a global rational arnoldi method for model reduction abstract in this paper we propose two new approaches for model order reduction of large scale multi input multi output mimo linear time invariant dynamical systems lti these methods are based on a generalization of the global arnoldi algorithm which is used to generate projection subspaces an adaptive procedure for the selection of shift parameters is proposed and some simple arnoldi relations are established in order to compute error bounds on the transfer function numerical experiments showing the effectiveness of these approaches are provided

spectral averaging techniques for jacobi matrices with matrix entries a jacobi matrix with matrix entries is a self adjoint block tridiagonal matrix with invertible blocks on the off diagonals averaging over boundary condi tions leads to explicit formulae for the averaged spectral measure which can potentially be useful for spectral analysis furthermore another variant of spectral averaging over coupling constants for these operators is presented

solution of the crow kimura and eigen models for alphabets of arbitrary size by schwinger spin coherent states to represent the evolution of nucleic acid and protein sequence we express the parallel and eigen models for molecular evolution in terms of a functional integral representation with an h letter alphabet lifting the two state purine pyrimidine assumption often made in quasi species theory for arbitrary h and a general mutation scheme we obtain the solution of this model in terms of a maximum principle euler s theorem for homogeneous functions is used to derive this thermodynamic formulation of evolution the general result for the parallel model reduces to known results for the purine pyrimidine h 2 alphabet and the nucleic acid h 4 alphabet for the kimura 3 st mutation scheme examples are presented for the h 4 and h 20 cases we also derive the maximum principle for the eigen model for general h the general result for the eigen model reduces to a known result for h 2 examples are presented for the nucleic acid h 4 and the amino acid h 20 alphabet an error catastrophe phase transition occurs in these models and the order of the phase transition changes from second to first order for smooth fitness functions when the alphabet size is increased beyond two letters to the generic case as examples we analyze the general analytic solution for sharp peak linear quadratic and quartic fitness functions

new algorithms for computing the real structured pseudospectral abscissa and the real stability radius of large and sparse matrices we present two new algorithms for investigating the stability of large and sparse matrices subject to real perturbations the first algorithm computes the real structured pseudospectral abscissa and is based on the algorithm for computing the pseudospectral abscissa proposed by guglielmi and overton siam j matrix anal appl 32 2011 pp 1166 1192 it entails finding the rightmost eigenvalues for a sequence of large matrices and we demonstrate that these eigenvalue problems can be solved in a robust manner by an unconventional eigenvalue solver we also develop an algorithm for computing the real stability radius of a real and stable matrix which utilizes a recently developed technique for detecting the loss of stability in a large dynamical system both algorithms are tested on large and sparse matrices

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

numerical low rank approximation of matrix differential equations abstract the efficient numerical integration of large scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications standard numerical methods applied to such problems require an unduly amount of computing time and memory in general based on a dynamical low rank approximation of the solution a new splitting integrator is proposed for a quite general class of stiff matrix differential equations this class comprises differential lyapunov and differential riccati equations that arise from spatial discretizations of partial differential equations the proposed integrator handles stiffness in an efficient way and it preserves the symmetry and positive semidefiniteness of solutions of differential lyapunov equations numerical examples that illustrate the benefits of this new method are given in particular numerical results for the efficient simulation of the weather phenomenon el nino are presented

sturm liouville operators in the half axis with local perturbations we give conditions which imply equivalence of the lebesgue measure with respect to a measure generated as an average of spectral measures corresponding to sturm liouville operators in the half axis we apply this to prove that some spectral properties of these operators hold for large sets of boundary conditions if and only if they hold for large sets of positive local perturbations

structure of principal eigenvectors and genetic diversity abstract the main concern of this paper is long term genotypic diversity genotypes are represented as finite sequences s 1 s 2 s n where the entries s i are drawn from a finite alphabet the mutation matrix is given in terms of hamming distances it is proved that the long time behavior of solutions for a class of genotype evolution models is governed by the principal eigenvectors of the sum of the mutation and fitness matrices it is proved that the components of principal eigenvectors are symmetric and monotonely decreasing in terms of hamming distances whenever the fitness matrix has those properties

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

low rank second order splitting of large scale differential riccati equations we apply first and second order splitting schemes to the differential riccati equation such equations are very important in e g linear quadratic regulator lqr problems where they provide a link between the state of the system and the optimal input the methods can also be extended to generalized riccati equations e g arising from lqr problems given in implicit form in contrast to previously proposed schemes such as bdf or rosenbrock methods the splitting schemes exploit the fact that the nonlinear and affine parts of the problem when considered in isolation have closed form solutions we show that if the solution possesses low rank structure which is frequently the case then this is preserved by the method this feature is used to implement the methods efficiently for large scale problems the proposed methods are expected to be competitive as they at most require the solution of a small number of linear equation systems per time step finally we apply our low rank implementations to the riccati equations arising from two lqr problems the results show that the rank of the solutions stay low and the expected orders of convergence are observed

on some krylov subspace based methods for large scale nonsymmetric algebraic riccati problems in the present paper we consider large scale nonsymmetric matrix riccati equations with low rank right hand sides these matrix equations appear in many applications such as transport theory wiener hopf factorization of markov chains applied probability and others we show how to apply directly krylov methods such as the extended block arnoldi algorithm to get low rank approximate solutions we also combine the newton method and block krylov subspace methods to get approximations of the desired minimal nonnegative solution we give some theoretical results and report some numerical experiments for the well known transport equation

probabilistic bounds for the matrix condition number with extended lanczos bidiagonalization reliable estimates for the condition number of a large sparse real matrix a are important in many applications to get an approximation for the condition number kappa a an approximation for the smallest singular value is needed standard krylov subspaces are usually unsuitable for finding a good approximation to the smallest singular value therefore we study extended krylov subspaces which turn out to be ideal for the simultaneous approximation of both the smallest and largest singular value of a matrix first we develop a new extended lanczos bidiagonalization method with this method we obtain a lower bound for the condition number moreover the method also yields probabilistic upper bounds for kappa a the user can select the probability with which the upper bound holds as well as the ratio of the probabilistic upper bound and the lower bound

power law subordinacy and singular spectra i half line operators we study hausdorff dimensional spectral properties of certain whole line quasiperiodic discrete schrodinger operators by using the extension of the gilbert pearson subordinacy theory that we previously developed in 19

a new iterative method for solving large scale lyapunov matrix equations in this paper we propose a new projection method to solve large scale continuous time lyapunov matrix equations the new approach projects the problem onto a much smaller approximation space generated as a combination of krylov subspaces in a and a 1 the reduced problem is then solved by means of a direct lyapunov scheme based on matrix factorizations the reported numerical results show the competitiveness of the new method compared to a state of the art approach based on the factorized alternating direction implicit iteration

maximum minimum and optimal mutation rates in dynamic environments we analyze the dynamics of the parallel mutation selection quasispecies model with a changing environment for an environment with the sharp peak fitness function in which the most fit sequence changes by k spin flips every period t we find analytical expressions for the minimum and maximum mutation rates for which a quasispecies can survive valid in the limit of large sequence size we find an asymptotic solution in which the quasispecies population changes periodically according to the periodic environmental change in this state we compute the mutation rate that gives the optimal mean fitness over a period we find that the optimal mutation rate per genome k t is independent of genome size a relationship which is observed across broad groups of real organisms

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

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

spectral methods in time for parabolic problems a pseudospectral explicit scheme for solving linear periodic parabolic problems is described it has infinite accuracy both in time and in space the high accuracy is achieved while the time resolution parameter m m o 1 delta t for time marching algorithm and the space resolution parameter n n o 1 delta x must satisfy m o n 1 varepsilon varepsilon 0 compared to the common stability condition m o n 2 which must be satisfied in any explicit finite order time algorithm

solvable biological evolution models with general fitness functions and multiple mutations in parallel mutation selection scheme in a recent paper phys rev e 69 046121 2004 we used the suzuki trottere formalism to study a quasispecies biological evolution model in a parallel mutation selection scheme with a single peak fitness function and a point mutation in the present paper we extend such a study to evolution models with more general fitness functions or multiple mutations in the parallel mutation selection scheme we give some analytical equations to define the error thresholds for some general cases of mean field like or symmetric mutation schemes and fitness functions we derive some equations for the dynamics in the case of a point mutation and polynomial fitness functions we derive exact dynamics for two point mutations asymmetric mutations and the four value spin model with a single peak fitness function the same method is applied for the model with a royal road fitness function we derive the steady state distribution for the single peak fitness function

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

short recurrences for computing extended krylov bases for hermitian and unitary matrices it is well known that the projection of a matrix a a onto a krylov subspace span left mathbf h a mathbf h a 2 mathbf h ldots a k 1 mathbf h right h ah a2h ak 1h with a in mathbb c n times n a cn n and mathbf h in mathbb c n h cn results in a hessenberg matrix we show that the projection of the matrix a a onto an extended krylov subspace which is of the form span left a k r mathbf h ldots a 2 mathbf h a 1 mathbf h mathbf h a mathbf h a 2 mathbf h ldots a k ell mathbf h right a krh a 2h a 1h h ah a2h aklh is a matrix of so called extended hessenberg form which can be characterized uniquely by its qr qr factorization this qr qr factorization will be presented by means of a pattern of 2 times 2 2 2 unitary rotations we will show how this rotation pattern leads to new insights and allows to elegantly predict the structure of the matrix in case a a is hermitian or unitary this extended hessenberg matrix is banded and structured allowing the design of short recurrence relations for the unitary case coupled two term recurrence relations are derived of which the coefficients capture all information necessary for a sparse factorization of the corresponding extended hessenberg matrix

quasispecies theory for multiple peak fitness landscapes we use a path integral representation to solve the eigen and crow kimura molecular evolution models for the case of multiple fitness peaks with arbitrary fitness and degradation functions in the general case we find that the solution to these molecular evolution models can be written as the optimum of a fitness function with constraints enforced by lagrange multipliers and with a term accounting for the entropy of the spreading population in sequence space the results for the eigen model are applied to consider virus or cancer proliferation under the control of drugs or the immune system

delocalization in random polymer models a random polymer model is a one dimensional jacobi matrix randomly composed of two finite building blocks if the two associated transfer matrices commute the corresponding energy is called critical such critical energies appear in physical models an example being the widely studied random dimer model it is proven that the lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport showing that it is almost surely overdiffusive even though the models are known to have pure point spectrum with exponentially localized eigenstates for almost every configuration of the polymers furthermore the level spacing is shown to be regular at the critical energy

quasispecies model of evolution with migration we consider the model of asexual evolution with migration which was proposed by waclaw et al phys rev lett 105 268101 2010 prltao0031 900710 1103 physrevlett 105 268101 this model setting is based on the standard mutation scheme from the quasispecies theory but with replicators moving from one habitat to another the primary goal is to solve exactly the infinite population genome length version of the model for the independent random distribution of fitnesses considered in the original paper moreover we propose the analytical solution for the single peak and the symmetric fitness landscape our analytical solution is exact at the limit of large n we found two phases the correlated phase with the identical distributions of mutations in both habitats and the uncorrelated phase where the second habitat is choosing another peak of distribution in sequence space compared to the peak in the first habitat

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

solvable biological evolution model with a parallel mutation selection scheme based on the connection between a quantum spin model and an asexual biological evolution model with a single peak fitness function in parallel mutation selection scheme we solve exactly both static and dynamics of the evolution model we find that relaxation in such a parallel scheme is faster than that in a connected scheme of eigen model our method can also be extended to other fitness functions

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

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

low rank methods for a class of generalized lyapunov equations and related issues in this paper we study possible low rank solution methods for generalized lyapunov equations arising in bilinear and stochastic control we show that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations since the theoretical tools strongly make use of a connection to the standard linear lyapunov equation we can even extend the result to the d dimensional case described by a tensorized linear system of equations we further provide some reasonable extensions of some of the most frequently used linear low rank solution techniques such as the alternating directions implicit adi iteration and the krylov plus inverted krylov k pik method by means of some standard numerical examples used in the area of bilinear model order reduction we will show the efficiency of the new methods

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

spectral averaging techniques for jacobi matrices spectral averaging techniques for one dimensional discrete schrodinger operators are revisited and extended in particular simultaneous averaging over several parameters is discussed special focus is put on proving lower bounds on the density of the averaged spectral measures these wegner type estimates are used to analyze stability properties for the spectral types of jacobi matrices under local perturbations

schwinger boson formulation and solution of the crow kimura and eigen models of quasispecies theory we express the crow kimura and eigen models of quasispecies theory in a functional integral representation we formulate the spin coherent state functional integrals using the schwinger boson method in this formulation we are able to deduce the long time behavior of these models for arbitrary replication and degradation functions we discuss the phase transitions that occur in these models as a function of mutation rate we derive for these models the leading order corrections to the infinite genome length limit

a reflection on the implicitly restarted arnoldi method for computing eigenvalues near a vertical line abstract in this article we will study the link between a method for computing eigenvalues closest to the imaginary axis and the implicitly restarted arnoldi method the extension to computing eigenvalues closest to a vertical line is straightforward by incorporating a shift without loss of generality we will restrict ourselves here to computing eigenvalues closest to the imaginary axis in a recent publication meerbergen and spence discussed a new approach for detecting purely imaginary eigenvalues corresponding to hopf bifurcations which is of interest for the stability of dynamical systems the novel method is based on inverse iteration inverse power method applied on a lyapunov like eigenvalue problem to reduce the computational overhead significantly a projection was added this method can also be used for computing eigenvalues of a matrix pencil near a vertical line in the complex plane we will prove in this paper that the combination of inverse iteration with the projection step is equivalent to sorensen s implicitly restarted arnoldi method utilizing well chosen shifts

fields of values and the adi method for non normal matrices the convergence of an iterative method is usually measured either by the spectral radius or by a matrix norm e g the spectral norm of the corresponding iteration operator however for nonnormal matrices each one of these approaches has its drawbacks the first one only describes the asymptotic rate of convergence the second one usually produces estimates which are much too pessimistic m eiermann has suggested using the field of values to judge the performance of an iterative method his results are generalized in this paper by upper bounds for the error reduction in the adi method based on the field of values in particular for the stationary adi method i e using the same parameters in each step we obtain the error estimate xm x 2 c max w v rm min w h rm x0 x 2 for the commutative case i e if hv vh is fulfilled furthermore we show that the field of values of a tridiagonal matrix becomes minimal if it is in complex symmetric form this result can be generalized to kronecker sums of tridiagonal matrices which occur in many practical applications e g discretized elliptic boundary value problems finally we present numerical experiments which show that the resulting estimates are useful for an a priori prediction of the convergence behavior and suggest the use of the field of values as a basis for the calculation of optimal adi parameters

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

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

anderson localization non linearity and stable genetic diversity in many models of genotypic evolution the vector of genotype populations satisfies a system of linear ordinary differential equations this system of equations models a competition between differential replication rates fitness and mutation mutation operates as a generalized diffusion process on genotype space in the large time asymptotics the replication term tends to produce a single dominant quasi species unless the mutation rate is too high in which case the asymptotic population becomes de localized we introduce a more macroscopic picture of genotypic evolution wherein a random fitness term in the linear model produces features analogous to anderson localization when coupled with density dependent non linearities which limit the population of any given genotype we obtain a model whose large time asymptotics display stable genotypic diversity

finite genome length corrections for the mean fitness and gene probabilities in evolution models using the hamilton jacobi equation approach to study genomes of length l we obtain 1 l corrections for the steady state population distributions and mean fitness functions for horizontal gene transfer model as well as for the diploid evolution model with general fitness landscapes our numerical solutions confirm the obtained analytic equations our method could be applied to the general case of nonlinear markov models

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

spin coherent state representation of the crow kimura and eigen models of quasispecies theory we present a spin coherent state representation of the crow kimura and eigen models of biological evolution we deal with quasispecies models where the fitness is a function of hamming distances from one or more reference sequences in the limit of large sequence length n we find exact expressions for the mean fitness and magnetization of the asymptotic quasispecies distribution in symmetric fitness landscapes the results are obtained by constructing a path integral for the propagator on the coset su 2 u 1 and taking the classical limit the classical limit gives a hamiltonian function on a circle for one reference sequence and on the product of 2m 1 circles for m reference sequences we apply our representation to study the schuster swetina phenomena where a wide lower peak is selected over a narrow higher peak the quadratic landscape with two reference sequences is also analyzed specifically and we present the phase diagram on the mutation fitness parameter phase space furthermore we use our method to investigate more biologically relevant system a model of escape from adaptive conflict through gene duplication and find three different phases for the asymptotic population distribution

value distribution and spectral analysis of differential operators the theory of value distribution is applied through a study of boundary value distribution for the weyl titchmarsh m function to the spectral analysis of differential operators on the half line particular consequences are the theoretical underpinning of a numerical approach to spectral analysis of differential operators exhibiting absolutely continuous or singular spectra and new identities for the spectral measure in terms of solutions of the associated differential equation

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

optimal mutation rates in dynamic environments the eigen model we consider the eigen quasispecies model with a dynamic environment for an environment with sharp peak fitness in which the most fit sequence moves by k spin flips each period t we find an asymptotic stationary state in which the quasispecies population changes regularly according to the regular environmental change from this stationary state we estimate the maximum and the minimum mutation rates for a quasispecies to survive under the changing environment and calculate the optimum mutation rate that maximizes the population growth interestingly we find that the optimum mutation rate in the eigen model is lower than that in the crow kimura model and at their optimum mutation rates the corresponding mean fitness in the eigen model is lower than that in the crow kimura model suggesting that the mutation process which occurs in parallel to the replication process as in the crow kimura model gives an adaptive advantage under changing environment

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

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

stability of singular spectral types under decaying perturbations we look at invariance of a e boundary condition spectral behavior under perturbations w of half line continuum or discrete schrodinger operators we extend the results of del rio simon stolz from compactly supported w s to suitable short range w we also discuss invariance of the local hausdorff dimension of spectral measures under such perturbations

eigen model as a quantum spin chain exact dynamics we map the eigen model of biological evolution naturwissenschaften 58 465 1971 into a quantum spin model with non hermitian hamiltonian based on such a connection we derive exact relaxation periods for the eigen model to approach static energy landscape from various initial conditions we also study a simple case of dynamic fitness function

a low rank in time approach to pde constrained optimization the solution of time dependent pde constrained optimization problems is a challenging task in numerical analysis and applied mathematics all at once discretizations and corresponding solvers provide efficient methods to robustly solve the arising discretized equations one of the drawbacks of this approach is the high storage demand for the vectors representing the discrete space time cylinder here we introduce a low rank in time technique that exploits the low rank nature of the solution the theoretical foundations for this approach originate in the numerical treatment of matrix equations and can be carried over to pde constrained optimization we illustrate how three different problems can be rewritten and used within a low rank krylov subspace solver with appropriate preconditioning

numerical methods for differential linear matrix equations via krylov subspace methods abstract in the present paper we present some numerical methods for computing approximate solutions to some large differential linear matrix equations in the first part of this work we deal with differential generalized sylvester matrix equations with full rank right hand sides using a global galerkin and a norm minimization approaches in the second part we consider large differential lyapunov matrix equations with low rank right hand sides and use the extended global arnoldi process to produce low rank approximate solutions we give some theoretical results and present some numerical examples

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

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

two parameter spectral averaging and localization for non monotonic random schrodinger operators we prove exponential localization at all energies for two types of one dimensional random schr6dinger operators the poisson model and the random displacement model as opposed to anderson type models these operators are not monotonic in the random parameters therefore the classical one parameter version of spectral averaging as used in localization proofs for anderson models breaks down we use the new method of two parameter spectral averaging and apply it to the poisson as well as the displacement case in addition we apply results from inverse spectral theory which show that two parameter spectral averaging works for sufficiently many energies all but a discrete set to conclude localization at all energies

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

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