a lanczos type method for multiple starting vectors given a square matrix and single right and left starting vectors the classical nonsymmetric lanczos process generates two sequences of bior thogonal basis vectors for the right and left krylov subspaces induced by the given matrix and vectors in this paper we propose a lanczos type algorithm that extends the classical lanczos process for single starting vectors to mul tiple starting vectors given a square matrix and two blocks of right and left starting vectors the algorithm generates two sequences of biorthogonal basis vectors for the right and left block krylov subspaces induced by the given data the algorithm can handle the most general case of right and left start ing blocks of arbitrary sizes while all previously proposed extensions of the lanczos process are restricted to right and left starting blocks of identical sizes other features of our algorithm include a built in deation procedure to detect and delete linearly dependent vectors in the block krylov sequences and the option to employ look ahead to remedy the potential breakdowns that may occur in nonsymmetric lanczos type methods

allele fixation probability in a moran model with fluctuating fitness landscapes evolution on changing fitness landscapes seascapes is an important problem in evolutionary biology we consider the moran model of finite population evolution with selection in a randomly changing dynamic environment in the model each individual has one of the two alleles wild type or mutant we calculate the fixation probability by making a proper ansatz for the logarithm of fixation probabilities this method has been used previously to solve the analogous problem for the wright fisher model the fixation probability is related to the solution of a third order algebraic equation for the logarithm of fixation probability we consider the strong interference of landscape fluctuations sampling and selection when the fixation process cannot be described by the mean fitness such an effect appears if the mutant allele has a higher fitness in one landscape and a lower fitness in another compared with the wild type and the product of effective population size and fitness is large we provide a generalization of the kimura formula for the fixation probability that applies to these cases when the mutant allele has a fitness dis advantage in both landscapes the fixation probability is described by the mean fitness

on the solution of the minimal rational interpolation problem it is shown that the reachability and the observability indices of appropriately defined pairs of matrices provide the key for solving the minimal rational bitangen tial interpolation problem

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

taming the diffusion approximation through a controlling factor wkb method the diffusion approximation da is widely used in the analysis of stochastic population dynamics from population genetics to ecology and evolution the da is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied this failure becomes severe in situations where the direction of selection switches sign here we employ the wkb wentzel kramers brillouin large deviations method which requires only the logarithm of a given quantity to be smooth over its state space combining the wkb scheme with asymptotic matching techniques we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations applicable for much wider regimes of parameters we also introduce a scalable independent of population size wkb based numerical technique the method is applied to a central problem in population genetics and evolution finding the chance of ultimate fixation in a zero sum two types competition

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

time domain model reduction of time delay systems via orthogonal polynomial expansions abstract this paper investigates model reduction of time delay systems in the time domain based on the expansion of systems over orthogonal polynomials time delay systems are first expanded over generalized laguerre polynomials the nice properties of generalized laguerre polynomials lead to a direct system expansion and a sylvester equation with special structures which enables an efficient calculation of laguerre coefficients of systems projection methods are then adopted to generate reduced models and we show that a desired number of laguerre coefficients are preserved by reduced models further we extend the proposed method to general orthogonal polynomials where the relationship between taylor expansion and orthogonal polynomial expansion is examined to achieve the expansion of time delay systems systems with multiple delays and delays appearing in the derivative of the states are also discussed two numerical examples are simulated to showcase the efficiency of our approach

model reduction by nice selections for linear switched systems a moment matching method for model reduction of linear switched systems lsss is presented the method can be seen as a non trivial extension of the krylov subspace methods for linear time invariant lti systems the procedure is based on the so called nice selections which represent a choice of basis in the reachability or observability space of the lss the framework can also be used for exact matching of the input output behavior of an lss with a reduced order lss for a specific switching sequence conditions for applicability of the method for model reduction are stated and finally the results are illustrated on numerical examples

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

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

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

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

a frequency weighted model order reduction technique and error bounds a new frequency weighted technique for balanced model reduction is proposed the proposed technique not only provides stable reduced order models for the case when both input and output weightings are included but also yields frequency response error bounds the method is illustrated using numerical examples and the results are compared with other frequency weighted model reduction techniques

two sided projection methods for nonlinear model order reduction in this paper we investigate a recently introduced approach for nonlinear model order reduction based on generalized moment matching using basic tensor calculus we propose a computationally efficient way of computing reduced order models we further extend the idea of two sided interpolation methods to this more general setting by employing the tensor structure of the hessian we investigate the use of oblique projections in order to preserve important system properties such as stability we test one sided and two sided projection methods for different semi discretized nonlinear partial differential equations and show their competitiveness when compared to proper orthogonal decomposition pod

molecular evolution under fitness fluctuations molecular evolution is a stochastic process governed by fitness mutations and reproductive fluctuations in a population here we study evolution where fitness itself is stochastic with random switches in the direction of selection at individual genomic loci as the correlation time of these fluctuations becomes larger than the diffusion time of mutations within the population fitness changes from an annealed to a quenched random variable we show that the rate of evolution has its maximum in the crossover regime where both time scales are comparable adaptive evolution emerges in the quenched fitness regime evidence for such fitness fluctuations has recently been found in genomic data the joint statistical theory of reproductive and fitness fluctuations establishes a conceptual connection between evolutionary genetics and statistical physics of disordered systems

structural dynamics analysis using an unsymmetric block lanczos algorithm this paper describes the way in which an unsymmetric block lanczos algorithm can be employed for approximate eigensolutions or dynamic response solutions for large systems having arbitrary damping and or repeated or closely spaced eigenvalues to reduce a square unsymmetric matrix which may have repeated eigenvalues to block tridiagonal form an unsymmetric block lanczos algorithm is developed right and left lanczos vectors generated by the reduction process are employed to define the reduced order model based on a discrete analytical model of general linear time invariant dynamic systems a special type of starting vector is also introduced which automatically includes the static displacement this reduced order model is employed to compute approximate eigenvalues vectors and dynamic responses of the original system to verify the proposed algorithm examples of an 8 dof beam rotor system are provided for eigensolutions and dynamic responses due to step and random external forces these examples showed promising results indicating that the proposed algorithm is worthy of further study

an algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to cr mappings we give a new algebraic characterization of holomorphic nondegeneracy for embedded real algebraic hypersurfaces in mathcal c n 1 n geq 1 we then use this criterion to prove the following result about real analyticity of smooth cr mappings any smooth cr mapping h between a real analytic hypersurface and a rigid polynomial holomorphically nondegenerate hypersurface is real analytic provided the map h is not totally degenerate in the sense of baouendi and rothschild

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

model reduction and perturbation analysis of wave finite element formulations for computing the forced response of coupled elastic systems involving junctions with uncertain eigenfrequencies abstract the wave finite element method is investigated for computing the low and mid frequency forced response of coupled elastic systems involving straight structures with junctions the relevance of the method is discussed when a component mode synthesis procedure is used for modeling the junctions a norm wise selection criterion is proposed so as to reduce efficiently the number of junction modes retained in the wave based formulations component wise perturbation bounds of the wave based displacement force solutions are also derived to address slight uncertainties for the junction eigenfrequencies numerical comparisons with standard finite element solutions as well as monte carlo simulations highlight the relevance of the formulation

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

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

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

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

on the decay rate of hankel singular values and related issues this paper investigates the decay rate of the hankel singular values of linear dynamical systems this issue is of considerable interest in model reduction by means of balanced truncation for instance since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error the decay rate involves a new set of invariants associated with a linear system which are obtained by evaluating a modified transfer function at the poles of the system these considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two lyapunov equations the related problem of determining the decay rate of the eigenvalues of the solution to one lyapunov equation will also be addressed very often these eigenvalues like the hankel singular values are rapidly decaying this fact has motivated the development of several algorithms for computing low rank approximate solutions to lyapunov equations however until now conditions assuring rapid decay have not been well understood such conditions are derived here by relating the solution to a numerically low rank cauchy matrix determined by the poles of the system bounds explaining rapid decay rates are obtained under some mild conditions

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

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

model reduction for large scale dynamical systems via equality constrained least squares in this paper we present a new method of model reduction for large scale dynamical systems which belongs to the svd krylov based method category it is a two sided projection where one side reflects the krylov part and the other side reflects the svd observability gramian part the reduced model matches the first r i markov parameters of the full order model and the remaining ones approximate in a least squares sense without being explicitly computed where r is the order of the reduced system and i is a nonnegative integer such that email protected i

a model with darwinian dynamics on a rugged landscape we discuss the population dynamics with selection and random diffusion keeping the total population constant in a fitness landscape associated with constraint satisfaction a paradigm for difficult optimization problems we obtain a phase diagram in terms of the size of the population and the diffusion rate with a glass phase inside which the dynamics keeps searching for better configurations and outside which deleterious mutations spoil the performance the phase diagram is analogous to that of dense active matter in terms of temperature and drive

speeding up evolutionary search by small fitness fluctuations we consider a fixed size population that undergoes an evolutionary adaptation in the weak mutation rate limit which we model as a biased langevin process in the genotype space we show analytically and numerically that if the fitness landscape has a small highly epistatic rough and time varying component then the population genotype exhibits a high effective diffusion in the genotype space and is able to escape local fitness minima with a large probability we argue that our principal finding that even very small time dependent fluctuations of fitness can substantially speed up evolution is valid for a wide class of models

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