restrictions on implicit filtering techniques for orthogonal projection methods abstract we consider the class of the orthogonal projection methods opm to solve iteratively large eigenvalue problems an opm is a method that projects a large eigenvalue problem on a smaller subspace in this subspace an approximation of the eigenvalue spectrum can be computed from a small eigenvalue problem using a direct method examples of opms are the arnoldi and the davidson method we show how an opm can be restarted implicitly and explicitly this restart can be used to remove a specific subset of vectors from the approximation subspace this is called explicit filtering an implicit restart can also be combined with an implicit filtering step i e the application of a polynomial or rational function on the subspace even if inaccurate arithmetic is assumed however the condition for the implicit application of a filter is that the rank of the residual matrix must be small

rational approximation preconditioners for sparse linear systems this paper presents a class of preconditioning techniques which exploit rational function approximations to the inverse of the original matrix the matrix is first shifted and then an incomplete lu factorization of the resulting matrix is computed the resulting factors are then used to compute a better preconditioner for the original matrix since the incomplete factorization is made on a shifted matrix a good lu factorization is obtained without allowing much fill in the result needs to be extrapolated to the nonshifted matrix thus the main motivation for this process is to save memory the method is useful for matrices whose incomplete lu factorizations are poor e g unstable

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