nonlinear parametric inversion using interpolatory model reduction nonlinear parametric inverse problems appear in several prominent applications one such application is diffuse optical tomography dot in medical image reconstruction such inverse problems present huge computational challenges mostly due to the need for solving a sequence of large scale discretized parametrized partial differential equations in the forward model in this paper we show how interpolatory parametric model reduction can significantly reduce the cost of the inversion process in dot by drastically reducing the computational cost of solving the forward problems the key observation is that function evaluations for the underlying optimization problem may be viewed as transfer function evaluations along the imaginary axis a similar observation holds for jacobian evaluations as well this motivates the use of system theoretic model order reduction methods we discuss the construction and use of interpolatory parametric reduced models as surrogates for the full forward model within the dot s

trajectory piecewise quadratic reduced order model for subsurface flow with application to pde constrained optimization abstract a new reduced order model based on trajectory piecewise quadratic tpwq approximations and proper orthogonal decomposition pod is introduced and applied for subsurface oil water flow simulation the method extends existing techniques based on trajectory piecewise linear tpwl approximations by incorporating second derivative terms into the reduced order treatment both the linear and quadratic reduced order methods referred to as pod tpwl and pod tpwq entail the representation of new solutions as expansions around previously simulated high fidelity full order training solutions along with pod based projection into a low dimensional space pod tpwq entails significantly more offline preprocessing than pod tpwl as it requires generating and projecting several third order hessian type terms the pod tpwq method is implemented for two dimensional systems extensive numerical results demonstrate that it provides consistently better accuracy than pod tpwl with speedups of about two orders of magnitude relative to high fidelity simulations for the problems considered we demonstrate that pod tpwq can be used as an error estimator for pod tpwl which motivates the development of a trust region based optimization framework this procedure uses pod tpwl for fast function evaluations and a pod tpwq error estimator to determine when retraining which entails a high fidelity simulation is required optimization results for an oil water problem demonstrate the substantial speedups that can be achieved relative to optimizations based on high fidelity simulation

a krylov subspace method for large scale second order cone linear complementarity problem in this paper we first show that the second order cone linear complementarity problem soclcp can be solved by finding a positive zero s in mathbb r of a particular rational function h s and we then propose a krylov subspace method to reduce h s to h ell s as in the model reduction the zero s of h s can be accurately approximated by that of h ell s 0 which itself can be cast as a small eigenvalue problem the new method is made possible by a complete characterization of the curve of h s and it has several advantages over the bisection newton sf bn iteration recently proposed by l h zhang and w h yang math comp 83 2013 pp 1701 1720 and shown to be very efficient for small to medium size problems the method is tested and compared against the sf bn iteration and two other state of the art packages sf sdpt3 and sf sedumi our numerical results show that the method is very efficient for both small to medium dense problems and large scale ones

model order reduction for nonlinear dynamical systems based on trajectory piecewise linear approximations in this paper we analyze and expand a recently developed approach to model order reduction mor for nonlinear dynamical systems based on trajectory piecewise linear tpwl approximations error estimates are given for solutions computed with tpwl reduced order models and problems of preserving stability and passivity are examined since the tpwl method has limited a priori guarantees on global accuracy its effectiveness is demonstrated on a range of examples including a micromachined switch two nonlinear electronic circuits and shock propagation modeled by burgers equation

coisotropic rigidity and c 0 symplectic geometry we prove that symplectic homeomorphisms in the sense of the celebrated gromov eliashberg theorem preserve coisotropic submanifolds and their characteristic foliations this result generalizes the gromov eliashberg theorem and demonstrates that previous rigidity results on lagrangians by laudenbach and sikorav and on characteristics of hypersurfaces by opshtein are manifestations of a single rigidity phenomenon to prove the above we establish a c0 dynamical property of coisotropic submanifolds which generalizes a foundational theorem in c0 hamiltonian dynamics uniqueness of generators for continuous analogues of hamiltonian flows

accelerating optimization of parametric linear systems by model order reduction design optimization problems are often formulated as an optimization problem whose objective is a function of the output of a large scale parametric linear system obtained from the discretization of a pde to reduce the high computational cost of the objective and its gradient model order reduction techniques can be used this paper uses interpolatory reduced models as surrogate models in an optimization procedure we replace the standard first order condition by the relaxed first order condition which is more suitable when algebraic reduced models are used as surrogate models the relaxed first order condition imposes that the approximation quality of the surrogate model at the interpolation point can be measured and refined and that the surrogate model is equipped with an error bound on the entire parameter space we propose two optimization algorithms one uses the error bound to define a trust region and the other penalizes the objective with the error bound we prove convergence of both methods un

real analytic actions of lattices this paper introduces a new approach to the study of lattice actions on compact manifolds the group sl n z acts by real analytic di eomorphisms on the sphere sny1 via the action on the space of in nite rays based at 0 2 r we believe this action is minimal in the following sense conjecture i any smooth action of a nite index subgroup of sl n z n 2 on a compact r manifold factors through a nite group action if r ny 1

design of fir digital phase networks the problem of the minimax design of fir digital filters with prescribed phase characteristics and unit magnitude is a nonlinear optimization problem in this paper it is approximated by a linear programming problem and it is shown that the solution of this linear program is optimal to first order that is if 0 and e 0 are optimal deviations of magnitude and phase characteristics then the actual deviations obtained from the linear program solution satisfy delta leq delta 0 epsilon min 0 max 2 and epsilon leq epsilon 0 1 delta 0 1 delta 0 numerical examples are given including design results for full band m term chirp filters which like linear phase filters can be implemented with m 1 2 multiplications per point

broadband model order reduction of polynomial matrix equations using single point well conditioned asymptotic waveform evaluation derivations and theory summary to eect a model order reduction more process on a system which has a polynomial matrix equation dependence on the more parameter researchers generally take one of two approaches therst is to linearize the system by introducing extra degrees of freedom and then to solve the resulting expanded linear system with a method such as lanczos or arnoldi the second approach is to work directly with the polynomial system and use a technique such as asymptotic waveform evaluation awe of course each approach has advantages and disadvantages in this paper a new technique will be presented which has some desirable characteristics from both approaches and which is able to circumvent simultaneously some of their disadvantages it can be shown that both the arnoldi and the awe methods are special cases of this new technique finally numerical results will show the viability of the new method which will be called the well conditioned asymptotic waveform evaluation wcawe method copyright 2003 john wiley sons ltd

solution of the time domain inverse resistivity problem in the model reduction framework part i one dimensional problem with siso data many time domain problems in engineering applications can be described by means of parameter dependent time invariant dynamic systems we are interested in parameter estimation by fitting available transient measurements using the nonlinear least square method as the main application we consider the control source electromagnetic csem method of geophysical exploration governed by the diffusion maxwell system where the unknown parameters describe the spatial distribution of electrical resistivity we propose a novel model reduction approach for constructing an efficient approximation of the jacobian the reduction is based on subspace projection of the state variable and it allows us to split the time and space dependence of the derivative we examine several popular rational krylov subspaces and single out the h 2 optimal subspace that not only minimizes the approximation error but completely annuls its influence on the inversion result preliminary numerical experiments with a simplified one dim

balancing and model reduction for second order form linear systems model reduction of second order form linear systems is considered where a second order form reduced model is desired the focus is on reduction methods that employ or mimic moore s balance and truncate 1981 first we examine second order form model reduction by conversion to first order form and obtain a complete solution for this problem then new gramians and input output i o invariants for second order systems are motivated and defined based on these direct second order balancing methods are developed this leads naturally to direct second order form analogs for the well known first order form balance and truncate model reduction method explicit algorithms are given throughout the paper

rotation vectors and fixed points of area preserving surface diffeomorphisms we consider the homological rotation vectors for area preserving diffeomorphisms of compact surfaces which are homotopic to the identity there are two main results the first is that if 0 is in the interior of the convex hull of the rotation vectors for such a diffeomorphism then f has a fixed point of positive index the second result asserts that if f has a vanishing mean rotation vector then f has a fixed point of positive index there are several applications including a new proof of the arnold conjecture for area preserving diffeomorphisms of compact surfaces in this paper we investigate area preserving diffeomorphisms of compact surfaces we are concerned with rotation vectors defined in terms of homology which can be associated to the points of the surface there are two main results the first is that if f is an area preserving diffeomorphism of a compact surface which is homotopic to the identity and 0 is in the interior of the convex hull of the rotation vectors of f then f has a fixed point of positive index the second result concerns the mean rotation vector of f it asserts that if this vector vanishes then f has a fixed point of positive index a special case of this second result when the surface in question is a surface of genus zero was proved in f5 in 6 we give several applications including a new proof of the arnold conjecture for area preserving diffeomorphisms of compact surfaces

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

a fast frequency sweep approach using pade approximations for solving helmholtz finite element models abstract in various engineering applications the solution of the helmholtz equation is required over a broad frequency range the simplest approach which consists in solving the system of equations obtained from a finite element discretization for each frequency becomes computationally prohibitive for fine increments particularly when dealing with large systems like those encountered when addressing mid frequency problems alternative approaches involving reduced order models built via pade approximations are now well established for systems exhibiting polynomial frequency dependency of second order kind and for frequency independent excitations this paper treats systems of more complicated wavenumber dependency likely to be encountered when applying frequency dependent boundary conditions and or loadings the well conditioned asymptotic waveform evaluation wcawe is selected as the method of choice and the approximated taylor coefficients are computed by differentiating the continuous frequency dependent models obtained through a fitting process of the system entries the method is benchmarked first against the second order arnoldi soar algorithm on a simple second order system then it is applied to realistic large scale interior and exterior helmholtz problems exhibiting high order polynomial or rational frequency behavior in either case the proposed methodology is shown to reduce the computational time of the frequency sweep by an order of magnitude when compared to the direct approach

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

complex chebyshev approximation for fir digital filters an adaptive approach to fir finite impulse response filter design is applied to a fully general complex specification showing it to be capable of giving an optimum chebyshev solution it is thought to be one of the first practical approaches to achieve this in the case of a complex function design examples include a bandpass filter with nearly linear phase and all pass filters with general shaping phase comparisons are drawn with equivalent results from other algorithms

frequency weighted model reduction via optimal projection the problem of order reduction with frequency weighting is considered necessary conditions characterizing the reduced order model which minimizes a quadratic error criterion in the frequency domain with either input or output frequency weighting are given the problem is transformed to the time domain where the weighting matrix is expressed as a shaping filter that generates colored noise at the input when the weighting matrix is singular at infinity the reduced order model has a direct transmission term even if the original system is strictly proper this additional degree of freedom can be used to obtain better approximation over a prespecified frequency range

periodic orbits of hamiltonian homeomorphisms of surfaces john franks and michael handel fh2 have recently proved that any nontrivial hamiltonian diffeomorphism of a closed surface of genus at least one has periodic orbits of arbitrarily large period they proved a similar result for a nontrivial area preserving diffeomorphism of a sphere with at least three fixed points we extend these results to the case of the homeomorphisms when the genus is at least one we prove moreover that the periodic orbits may be chosen contractible if the set of contractible fixed points is contained in a disk when the surface is a sphere we extend the result to the case of a nontrivial homeomorphism with no wandering points the proofs make use of an equivariant foliated version of brouwer s plane translation theorem see b proposition 2 1 and some properties of the linking number of fixed points

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 optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

interpolatory projection methods for parameterized model reduction we provide a unifying projection based framework for structure preserving interpolatory model reduction of parameterized linear dynamical systems i e systems having a structured dependence on parameters that we wish to retain in the reduced order model the parameter dependence may be linear or nonlinear and is retained in the reduced order model moreover we are able to give conditions under which the gradient and hessian of the system response with respect to the system parameters is matched in the reduced order model we provide a systematic approach built on established interpolatory mathcal h 2 optimal model reduction methods that will produce parameterized reduced order models having high fidelity throughout a parameter range of interest for single input single output systems with parameters in the input output maps we provide reduced order models that are optimal with respect to an mathcal h 2 otimes mathcal l 2 joint error measure the capabilities of these approaches are illustrated by several numerical examples from technical applications

model order reduction of mimo bilinear systems by multi order arnoldi method abstract in this paper we present a time domain model order reduction method for multi input multi output mimo bilinear systems by general orthogonal polynomials the proposed method is based on a multi order arnoldi algorithm applied to construct the projection matrix the resulting reduced model can match a desired number of expansion coefficient terms of the original system the approximate error estimate of the reduced model is given and we also briefly discuss the stability preservation of the reduced model in some cases additionally in combination with krylov subspace methods we propose a two sided projection method to generate reduced models which capture properties of the original system in the time and frequency domain simultaneously the effectiveness of the proposed methods is demonstrated by two numerical examples

pod deim reduced order strategies for efficient four dimensional variational data assimilation this work studies reduced order modeling rom approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models it is shown that a key requirement for a successful reduced order solution is that reduced order karush kuhn tucker conditions accurately represent their full order counterparts in particular accurate reduced order approximations are needed for the forward and adjoint dynamical models as well as for the reduced gradient new strategies to construct reduced order based are developed for proper orthogonal decomposition pod rom data assimilation using both galerkin and petrov galerkin projections for the first time pod tensorial pod and discrete empirical interpolation method deim are employed to develop reduced data assimilation systems for a geophysical flow model namely the two dimensional shallow water equations numerical experiments confirm the theoretical framework for galerkin projection in the case of petrov galerkin projection stabilization strategies must be considered for the reduced order models the new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time

an on line time dependent parametric model order reduction scheme with focus on dynamic stress recovery in view of the tendency towards ever lighter and more powerful machines and even shorter design cycles it becomes essential to have virtual prototyping tools that allow for fast and reliable numerical simulations current state of the art structural dynamics and flexible multibody simulation techniques usually involve the solution of matrix systems with thousands to millions of variables model order reduction schemes are used to keep computational effort affordable at the expense of a minimal loss of accuracy these techniques typically face difficulties with systems in which flexible bodies can be loaded in many degrees of freedom and rarely allow for accurate local stress and strain evaluation the present work proposes to address both issues for the particular but frequent case of moving loads or boundary conditions this behavior is found in most of the contact problems and systems that include sliding components for which many loading or boundary locations are possible but only a few of them are active at a certain moment in time the proposed scheme exploits a reduction vector space that continuously varies in time by means of a parametric definition of the external load position contrary to the majority of the parametric model order reduction schemes that allow mainly for quasi static parametric variations the proposed approach can be used efficiently for time simulation of dynamically varying parametric models this is achieved by considering the implicit time dependency of the reduction vector space using galerkin projections or alternatively by direct substitution of the reduced kinetic energy potential energy and generalized forces in the lagrange equations it is shown by developing a consistent mathematical framework that the price to pay for the very compact reduction space obtained is the evaluation of some extra terms in the equations of motion numerical examples are used to assess the accuracy of the proposed method results show the potential of this strategy with particular focus on displacement and stress fields and furthermore highlight its real time potential moreover the developed framework together with the numerical results allow for a deeper physical understanding of the complex phenomena related to this category of time varying multiple input multiple output systems

growth of maps distortion in groups and symplectic geometry in the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism the uniform norm of the differential of its n th iteration the word length of its n th iteration where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms we find lower bounds for the growth rates of these sequences in a number of situations these bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism they are obtained by using recent results of schwarz on floer homology as an application we prove non existence of certain non linear symplectic representations for finitely generated groups

a two step model reduction approach for mechanical systems with moving loads we consider model order reduction of mechanical systems with moving loads such systems have a time varying input matrix that makes the direct application of standard model reduction methods difficult in this paper we present a two step model reduction approach for systems with moving loads which is based on a low rank approximation of the input matrix and applying krylov subspace methods to the resulting linear time invariant system with a modified input numerical results demonstrate the properties of the proposed model reduction method

using krylov pade model order reduction for accelerating design optimization of structures and vibrations in the frequency domain summary in many engineering problems the behavior of dynamical systems depends on physical parameters in design optimization these parameters are determined so that an objective function is minimized for applications in vibrations and structures the objective function depends on the frequency response function over a given frequency range and we optimize it in the parameter space because of the large size of the system numerical optimization is expensive in this paper we propose the combination of quasi newton type line search optimization methods and krylov pade type algebraic model order reduction techniques to speed up numerical optimization of dynamical systems we prove that krylov pade type model order reduction allows for fast evaluation of the objective function and its gradient thanks to the moment matching property for both the objective function and the derivatives towards the parameters we show that reduced models for the frequency alone lead to significant speed ups in addition we show that reduced models valid for both the frequency range and a line in the parameter space can further reduce the optimization time copyright 2012 john wiley sons ltd

model reduction for large scale systems with high dimensional parametric input space a model constrained adaptive sampling methodology is proposed for the reduction of large scale systems with high dimensional parametric input spaces our model reduction method uses a reduced basis approach which requires the computation of high fidelity solutions at a number of sample points throughout the parametric input space a key challenge that must be addressed in the optimization control and probabilistic settings is the need for the reduced models to capture variation over this parametric input space which for many applications will be of high dimension we pose the task of determining appropriate sample points as a pde constrained optimization problem which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters the methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21 which describe thermal analysis and design of a heat conduction fin and compared with statistically based sampling methods for these examples the model constrained adaptive sampling leads to reduced models that for a given basis size have error several orders of magnitude smaller than that obtained using the other methods

krylov subspace methods for reduced order modeling in circuit simulation the simulation of electronic circuits involves the numerical solution of very large scale sparse in general nonlinear systems of differential algebraic equations often the size of these systems can be reduced considerably by replacing the equations corresponding to linear subcircuits by approximate models of much smaller state space dimension in this paper we describe the use of krylov subspace methods for generating such reduced order models of linear subcircuits particular emphasis is on reduced order modeling techniques that preserve the passivity of linear rlc subcircuits

contragredient transformations applied to the optimal projection equations the optimal projection approach to solving the h2 reduced order model problem produces two coupled highly nonlinear matrix equations with rank conditions as constraints it is not obvious from their original form how they can be differentiated and how some algorithm for solving nonlinear equations can be applied to them a contragredient transformation a transformation which simultaneously diagonalizes two symmetric positive semi definite matrices is used to transform the equations into forms suitable for algorithms for solving nonlinear problems three different forms of the equations obtained using contragredient transformations are given an svd based algorithm for the contragredient transformation and a homotopy algorithm for the transformed equations are given together with a numerical example

an approximate approach to h sup 2 optimal model reduction deals with the problem of computing an h sub 2 optimal reduced order model for a given stable multivariable linear system by way of orthogonal projection the problem is formulated as that of minimizing the h sub 2 model reduction cost over the stiefel manifold so that the stability constraint on reduced order models is automatically satisfied and thus totally avoided in the new problem formulation the closed form expression for the gradient of the cost over the manifold is derived from which a gradient flow results as an ordinary differential equation ode a number of nice properties about such a flow are established furthermore two explicit iterative convergent algorithms are developed from the flow one has a constant step size and the other has a varying step size and is much more efficient both of them inherit the properties that the iterates remain on the manifold starting from any orthogonal initial point and that the model reduction cost is decreasing to minima along the iterates a procedure for closing the gap between the original and modified problem is proposed in the symmetric case the two problems are shown to be equivalent numerical examples are presented to illustrate the effectiveness of the proposed algorithms as well as convergence

commuting hamiltonians and hamilton jacobi multi time equations the aim of this paper is twofold first of all we show that the c 0 limit of a pair of commuting hamiltonians commute this means on one hand that if the limit of the hamiltonians is smooth the poisson bracket of their limit still vanishes and on the other hand that we may define commutation for c 0 functions the second part of the paper deals with solving multi time hamilton jacobi equations using variational solutions this extends the work of barles and tourin in the viscosity case to include the case of c 0 hamiltonians and removes their convexity assumption provided we are in the framework of variational solutions

some results of hamiltonian homeomorphisms on closed aspherical surfaces abstract on closed symplectically aspherical manifolds by using floer homology schwarz proved a classical result i e that the action function of a nontrivial hamiltonian diffeomorphism is not constant in this article we generalize schwarz s theorem to the c 0 case on closed aspherical surfaces our methods are purely topological and involve the theory of transverse foliations for dynamical systems of surfaces and the recent progress inspired by le calvez as an application we prove that the contractible fixed points set and consequently the fixed points set of a nontrivial hamiltonian homeomorphism is not connected we also get a similar result for an orientation preserving nonwandering point homeomorphism of the two sphere in the end we give further applications based on the c 0 schwarz theorem

a c 0 counterexample to the arnold conjecture the arnold conjecture states that a hamiltonian diffeomorphism of a closed and connected symplectic manifold m omega must have at least as many fixed points as the minimal number of critical points of a smooth function on m it is well known that the arnold conjecture holds for hamiltonian homeomorphisms of closed symplectic surfaces the goal of this paper is to provide a counterexample to the arnold conjecture for hamiltonian homeomorphisms in dimensions four and higher more precisely we prove that every closed and connected symplectic manifold of dimension at least four admits a hamiltonian homeomorphism with a single fixed point

accelerating pde constrained optimization by the reduced basis method application to batch chromatography summary in this work we show that the reduced basis method accelerates a partial differential equation constrained optimization problem where a nonlinear discretized system with a large number of degrees of freedom must be repeatedly solved during optimization such an optimization problem arises for example from batch chromatography to reduce the computational burden of repeatedly solving the large scale system under parameter variations a parametric reduced order model with a small number of equations is derived by using the reduced basis method as a result the small reduced order model rather than the full system is solved at each step of the optimization process an adaptive technique for selecting the snapshots is proposed so that the complexity and runtime for generating the reduced basis are largely reduced an output oriented error bound is derived in the vector space whereby the construction of the reduced model is managed automatically an early stop criterion is proposed to circumvent the stagnation of the error and to make the construction of the reduced model more efficient numerical examples show that the adaptive technique is very efficient in reducing the offline time the optimization based on the reduced model is successful in terms of the accuracy and the runtime for acquiring the optimal solution copyright 2015 john wiley sons ltd

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

design of fir filters in the complex domain the design of fir digital filters with a complex valued desired frequency response using the chebyshev error is investigated the complex approximation problem is converted into a real approximation problem which is nearly equivalent to the complex problem a standard linear programming algorithm for the chebyshev solution of overdetermined equations is used to solve the real approximation problem additional constraints are introduced which allow weighting of the phase and or group delay of the approximation digital filters are designed which have nearly constant group delay in the passbands the desired constant group delay which gives the minimum chebyshev error is found to be smaller than that of a linear phase filter of the same length these filters in addition to having a smaller approximately constant group delay have better magnitude characteristics than exactly linear phase filters with the same length the filters have nearly equiripple magnitude and group delay

implicit volterra series interpolation for model reduction of bilinear systems we propose a new interpolatory framework for model reduction of large scale bilinear systems the input output representation of a bilinear system in frequency domain involves a series of multivariate transfer functions each representing a subsystem of the bilinear system if a weighted sum of these multivariate transfer functions associated with a reduced bilinear system interpolates a weighted sum of the original multivariate transfer functions we say that the reduced system satisfies a volterra series interpolation flagg and gugercin 2015 these interpolatory conditions can also ensure the necessary conditions for h 2 optimal model reduction flagg and gugercin 2015 benner and breiten 2012 we observe that by carefully selecting the weights of the series the volterra series interpolatory conditions are transformed to the problem of interpolating a linear system with an affine parameter dependence such linear parametric systems can then be reduced by some method for parametric model order reduction linear systems where the affine parameter dependence is given as low rank variation in the state matrix can be mapped into a non parameterized multi input multi output linear system this allows us to utilize the standard non parametric linear irka gugercin et al 2008 for the problem of parameterized bilinear interpolation numerical results show that the approximations are of comparable accuracy to those obtained from the bilinear iterative rational krylov algorithm benner and breiten 2012 the proposed approach however has the advantage that it reduces the computational costs as it involves computations associated with solving linear systems only

erratum to generalizations of the poincare birkhoff theorem other results of 1 which rely on theorem 2 1 are correct as stated and follow from the result above a disk u in the covering space a is called positively returning for f if f u u and fn u t k u for some n k 0 where t is the generator of the infinite cyclic group of covering translations negatively returning disks are defined similarly the difference in the statement of the theorem above and the statement of theorem 2 1 of 1 is the additional requirement in item 3 that the positively and negatively returning disks in the covering space are lifts of disks in the annulus equivalently the positively and negatively returning disks in the covering space must be disjoint from their image under t k k 0 standard arguments show it suffices that t u u this additional hypothesis is satisfied in other results from 1 which use theorem 2 1 i do not know whether theorem 2 1 of 1 as originally stated is true or not

a structured quasi arnoldi procedure for model order reduction of second order systems abstract existing krylov subspace based structure preserving model order reduction methods for the second order systems proceed in two stages the first stage is to generate a basis matrix of the underlying krylov subspace the second stage is to employ an explicit subspace projection to obtain a reduced order model with a moment matching property an open problem is how to avoid explicit projection so that it will be efficient for truly large scale systems in addition it is also desired that a structure preserving reduced system of order n matches maximum 2 n moments in this paper we propose a new procedure to compute a so called structured quasi arnoldi sqa decomposition once the sqa decomposition is computed a structure preserving reduced order model can be defined immediately from the decomposition without the explicit subspace projection furthermore the reduced model of order n matches maximum 2 n moments numerical examples demonstrate that the transpose free sqa based reduced model is compatible with the two sided structure preserving explicit projection methods and is more accurate than the one sided structure preserving explicit projection methods due to the higher number of matched moments

linear system approximation via covariance equivalent realizations abstract a truncation technique for model reduction to simultaneously match a specified number of output covariance derivatives is described these covariance equivalent realizations which match q 1 covariance derivatives are called q covers in general q covers are not unique the additional freedom is used herein so that the q cover obtained also matches q markov parameters the truncation technique uses a form of the observability matrices of the full order system to determine a priori the order required of the reduced order model to match a specified number of output covariance derivatives and markov parameters the resulting realization is shown to be independent of the basis of the complete model to within a unitary transformation stability conditions for the reduced order model are also described and the relationships are established between stochastically equivalent realizations and the q covers

strictly toral dynamics this article deals with nonwandering e g area preserving homeomorphisms of the torus mathbb t 2 which are homotopic to the identity and strictly toral in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane this includes all homeomorphisms which have a rotation set with nonempty interior we define two types of points inessential and essential the set of inessential points operatorname ine f is shown to be a disjoint union of periodic topological disks elliptic islands while the set of essential points operatorname ess f is an essential continuum with typically rich dynamics the chaotic region this generalizes and improves a similar description by jager the key result is boundedness of these elliptic islands which allows among other things to obtain sharp uniform bounds of the diffusion rates we also show that the dynamics in operatorname ess f is as rich as in mathbb t 2 from the rotational viewpoint and we obtain results relating the existence of large invariant topological disks to the abundance of fixed points

generalizations of the poincare birkhoff theorem in this article we prove some generalizations of the classical poincar birkhoff theorem on area preserving homeomorphisms of the annulus which satisfy a boundary twist condition the work of g d birkhoff on this theorem and its applications can be found in b1 b2 and chapter v of b3 a more modern treatment can be found in b n we prove a theorem for the open annulus a s1 x 0 1 since as we will see the theorem for the closed annulus can easily be obtained from this result for the open annulus however it is not immediately obvious what should be the analogue of the twist condition it turns out that the most general hypothesis and the most natural from the point of view of our proof involves the notion of positively and negatively returning disks for some lift of f to the covering space a r x 0 1 more precisely if f a a is a lift of f a a we will say that the e is a positively returning disk for f if there is an open disk u c a such that f u n u 0 and fp u n u k 0 forsome n k 0 here u k denotes the set x k t i x t e u thus u is disjoint from its image but under iteration by f returns to a positive translate of itself a negatively returning disk is defined similarly but with k 0 such that fl u n u 0 in section 2 we prove the following

a multiscale method for model order reduction in pde parameter estimation abstract estimating parameters of partial differential equations pdes is of interest in a number of applications such as geophysical and medical imaging parameter estimation is commonly phrased as a pde constrained optimization problem that can be solved iteratively using gradient based optimization a computational bottleneck in such approaches is that the underlying pdes need to be solved numerous times before the model is reconstructed with sufficient accuracy one way to reduce this computational burden is by using model order reduction mor techniques such as the multiscale finite volume method msfv in this paper we apply msfv for solving high dimensional parameter estimation problems given a finite volume discretization of the pde on a fine mesh the msfv method reduces the problem size by computing a parameter dependent projection onto a nested coarse mesh a novelty in our work is the integration of msfv into a pde constrained optimization framework which updates the reduced space in each iteration we present a computationally tractable way of explicitly differentiating the mor solution that acknowledges the change of basis as we demonstrate in our numerical experiments our method leads to computational savings for large scale parameter estimation problems where iterative pde solvers are necessary and offers potential for additional speed ups through parallel implementation

recycling bicgstab with an application to parametric model order reduction krylov subspace recycling is a process for accelerating the convergence of sequences of linear systems based on this technique the recycling bicg algorithm has been developed recently here we now generalize and extend this recycling theory to bicgstab recycling bicg focuses on efficiently solving sequences of dual linear systems while the focus here is on efficiently solving sequences of single linear systems assuming nonsymmetric matrices for recycling bicg and recycling bicgstab as compared with other methods for solving sequences of single linear systems with nonsymmetric matrices e g recycling variants of gmres bicg based recycling algorithms like recycling bicgstab have the advantage that they involve a short term recurrence and hence do not suffer from storage issues and are also cheaper with respect to the orthogonalizations we modify the bicgstab algorithm to use a recycle space which is built from left and right approximate invariant subspaces using our algorithm for a parametr

an online coupled state input parameter estimation approach for structural dynamics abstract in many practical structural applications unknown states inputs and parameters are present however most methods require one or more of these variables to be known in order to estimate the other s in this research an estimation technique which employs physical models is proposed to perform coupled state input parameter estimation in order to obtain a modeling technique which allows the estimation of a wide range of parameters in a generic fashion at a minimal computational cost even real time the use of a parametric model reduction technique is proposed the reduced model is coupled to an extended kalman filter ekf with augmented states for the unknown inputs and parameters this leads to a very efficient framework for estimation in structural dynamics problems special attention is also given to the measurement requirements in order to obtain an adequate observability of all unknown quantities and the necessity for at least one displacement level measurement is shown the proposed methodology is validated numerically and experimentally the approach is shown to be easy to tune and provides good results with different measurement methods

sample selection based on sensitivity analysis in parameterized model order reduction modeling of scientific or engineering applications often yields high dimensional dynamical systems due to techniques of computer aided design for example thus a model order reduction is required to decrease the dimensionality and to enable an efficient numerical simulation in addition methods of parameterized model order reduction pmor are often used to preserve the physical or geometric parameters as independent variables in the reduced order models we consider linear dynamical systems in the form of ordinary differential equations in the domain of the parameters often samples are chosen to construct a reduced order model for each sample point a common technique for model order reduction can be applied to compute a local basis moment matching or balanced truncation are feasible for example a global basis for pmor can be constructed from the local bases by a singular value decomposition we investigate approaches for an appropriate selection of a finite set of samples the transfer function of the dynamical system is examined in the frequency domain and our focus is on moment matching techniques using the arnoldi procedure we use a sensitivity analysis of the transfer function with respect to the parameters as a tool to select sample points simulation results are shown for two examples

a parameterised model order reduction method for parametric systems based on laguerre polynomials abstractthis paper presents a laguerre polynomials based parametrised model order reduction method for the parametric system in time domain the method allows that the parametric dependence in system matrices is nonaffine the method is presented via reducing an approximate polynomial parametric system based on taylor expansion and laguerre polynomials resulting in a parametric reduced system that can accurately approximate the time response of the original parametric system over a wide range of parameter the reduced parametric system obtained by proposed method can be implemented by two algorithms algorithm 1 is a direct way that is suitable for single input multi output parametric systems algorithm 2 is presented based on a connection to the krylov subspace which is efficient and suitable for multi input multi output parametric systems the effectiveness of the proposed method is illustrated with two benchmarks in practical applications

a real time procedure for affinely dependent parametric model order reduction using interpolation on grassmann manifolds summary model order reduction helps to reduce the computational time in dealing with large dynamical systems for example during simulation control optimization in many cases the considered model depends on parameters model order reduction techniques are therefore preferred to symbolically preserve this dependence or to be adaptive to the change of the model caused by the variation in the values of the parameters in this paper we first present the application of the interpolation technique on grassmann manifolds to this problem we then improve the method for the models whose system matrices depend affinely on parameters by considerably reducing the computational complexity on the basis of analyzing the structure of sums of singular value decompositions and decomposing the whole procedure into offline and online stages a numerical example is shown to illustrate the method as well as to prove its effectiveness copyright 2012 john wiley sons ltd

a low order reduced model for the long range propagation of infrasounds in the atmosphere this paper considers a class of low order range dependent propagation models obtained from the normal mode decomposition of infrasounds in complex atmospheres the classical normal mode method requires calculating eigenvalues for large matrices making the computation expensive even though some modes have little influence on the numerically obtained results by decomposing atmospheric perturbations into a wavelet basis it is shown that the most sensitive eigenvalues provide the best reduced model for infrasound propagation these eigenvalues lie on specific curves in the complex plane that can be directly deduced from atmospheric data through a wkb approach the computation cost can be reduced by computing the invariant subspace associated with the most sensitive eigenvalues the reduction method is illustrated in the case of the fukushima explosion 12 march 2011 the implicitly restarted arnoldi algorithm is used to compute the three most sensitive modes and the correct tropospheric arrival is found with a cost of 2 of the total run time the cost can be further reduced by using a stationary phase technique finally it is shown that adding uncertainties triggers a stratospheric arrival even though the classical criteria based on the ratio of stratospheric sound speed to that at ground level is not satisfied

simultaneous design in both magnitude and group delay of iir and fir filters based on multiple criterion optimization this work considers the simultaneous design in both magnitude and group delay of digital transfer functions on the basis of multiple criterion optimization both causal iir filters and nonlinear phase fir filters are studied examples of the optimal tradeoff filters both fir filters and iir filters are presented and their characteristics are analyzed

finite element model updating for structural applications abstract a novel method for performing model updating on finite element models is presented the approach is particularly tailored to modal analyses of buildings by which the lowest frequencies obtained by using sensors and system identification approaches need to be matched to the numerical ones predicted by the model this is done by optimizing some unknown material parameters such as mass density and young s modulus of the materials and or the boundary conditions which are often known only approximately in particular this is the case when considering historical buildings the straightforward application of a general purpose optimizer can be impractical given the large size of the model involved in the paper we show that by slightly modifying the projection scheme used to compute the eigenvalues at the lowest end of the spectrum one can obtain local parametric reduced order models that embedded in a trust region scheme form the basis for a reliable and efficient specialized algorithm we describe an optimization strategy based on this approach and we provide numerical experiments that confirm its effectiveness and accuracy

interpolatory projection methods for structure preserving model reduction abstract we present a framework for interpolatory model reduction that treats systems having a generalized coprime factorization c s s 1 b s d this includes rational krylov based interpolation methods as a special case the broader framework allows retention of special structure in reduced models such as symmetry second and higher order structure state constraints internal delays and infinite dimensional subsystems two numerical examples are provided for systems associated with viscoelastic dynamics and with an internal state delay that demonstrate the effectiveness and flexibility of the approach

the reduced basis element method application to a thermal fin problem the reduced basis element method is a new approach for the approximation of partial differential equations that takes its roots in the domain decomposition method and in the reduced basis discretization the basic idea is to decompose the domain of computation into a series of subdomains the elements that are similar to a few reference domains these reference domains are actually filled with reduced basis functional spaces that are mapped to each subdomain together with the geometry the discrete approximation space is then composed of functions with the property that a function restricted to a subdomain belongs to the mapped reduced spaces finally a mortar type method is applied to glue the various local functions in this paper we focus on the definition of the reference shapes and together with theoretical and numerical justifications of the method we provide a posteriori error analysis tools that allow us to certify the computational results

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

h2 norm optimal model reduction for large scale discrete dynamical mimo systems modeling strategies often result in dynamical systems of very high dimension it is then desirable to find systems of the same form but of lower complexity whose input output behavior approximates the behavior of the original system here we consider linear time invariant discrete time dynamical systems the cornerstone of this paper is a relation between optimal model reduction in the h 2 norm and tangential rational hermite interpolation first order necessary conditions for h 2 optimal model reduction are presented for discrete multiple input multiple output mimo systems these conditions suggest a specific choice of interpolation data and a novel algorithm aiming for anh 2 optimal model reduction for mimo systems it is also shown that the conditions are equivalent to two known gramian based first order necessary conditions numerical experiments demonstrate the approximation quality of the method

a technique for multiple criterion approximation of fir filters in magnitude and group delay in spite of the attention received by nonlinear phase nlp fir filter design the best way to solve this problem is still open to debate the formulation of nonlinear phase fir filter design in terms of simultaneous magnitude and group delay approximation addresses the two parameters of ultimate interest for applications the simultaneous minimization of these two functions one of which the group delay rational is approached on the basis of an original extension of the differential correction algorithm the proposed design tool enjoys interesting theoretical properties and works very effectively the filters obtained according to the multiple criterion optimization mco formulation are compared against filters that are optimal in the complex chebychev norm then the flexibility characteristic of mco formulations of trading off between magnitude and group delay performance is exemplified

design of finite impulse response digital filters with nonlinear phase response most of the existing literature on fir digital filters is concerned with linear phase lp filters however several papers have appeared on the subject of nonlinear phase nlp filters mainly proposing methods for designing minimum phase filters or approximating a desired phase response in this paper an investigation is made of one such method based upon a selection of zeros from a prototype lp filter it is shown that with respect to minimizing the order of a filter subject to given gain response specifications this is the most efficient method for designing fir filters coefficient quantization error is analyzed for filters generated by this method a practical comparison is given between the resulting filter and the corresponding minimal order lp filter it is shown that while most lp filters can be implemented more efficiently than nlp filters by taking into account the symmetry of their coefficients for filters with very wide passband and for certain special purpose filters such as ccd and those used for filtering a delta modulated or adpcm signal an nlp implementation is usually more efficient in addition an alternate design algorithm is proposed for nlp filters which decreases ripple magnitude the resulting filters while not of minimal order can be efficiently implemented by decomposing the filter into lp stopband and nlp passband sections which is especially attractive for narrow passbands

reduction of second order network systems with structure preservation this paper proposes a general framework for structure preserving model reduction of a second order network system based on graph clustering in this approach vertex dynamics are captured by the transfer functions from inputs to individual states and the dissimilarities of vertices are quantified by the mathcal h 2 norms of the transfer function discrepancies a greedy hierarchical clustering algorithm is proposed to place those vertices with similar dynamics into same clusters then the reduced order model is generated by the petrov galerkin method where the projection is formed by the characteristic matrix of the resulting network clustering it is shown that the simplified system preserves an interconnection structure i e it can be again interpreted as a second order system evolving over a reduced graph furthermore this paper generalizes the definition of network controllability gramian to second order network systems based on it we develop an efficient method to compute mathcal h 2 norms and derive the approximation error between the full order and reduced order models finally the approach is illustrated by the example of a small world network

on structure preserving model reduction for damped wave propagation in transport networks we consider the discretization and subsequent model reduction of a system of partial differential algebraic equations describing the propagation of pressure waves in a pipeline network important properties like conservation of mass dissipation of energy passivity existence of steady states and exponential stability can be preserved by an appropriate semidiscretization in space via a mixed finite element method and also during the further dimension reduction by structure preserving galerkin projection which is the main focus of this paper krylov subspace methods are employed for the construction of the reduced models and we discuss certain modifications needed to satisfy some algebraic compatibility conditions which are required to ensure the well posedness of the reduced models and the preservation of the key properties our arguments are based on a careful analysis of the underlying infinite dimensional problem and its galerkin approximations the proposed algorithms therefore have a direct inte

on the design of fir filters by complex chebyshev approximation the long standing problem of approximating a complex valued desired function with a finite impulse response fir filter is considered it is formulated as an equalization to be solved using complex valued filters the proposed algorithm deals directly with the complex error function which depends linearly on the coefficients of the filter to be designed the magnitude of this error function is minimized in the chebshev sense using a generalization of the remez exchange algorithm the method can be used to design complex or real valued selective systems as well the well known design of optimal fir filters with linear phase is included here as a special case

fast algorithms for hyperspectral diffuse optical tomography the image reconstruction of chromophore concentrations using diffuse optical tomography dot data can be described mathematically as an ill posed inverse problem recent work has shown that the use of hyperspectral dot data as opposed to data sets comprising a single or at most a dozen wavelengths has the potential for improving the quality of the reconstructions the use of hyperspectral diffuse optical data in the formulation and solution of the inverse problem poses a significant computational burden the forward operator is in actuality nonlinear however under certain assumptions a linear approximation called the born approximation provides a suitable surrogate for the forward operator and we assume this to be true in the present work computation of the born matrix requires the solution of thousands of large scale discrete pdes and the reconstruction problem requires matrix vector products with the dense born matrix in this paper we address both of these difficulties thus making the

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

reduced order modeling of multiscreen frequency selective surfaces using krylov based rational interpolation a method is presented for generating a broad band rational interpolant approximation of the reflection coefficient of multiple screen frequency selective surfaces fsss the technique is structured around a linearization of the system provided by a spectral domain moment method based analysis of the fss followed by a model order reduction of the linearized system using the dual rational arnoldi method this process creates a rational interpolant of the linearized system that matches its transfer function and its derivatives at several expansion points in the laplace domain numerical results indicate that a reduced order model with a system matrix of dimension less than 20 spl times 20 can accurately reproduce the broad band behavior of multiscreen fsss originally modeled with several hundreds or thousands of unknowns

families of moment matching based low order approximations for linear systems the problem of finding a framework for linear moment matching based approximation of linear systems is discussed it is shown that it is possible to define five equivalent notions of moments i e one from a complex s domain point of view two using krylov projections and two from a time domain perspective based on these notions classes of parameterized reduced order models that achieve moment matching are obtained we analyze the controllability and observability properties of the models belonging to each of the classes of reduced order models we find the subclasses of models of orders lower than the number of matched moments i e we find the sets of parameters that allow for pole zero cancellations to occur furthermore we compute the subclass of minimal order model s the minimal order is equal to half of the number of matched moments i e generically the largest number of possible pole zero cancellations is half of the number of matched moments finally we present classes of reduced order models that match larger numbers of moments based on interconnections between reduced order models

a new algorithm for l 2 optimal model reduction in this paper the quadratically optimal model reduction problem for single input single output systems is considered the reduced order model is determined by minimizing the integral of the magnitude squared of the transfer function error it is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and on this basis a two step iterative algorithm is developed combining a least squares solver with a gradient minimizer convergence of the proposed algorithm to stationary values of the quadratic cost function is proved the formulation is extended to handle the frequency weighted optimal model reduction problem three examples demonstrate the optimization algorithm

cuplength estimates on lagrangian intersections on demontre l estimation suivante sur des intersections lagrangiennes si l est une sous variete lagrangienne de p avec 2 p l 0 et l est obtenue a partir de l par un diffeomorphisme exact de p alors le nombre d elements de l l est superieure ou egale a la longueur de calotte de p

a two directional arnoldi process and its application to parametric model order reduction we consider a two directional krylov subspace k k a j b j where besides the dimensionality k of the subspace increases the matrix a j and vector b j which induce the subspace may also augment specifically we consider the case where the matrix a j and the vector b j are augmented by block triangular bordering we present a two directional arnoldi process to efficiently generate a sequence of orthonormal bases q k j of the krylov subspaces the concept of a two directional krylov subspace and an arnoldi process is triggered by the need of a multiparameter moment matching based model order reduction technique for parameterized linear dynamical systems numerical examples illustrate computational efficiency and flexibility of the proposed two directional arnoldi process

a reduced order matrices fitting scheme with log euclidean metrics for fast approximation of dynamic response of parametric structural systems abstract structural engineering practice often involves tasks such as design optimization or statistical analysis where iterative solutions of dynamical systems are required varying some parameter which affect the system matrices in a generally non linear way the approach of interpolating among matrices of reduced order models roms is not new in the literature and it is very promising in order to speed up the calculations in particular the method shows a great potential since it is applicable to any kind of fe model and of geometry with no restrictions regarding the element type constituting the full order model or the range of parameter variability yet dealing with symmetric positive definite spd full order matrices the spd nature of the reduced matrices must be preserved to this end a special mathematical framework must be used which allows to linearize the curved space of spd matrices in the literature of roms interpolation the common choice is to rely on the principles of riemannian geometry through the concept of the tangent plane to the spd manifold although mathematically robust this approach brings a marked distortion in the space of spd matrices also it shows numerical instabilities when dealing with strongly anisotropic matrices in this work a polynomial fitting of the reduced matrices is proposed with log euclidean metrics these metrics were obtained by giving the spd manifold the structure of a lie group while maintaining the already proposed approach of interpolating between a number of sampling roms the advantages of using log euclidean metrics are illustrated in details in particular a great improvement is shown in case of a unique reduction matrix whose columns span the solution of the parameter space of interest in an accurate enough way and in case of substructuring reduction techniques the reported examples of applications show excellent accuracy while maintaining the dramatic decrease in the computational time of the interpolatory scheme

dimension reduction of large scale second order dynamical systems via a second order arnoldi method a structure preserving dimension reduction algorithm for large scale second order dynamical systems is presented it is a projection method based on a second order krylov subspace a second order arnoldi soar method is used to generate an orthonormal basis of the projection subspace the reduced system not only preserves the second order structure but also has the same order of approximation as the standard arnoldi based krylov subspace method via linearization the superior numerical properties of the soar based method are demonstrated by examples from structural dynamics and microelectromechanical systems