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

projected nonsymmetric algebraic riccati equations and refining estimates of invariant and deflating subspaces we consider the numerical solution of the projected nonsymmetric algebraic riccati equations or their associated sylvester equations via newton s method arising in the refinement of estimates of invariant or deflating subspaces for a large and sparse real matrix a or pencil a b the engine of the method is the inversion of the matrix p 2 p 2 a i n or p l 2 p l 2 a b for some orthonormal p 2 or p l 2 from r n n m making use of the structures in a or a b and the sherman morrison woodbury formula our algorithms are efficient under appropriate assumptions as shown in our error analysis and illustrated by numerical examples

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

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

positivity of the universal pairing in 3 dimensions associated to a closed oriented surface s is the complex vector space with basis the set of all compact oriented 3 manifolds which it bounds gluing along s defines a hermitian pairing on this space with values in the complex vector space with basis all closed oriented 3 manifolds the main result in this paper is that this pairing is positive i e that the result of pairing a nonzero vector with itself is nonzero this has bearing on the question of what kinds of topological information can be extracted in principle from unitary 2 1 dimensional tqfts the proof involves the construction of a suitable complexity function c on all closed 3 manifolds satisfying a gluing axiom which we call the topological cauchy schwarz inequality namely that c ab max c aa c bb for all a b which bound s with equality if and only if a b the complexity function c involves input from many aspects of 3 manifold topology and in the process of establishing its key properties we obtain a number of results of independent interest for example we show that when two finite volume hyperbolic 3 manifolds are glued along an incompressible acylindrical surface the resulting hyperbolic 3 manifold has minimal volume only when the gluing can be done along a totally geodesic surface this generalizes a similar theorem for closed hyperbolic 3 manifolds due to agol storm thurston

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

using spectral discretisation for the optimal ℋ 2 design of time delay systems the stabilisation and robustification of a time delay system is the topic of this article more precisely we want to minimise the 2 norm of the transfer function corresponding to the class of linear time invariant input output systems with fixed time delays in the states due to the presence of the delays the transfer function is a nonrational nonlinear function and the classical procedure which involves solving lyapunov equations is no longer applicable we therefore propose an approach based on a spectral discretisation applied to a reformulation of the time delay system as an infinite dimensional standard linear system in this way we obtain a large delay free system which serves as an approximation to the original time delay system and which allows the application of standard 2 norm optimisation techniques we give an interpretation of this approach in the frequency domain and relate it to the approximation of the nonlinear terms in the time delay transfer function by means of a rational func

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

block arnoldi based methods for large scale discrete time algebraic riccati equations in the present paper we present block arnoldi based methods for the computation of low rank approximate solutions of large discrete time algebraic riccati equations dare the proposed methods are projection methods onto block or extended block krylov subspaces we give new upper bounds for the norm of the error obtained by applying these block arnoldi based processes we also introduce the newton method combined with the block arnoldi algorithm and present some numerical experiments with comparisons between these methods

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

the existence of least area surfaces in 3 manifolds this paper presents a new and unified approach to the existence theorems for least area surfaces in 3 manifolds introduction a surface f smoothly embedded or immersed in a riemannian manifold m is minimal if it has mean curvature zero at all points it is a least area surface in a class of surfaces if it has finite area which realizes the infimum of all possible areas for surfaces in this class the connection between these two ideas is that a surface which is of least area in a reasonable class of surfaces must be minimal the converse is false minimal surfaces are in general only critical points for the area function there are close analogies between these two concepts and the theory of geodesics in a riemannian manifold minimal surfaces correspond to geodesics and least area surfaces correspond to geodesic arcs or closed geodesics which have shortest length in some class of paths any geodesic a in a riemannian manifold m has the property that it is locally shortest i e if p and q are nearby points on a then the subarc of a which joins p and q is the shortest path in m from p to q it can also be proved that minimal surfaces are locally of least area but the proof is difficult and involves substantial knowledge of the theory of partial differential equations there are now a large number of theorems asserting the existence of surfaces of least area in various classes surfaces of this type have become an important tool in 3 dimensional topology in this paper we present a new approach to the proofs of these existence theorems this yields a simplified and unified method for the proof of the existence of minimal surfaces in 3 dimensional riemannian manifolds in 1930 douglas do and rado ra independently showed that a simple closed curve in ri which bounds a disk of finite area bounds a disk of least possible area this result was extended by morrey moi in 1948 to a general class of riemannian manifolds the homogeneously regular manifolds a class which includes all closed manifolds work of osserman 0 and gulliver g later showed that least area disks in 3 manifolds were immersed in their interiors more recently there has been a series of new existence results for closed surfaces of least area in manifolds of any dimension it follows from work of sacks and uhlenbeck s ui that if m is closed and 7r2 m is nonzero then there is an essential map of the 2 sphere into m which has least area among all essential maps sacks and uhlenbeck s uii and schoen and yau s y independently showed that if f f m is a map of a closed orientable surface f not the 2 sphere into a closed riemannian manifold m such received by the editors june 23 1986 1980 mathematics subject classification 1985 revision primary 53a10 secondary 57m35 the first author was supported by nsf postdoctoral fellowship dms84 14097 1988 american mathematical society 0002 9947 88 1 00 25 per page

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

radi a low rank adi type algorithm for large scale algebraic riccati equations this paper introduces a new algorithm for solving large scale continuous time algebraic riccati equations care the advantage of the new algorithm is in its immediate and efficient low rank formulation which is a generalization of the cholesky factored variant of the lyapunov adi method we discuss important implementation aspects of the algorithm such as reducing the use of complex arithmetic and shift selection strategies we show that there is a very tight relation between the new algorithm and three other algorithms for care previously known in the literature all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters they are algorithmically different descriptions of the same approximation sequence to the riccati solution

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

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

intersection numbers of twisted cycles associated with the selberg integral and an application to the conformal field theory intersection numbers of twisted or loaded cycles associated with the selberg integral are studied in particular the self intersection number of the cycle which is invariant under the action of the symmetric group is expressed by the product of trigonometric functions this formula reproduces the four point correlation functions in the conformal field theory calculated by dotsenko fateev in 3 in our study a compact non singular model terada model of the configuration space of n 3 points on the real projective line and a q analogue of the chu vadermonde formula for the hypergeometric series play a crucial role intersection numbers of the corresponding cocycles are also studied

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

monotone convergence of the extended krylov subspace method for laplace stieltjes functions of hermitian positive definite matrices abstract the extended krylov subspace method is known to be very efficient in many cases in which one wants to approximate the action of a matrix function f a on a vector b in particular when f belongs to the class of laplace stieltjes functions we prove that the euclidean norm of the error decreases strictly monotonically in this situation when a is hermitian positive definite similar results are known for the polynomial lanczos method for f a b and we demonstrate how the techniques of proof used in the polynomial krylov case can be transferred to the extended krylov case

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

fast tensor product solvers for optimization problems with fractional differential equations as constraints fractional differential equations have recently received much attention within computational mathematics and applied science and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms an optimization problem with constraints given by fractional differential equations is considered which in its discretized form leads to a high dimensional tensor equation to reduce the computation time and storage the solution is sought in the tensor train format we compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned krylov solvers or tailored alternating schemes the competitiveness of these approaches is presented using several examples with constant and variable coefficients

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

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

lower bounds on volumes of hyperbolic haken 3 manifolds we prove a volume inequality for 3 manifolds having c o metrics bent along a surface and satisfying certain curvature conditions the result makes use of perelman s work on the ricci flow and geometrization of closed 3 manifolds corollaries include a new proof of a conjecture of bonahon about volumes of convex cores of kleinian groups improved volume estimates for certain haken hyperbolic 3 manifolds and a lower bound on the minimal volume of orientable hyperbolic 3 manifolds an appendix compares estimates of volumes of hyperbolic 3 manifolds drilled along a closed embedded geodesic with experimental data

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

on the eigenvalue decay of solutions to operator lyapunov equations this paper is concerned with the eigenvalue decay of the solution to operator lyapunov equations with right hand sides of finite rank we show that the kth generalized eigenvalue decays exponentially in root k provided that the involved operator a generates an exponentially stable analytic semigroup and a is either self adjoint or diagonalizable with its eigenvalues contained in a strip around the real axis numerical experiments with discretizations of 1d and 2d pde control problems confirm this decay c 2014 elsevier b v all rights reserved

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

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

the asymptotics of quantum max flow min cut the quantum max flow min cut conjecture relates the rank of a tensor network to the minimum cut in the case that all tensors in the network are identical in calegari et al j am math soc 23 1 107 188 2010 this conjecture was shown to be false in cui et al j math phys 57 062206 2016 by an explicit counter example here we show that the conjecture is almost true in that the ratio of the quantum max flow to the quantum min cut converges to 1 as the dimension n of the degrees of freedom on the edges of the network tends to infinity the proof is based on estimating moments of the singular values of the network we introduce a generalization of rainbow diagrams to tensor networks to estimate the dominant diagrams a direct comparison of second and fourth moments lower bounds the ratio of the quantum max flow to the quantum min cut by a constant to show the tighter bound that the ratio tends to 1 we consider higher moments in addition we show that the limiting moments as n agree with that in a different ensemble where tensors in the network are chosen independently this is used to show that the distributions of singular values in the two different ensembles weakly converge to the same limiting distribution we present also a numerical study of one particular tensor network which shows a surprising dependence of the rank deficit on n mod 4 and suggests further conjecture on the limiting behavior of the rank

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

quantum max flow min cut the classical max flow min cut theorem describes transport through certain idealized classical networks we consider the quantum analog for tensor networks by associating an integral capacity to each edge and a tensor to each vertex in a flow network we can also interpret it as a tensor network and more specifically as a linear map from the input space to the output space the quantum max flow is defined to be the maximal rank of this linear map over all choices of tensors the quantum min cut is defined to be the minimum product of the capacities of edges over all cuts of the tensor network we show that unlike the classical case the quantum max flow min cut conjecture is not true in general under certain conditions e g when the capacity on each edge is some power of a fixed integer the quantum max flow is proved to equal the quantum min cut however concrete examples are also provided where the equality does not hold we also found connections of quantum max flow min cut with entropy of entanglement and the quantum satisfiability problem we speculate that the phenomena revealed may be of interest both in spin systems in condensed matter and in quantum gravity

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

recursion relations for the extended krylov subspace method abstract the evaluation of matrix functions of the form f a v where a is a large sparse or structured symmetric matrix f is a nonlinear function and v is a vector is frequently subdivided into two steps first an orthonormal basis of an extended krylov subspace of fairly small dimension is determined and then a projection onto this subspace is evaluated by a method designed for small problems this paper derives short recursion relations for orthonormal bases of extended krylov subspaces of the type k m mi 1 a span a m 1 v a 1 v v a v a mi v m 1 2 3 with i a positive integer and describes applications to the evaluation of matrix functions and the computation of rational gauss quadrature rules

on optimality of approximate low rank solutions of large scale matrix equations abstract in this paper we discuss some optimality results for the approximation of large scale matrix equations in particular this includes the special case of lyapunov and sylvester equations respectively we show a relation between the iterative rational krylov algorithm and a riemannian optimization method which recently has been shown to locally minimize a certain energy norm of the underlying lyapunov operator moreover we extend the results for a more general setting leading to a slight modification of irka by means of some numerical test examples we show the efficiency of the proposed methods

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

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

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

vector space of feynman integrals and multivariate intersection numbers feynman integrals obey linear relations governed by intersection numbers which act as scalar products between vector spaces we present a general algorithm for the construction of multivariate intersection numbers relevant to feynman integrals and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections and to directly derive functional equations fulfilled by the latter we apply it to the decomposition of a few feynman integrals at one and two loops as first steps toward potential applications to generic multiloop integrals the proposed method can be more generally employed for the derivation of contiguity relations for special functions admitting multifold integral representations

heegaard splittings the virtually haken conjecture and property τ the behaviour of finite covers of 3 manifolds is a very important but poorly understood topic there are three increasingly strong conjectures in the field that have remained open for over twenty years the virtually haken conjecture the positive virtual b1 conjecture and the virtually fibred conjecture any of these would have profound ramifications for 3 manifold theory in this paper we explore the interaction of these conjectures with the following seemingly unrelated areas eigenvalues of the laplacian and heegaard splittings

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

matrix product states and the quantum max flow min cut conjectures in this note we discuss the geometry of matrix product states with periodic boundary conditions and provide three infinite sequences of examples where the quantum max flow is strictly less than the quantum min cut in the first we fix the underlying graph to be a 4 cycle and verify a prediction of hastings that inequality occurs for infinitely many bond dimensions in the second we generalize this result to a 2d cycle in the third we show that the 2d cycle with periodic boundary conditions gives inequality for all d when all bond dimensions equal two namely a gap of at least 2d 2 between the quantum max flow and the quantum min cut

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

higher algebraic structures and quantization we derive quasi quantum groups in 2 1 dimensional topological field theory directly from the classical action and the path integral detailed computations are carried out for the chern simons theory with finite gauge group the principles behind our computations are presumably more general we extend the classical action in ad 1 dimensional topological theory to manifolds of dimension less thand 1 we then construct a generalized path integral which ind 1 dimensions reduces to the standard one and ind dimensions reproduces the quantum hilbert space in a 2 1 dimensional topological theory the path integral over the circle is the category of representations of a quasi quantum group in this paper we only consider finite theories in which the generalized path integral reduces to a finite sum new ideas are needed to extend beyond the finite theories treated here

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

an alternative extended block arnoldi method for solving low rank sylvester equations abstract projection methods based on krylov subspaces are an effective tool for solving low rank sylvester matrix equations in the large scale cases the initial problem is projected onto a sequence of nested subspaces at each step of the process a reduced sylvester equation whose coefficient matrices are restrictions of the original coefficient matrices to the projection subspaces are solved by a direct method in this paper after recalling some important properties about the extended block arnoldi process we propose an alternative approach that enables us to determine approximate solutions by solving projected and reduced sylvester equations whose coefficient matrices are restrictions of the inverse of the original matrices some numerical experiments are given in order to illustrate the performance of our approach

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

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

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

an extended block golub kahan algorithm for large algebraic and differential matrix riccati equations abstract in this paper we propose a new projection method to solve both large scale continuous time matrix riccati equations and differential matrix riccati equations the new approach projects the problem onto an extended block krylov subspace and gets a low dimensional equation we use the block golub kahan procedure to construct the orthonormal bases for the extended krylov subspaces for matrix riccati equations the reduced problem is then solved by means of a direct riccati scheme such as the schur method when we solve differential matrix riccati equations the reduced problem is solved by the backward differentiation formula bdf method and the obtained solution is used to build the low rank approximate solution of the original problem finally we give some theoretical results and present numerical experiments

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

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

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

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

intersection numbers of twisted cycles and the correlation functions of the conformal field theory the coefficients of the four point correlation functions calculated by dotsenko fateev are interpreted naturally in terms of the intersection numbers of twisted or loaded cycles this is an answer to the long standing problem of clarifying the meaning of such coefficients appearing in correlation functions

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

convergence analysis of projection methods for the numerical solution of large lyapunov equations the numerical solution of large scale continuous time lyapunov matrix equations is of great importance in many application areas assuming that the coefficient matrix is positive definite but not necessarily symmetric in this paper we analyze the convergence of projection type methods for approximating the solution matrix under suitable hypotheses on the coefficient matrix we provide new asymptotic estimates for the error matrix when a galerkin method is used in a krylov subspace numerical experiments confirm the good behavior of our upper bounds when linear convergence of the solver is observed

proof of the riemannian penrose inequality using the positive mass theorem we prove the riemannian penrose conjecture an important case of a con jecture 41 made by roger penrose in 1973 by defining a new flow of metrics this flow of metrics stays inside the class of asymptotically flat riemannian 3 manifolds with nonnegative scalar curvature which contain minimal spheres in particular if we consider a riemannian 3 manifold as a totally geodesic submanifold of a space time in the context of general relativity then outermost minimal spheres with total area a correspond to apparent horizons of black holes contributing a massa 16 scalar cur vature corresponds to local energy density at each point and the rate at which the metric becomes flat at infinity corresponds to total mass also called the adm mass the riemannian penrose conjecture then states that the total mass of an asymptotically flat 3 manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes the flow of metrics we define continuously evolves the original 3 metric to a schwarzschild 3 metric which represents a spherically symmetric black hole in vacuum we define the flow such that the area of the minimal spheres which flow outward and hence the mass contributed by the black holes in each of the metrics in the flow is constant and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing then since the total mass equals the mass of the black hole in a schwarzschild metric the riemannian penrose conjecture follows we also refer the reader to the beautiful work of huisken and ilmanen 30 who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole in sections 1 and 2 we motivate the problem discuss important quantities like total mass and horizons of black holes and state the positive mass theorem and the penrose conjecture for riemannian 3 manifolds in section 3 we give the proof of the riemannian penrose

continuation of probability density functions using a generalized lyapunov approach techniques from numerical bifurcation theory are very useful to study transitions between steady fluid flow patterns and the instabilities involved here we provide computational methodology to use parameter continuation in determining probability density functions of systems of stochastic partial differential equations near fixed points under a small noise approximation key innovation is the efficient solution of a generalized lyapunov equation using an iterative method involving low rank approximations we apply and illustrate the capabilities of the method using a problem in physical oceanography i e the occurrence of multiple steady states of the atlantic ocean circulation