dynamical system for computing the eigenvectors associated with the largest eigenvalue of a positive definite matrix a dynamical system for computing the largest eigenpair of a given positive matrix is introduced certain qualitative properties of the proposed system are analyzed in detail the authors results suggest that the weight bounding term in pca algorithms for extracting the first principal component could assume a class of forms

qualitative behavior of non abelian toda like flows in this paper we analyze and compare the long term behavior of certain non abelian generalizations of the integrable toda lattice flow in particular we show that these flows exhibit generalized sorting behavior analogous to the sorting behavior of the original tridiagonal toda lattice we show that one of these generalizations the block double bracket equation is gradient with respect to a normal metric and exactly solvable this provides a new solvable case of the double bracket system

an object oriented and quadrilateral mesh based solution adaptive algorithm for compressible multi fluid flows in this paper an object oriented and quadrilateral mesh based solution adaptive algorithm for the simulation of compressible multi fluid flows is presented the hllc scheme harten lax and van leer approximate riemann solver with the contact wave restored is extended to adaptively solve the compressible multi fluid flows under complex geometry on unstructured mesh it is also extended to the second order of accuracy by using muscl extrapolation the node edge and cell are arranged in such an object oriented manner that each of them inherits from a basic object a home made double link list is designed to manage these objects so that the inserting of new objects and removing of the existing objects nodes edges and cells are independent of the number of objects and only of the complexity of o 1 in addition the cells with different levels are further stored in different lists this avoids the recursive calculation of solution of mother non leaf cells thus high efficiency is obtained due to these features besides as compared to other cell edge adaptive methods the separation of nodes would reduce the memory requirement of redundant nodes especially in the cases where the level number is large or the space dimension is three five two dimensional examples are used to examine its performance these examples include vortex evolution problem interface only problem under structured mesh and unstructured mesh bubble explosion under the water bubble shock interaction and shock interface interaction inside the cylindrical vessel numerical results indicate that there is no oscillation of pressure and velocity across the interface and it is feasible to apply it to solve compressible multi fluid flows with large density ratio 1000 and strong shock wave the pressure ratio is 10 000 interaction with the interface

differential equations for roaming pseudospectra paths to extremal points and boundary tracking when studying the varepsilon pseudospectrum of a matrix one is often interested in computing the extremal points having maximum real part or modulus this is a crucial step for example when computing the distance to instability of a stable system using the property that the pseudospectrum is determined via perturbations by rank 1 matrices we derive differential equations on the manifold of normalized rank 1 matrices whose solutions tend to the critical rank 1 perturbations associated with the extremal points of locally maximum real part and modulus this approach also allows us to track the boundary contour of the pseudospectrum in a neighborhood of the extremal points the technique we propose is related to an idea recently developed by guglielmi and overton who derived discrete dynamical systems instead of the continuous ones we present the method turns out to be fast in comparison with those previously proposed in the literature and appears to be promising in dealing with large sparse problems

finite volume treatment of dispersion relation preserving and optimized prefactored compact schemes for wave propagation in developing suitable numerical techniques for computational aero acoustics the dispersion relation preserving drp scheme by tam and co workers and the optimized prefactored compact opc scheme by ashcroft and zhang have shown desirable properties of reducing both dissipative and dispersive errors these schemes originally based on the finite difference attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre determined formal orders of accuracy in the present study finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications linear and nonlinear wave equations with and without viscous dissipation have been adopted as the test problems highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases the performance of these approaches is documented for the linear wave equation there is no major difference between the drp and opc schemes for the nonlinear wave equations the finite volume version of both drp and opc schemes offers substantially better solutions in regions of high gradient or discontinuity

an h r adaptive approximate riemann solver for the euler equations in two dimensions a new adaptive strategy for solving the euler equations of compressible flow is presented the method of roe j comput phys 43 1981 pp 357 372 is extended into two dimensions for an arbitr

orthogonal eigensubspace estimation using neural networks we present a neural network nn approach for simultaneously estimating all or some of the orthogonal eigenvectors of a symmetric nonindefinite matrix corresponding to its repeated minimum in magnitude eigenvalue this problem has its origin in the constrained minimization framework and has extensive applications in signal processing we recast this problem into the nn framework by constructing an appropriate energy function which the nn minimizes the nn is of feedback type with the neurons having sigmoidal activation function the proposed approach is analyzed to characterize the nature of the minimizers of the energy function the main result is that the matrix w is a minimizer of the energy function if and only if the columns of w are the orthogonal eigenvectors with a given norm corresponding to the smallest eigenvalue of the given matrix further all minimizers are global minimizers bounds on the integration time step that is required to numerically solve the system of differential equations which define the dynamics of the nn have also been derived results of computer simulations are presented to support our analysis

singular value decomposition via gradient and self equivalent flows the task of finding the singular value decomposition svd of a finite dimensional complex linear operator is here addressed via gradient flows evolving on groups of complex unitary matrices and associated self equivalent flows the work constitutes a generalization of that of brockett on the diagonalization of real symmetric matrices via gradient flows on orthogonal matrices and associated isospectral flows it complements results of symes chu and other authors on continuous analogs of the classical qr algorithm as well as earlier work by the authors on svd via gradient flows on positive definite matrices

recurrent neural networks for computing pseudoinverses of rank deficient matrices three recurrent neural networks are presented for computing the pseudoinverses of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of moore penrose inversion under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a pseudoinverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a pseudoinverse matrix and a lagrangian matrix with constant connections all three recurrent neural networks are also composed of a number of independent subnetworks corresponding to the rows or columns of a pseudoinverse the proposed recurrent neural networks are shown to be capable of computing the pseudoinverses of rank deficient matrices

the constrained newton method on a lie group and the symmetric eigenvalue problem the newton method for estimating a critical point of a real function is formulated in a coordinate free manner on an arbitrary lie group convergence proofs for the numerical method are given an application of the general approach to computing the eigenvalues of a symmetric matrix is given and the resultant algorithm is compared with the classical shifted qr algorithm properties of the method described suggest that it is of interest for certain computations in online and adaptive environments

vector fields global flows and the calculus of pulses the point of this paper is to show a remarkable property of functional expansions in the form of iterated integrals in capturing the dynamics of digital computation when realized in the form of pulse spike driven controlled smooth dynamical systems the key features of the structure thus captured by the formalism are a the combinatorics of spike trains in various input channels and their effect on the discrete time dynamics of the realized automaton and b the relative timing precedence of the spike trains in various input channels we do this by providing globally valid functional expansions for a certain kind of hybrid dynamical system when subject to inputs which resemble pulse trains we give these functional expansions in the form of a product of a nonlinear holonomy factor and a local fliess functional expansion we also obtain a computational scheme to determine the response of finite automata to input sequences which is meant as a substitute for the naive iteration of the next state maps we obtain this by embedding finite automata in suitable controlled dynamical systems on the special orthogonal group incidentally we show that our realization of automata can also be modified to realize hybrid machines in which a finite automaton is interacting with an external system some of whose analog variables are converted into interspike time intervals which in turn are simultaneously tokenized and acted upon by the modified realization of the finite automaton the latter is achieved intrinsically by exploiting the dynamics of the realization

a trace inequality with a subtracted term abstract for fixed real or complex matrices a and b the well known von neumann trace inequality identifies the maximum of tr uavb z sfnc as u and v range over the unitary group the maximum being a bilinear expression in the singular values of a and b this paper establishes the analogue of this inequality for real matrices a and b when u and v range over the proper real orthogonal group the maximum is again a bilinear expression in the singular values but there is a subtracted term when a and b have determinants of opposite sign

low rank differential equations for hamiltonian matrix nearness problems for a hamiltonian matrix with purely imaginary eigenvalues we aim to determine the nearest hamiltonian matrix such that some or all eigenvalues leave the imaginary axis conversely for a hamiltonian matrix with all eigenvalues lying off the imaginary axis we look for a nearest hamiltonian matrix that has a pair of imaginary eigenvalues the hamiltonian matrices can be allowed to be complex or restricted to be real such hamiltonian matrix nearness problems are motivated by applications such as the analysis of passive control systems they are closely related to the problem of determining extremal points of hamiltonian pseudospectra we obtain a characterization of optimal perturbations which turn out to be of low rank and are attractive stationary points of low rank differential equations that we derive we use a two level approach where in the inner level we determine extremal points of the hamiltonian varepsilon pseudospectrum for a given varepsilon by following the low rank differential equations into a stationary point and on the outer level we optimize for varepsilon this permits us to give fast algorithms exhibiting quadratic convergence for solving the considered hamiltonian matrix nearness problems

applying compactness constraints to differential traveltime tomography tomographic imaging problems are typically ill posed and often require the use of regularization techniques to guarantee a stable solution minimization of a weighted norm of model length is one commonly used secondary constraint tikhonov methods exploit low order differential operators to select for solutions that are small flat or smooth in one or more dimensions this class of regularizing functionals may not always be appropriate particularly in cases where the anomaly being imaged is generated by a nonsmooth spatial process time lapse imaging of flow induced velocity anomalies is one such case flow features are often characterized by spatial compactness or connectivity by performing inversions on differenced arrival time data the properties of the time lapse feature can be directly constrained we develop a differential traveltime tomography algorithm whichselects for compact solutions i e models with a minimum area of support through application of model space iteratively reweighted leas

nonholonomic double bracket equations and the gauss thermostat in this rapid communication we consider certain equations that arise from imposing a constant kinetic energy constraint on a one dimensional set of oscillators this is a nonlinear nonholonomic constraint on these oscillators and the dynamics are consistent with gauss s law of least constraint dynamics of this sort are of interest in nonequilibrium molecular dynamics we show that under certain choices of external potential these equations give rise to a generalization of the so called double bracket equations which are of interest in studying gradient flows and integrable systems such as the toda lattice in the case of harmonic potentials the flow is described by a symmetric bracket and periodic solutions are obtained

conservative smoothing on an adaptive quadrilateral grid the lax wendroff scheme can be freed of spurious oscillations by introducing conservative smoothing in this paper the approach is first tested in 1 d modeling equations and then extended to multidimensional flows by the finite volume method the scheme is discretized by a space splitting method on an adaptive quadrilateral grid the artificial viscosity coefficients in the conservative smoothing step are specially designed to capture slipstreams and vortices algorithms are programmed using a vectorizable data structure under which not only the flow solver but also the adaptation procedure is well vectorized the good resolution and high efficiency of the approach are demonstrated in calculating both unsteady and steady compressible flows with either weak or strong shock waves

a box scheme for transcritical flow the accurate computer simulation of river and pipe flow is of great importance in the design of urban drainage networks the use of implicit numerical schemes allows the time step to be chosen on the basis of accuracy rather than stability offering a potential computational saving over explicit methods the highly successful box scheme is an implicit method which can be used to model a wide range of subcritical and supercritical flows however care must be taken over the modelling of transcritical flows since unless the correct internal boundary conditions are imposed the scheme becomes unstable the necessity of accurately tracking all the critical interfaces and treating them accordingly can be algorithmically complex and in practice the underlying mathematical model is often modified to ensure that the flow remains essentially subcritical such a modification however inevitably leads to additional errors and incorrect qualitative behaviour can be observed in this paper we show how the technique of residual distribution can be successfully implemented in order to accurately model unsteady transcritical flow without the need to know a priori which regions of the computational domain correspond to subcritical and supercritical flow when used in conjunction with a form of artificial smoothing the resulting method generates very high resolution results even for transcritical problems involving shocks as can be seen in the numerical results

evolution equations for continuous scale morphological filtering multiscale signal analysis has emerged as a useful framework for many computer vision and signal processing tasks morphological filters can be used to develop nonlinear multiscale operations that have certain advantages over linear multiscale approaches in that they preserve important signal features such as edges the authors discuss several nonlinear partial differential equations that model the scale evolution associated with continuous space multiscale morphological erosions dilations openings and closings these equations relate the rate of change of the multiscale signal ensemble as scale increases to a nonlinear operator acting on the space of signals the nonlinear operator is characterized by the shape and dimensionality of the structuring element used by the morphological operators generally taking the form of a nonlinear function of certain partial differential operators

dynamical methods for polar decomposition and inversion of matrices we show how one may obtain polar decomposition as well as inversion of fixed and time varying matrices using a class of nonlinear continuous time dynamical systems first we construct a dynamic system that causes an initial approximation of the inverse of a time varying matrix to flow exponentially toward the true time varying inverse using a time parameterized homotopy from the identity matrix to a fixed matrix with unknown inverse and applying our result on the inversion of time varying matrices we show how any positive definite fixed matrix may be dynamically inverted by a prescribed time without an initial guess at the inverse we then construct a dynamical system that solves for the polar decomposition factors of a time varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition as a byproduct this method gives another method of inverting time varying matrices finally using homotopy again we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time

on the universality of the probability distribution of the product b 1x of random matrices consider random matrices a of dimension m x m n drawn from an ensemble with probability density f tr aa with f x a given appropriate function break a b x into an m x m block b and the complementary m x n block x and define the random matrix z b 1 x we calculate the probability density function p z of the random matrix z and find that it is a universal function independent of f x the universal probability distribution p z is a spherically symmetric matrix variate t distribution universality of p z is essentially a consequence of rotational invariance of the probability ensembles we study as an application we study the distribution of solutions of systems of linear equations with random coefficients and extend a classic result due to girko

hessian riemannian gradient flows in convex programming in view of solving theoretically constrained minimization problems we investigate the properties of the gradient flows with respect to hessian riemannian metrics induced by legendre functions the first result characterizes hessian riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities giving a new motivation for the introduction of bregman type distances then the general evolution problem is introduced and global convergence is established under quasi convexity conditions with interesting refinements in the case of convex minimization some explicit examples of these gradient flows are discussed dual trajectories are identified and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints some convergence rate results are established in the case of a linear objective function several optimality characterizations of the orbits are given optimal path of viscosi

a sort jacobi algorithm for semisimple lie algebras the classical jacobi algorithm is extended to an unified lie algebraic approach the conventional jacobi algorithm minimize the distance to diagonality they reduce the off norm i e the sum of squares of off diagonal entries sorting the diagonal elements after each step would accelerate the convergence but there are difficulties to apply this sorting to the off norm that has to minimize using the gradient flow of a trace function it r w brockett dynamical systems that sort lists diagonalize matrices and solve linear programming problems linear algebra appl 146 79 91 1991 zbl 0719 90045 as a more appropriate distance measure matrices can be diagonalized and the eigenvalues can be simultaneously sorted in this paper the sort jacobi algorithm is extended to a large class of structured matrices besides the semisimple lie algebra cases such as e g the symmetric hermitian and skew symmetric eigenvalue problem and the real and complex singular value decomposition par the local quadratic convergence of the sort jacobi method is proved for the regular case but not for the case in which the eigenvalues or singular values occur in clusters for these more complicated irregular problem a future publication is announced par the results are illustrated considering the lie algebra of derivations of the complex octonions

non perturbative flow equations from continuous unitary transformations we use a novel parametrization of the flowing hamiltonian to show that the flow equations based on continuous unitary transformations as proposed by wegner can be implemented through a nonlinear partial differential equation involving one flow parameter and two system specific auxiliary variables the implementation is non perturbative as the partial differential equation involves a systematic expansion in fluctuations controlled by the size of the system rather than the coupling constant the method is applied to the lipkin model to construct a mapping which maps the non interacting spectrum onto the interacting spectrum to a very high accuracy this function is universal in the sense that the full spectrum for any large number of particles can be obtained from it in a similar way expectation values for a large class of operators can be obtained which also makes it possible to probe the structure of the eigenstates

an introduction to lie group integrators basics new developments and applications we give a short and elementary introduction to lie group methods a selection of applications of lie group integrators are discussed finally a family of symplectic integrators on cotangent bundles of lie groups is presented and the notion of discrete gradient methods is generalised to lie groups

spectral analysis of fokker planck and related operators arising from linear stochastic differential equations we study spectral properties of certain families of linear second order differential operators arising from linear stochastic differential equations we construct a basis in the hilbert space of square integrable functions using modified hermite polynomials and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed in particular we completely describe the spectrum of the fokker planck operator on an appropriate invariant subspace of rapidly decaying functions the eigenvalues of the fokker planck operator provide information about the speed of convergence of the corresponding probability distribution to steady state which is important for stochastic estimation and control applications we show that the operator families under consideration can be realized as solutions of differential equations in the double bracket form on an operator lie algebra which leads to a simple expression for the flow of their eigenfunctions

enhancing the resolving power of least squares inversion with active constraint balancing most geophysical inverse problems are solved using least squares inversion schemes with damping or smoothness constraints to improve stability and convergence rate since the lagrangian multiplier controls resolution and stability of the inverse problem we always want to use the optimum multiplier which is not easy to get and is usually obtained by experience or a time consuming optimization process we present a new regularization approach in which the lagrangian multiplier is set as a spatial variable at each parameterized block and automatically determined via the parameter resolution matrix and spread function analysis for highly resolvable parameters a small value of the lagrangian multiplier is assigned and vice versa this approach named active constraint balancing acb tries to balance the constraints of the least squares inversion according to sensitivity for a given problem so that it enhances the resolution as well as the stability of the inversion process we demonstrate the performa

numerical solution of isospectral flows in this paper we are concerned with the problem of solving numerically isospectral flows these flows are characterized by the differential equation l b l l l 0 l 0 where l 0 is a d x d symmetric matrix b l is a skew symmetric matrix function of l and b l is the lie bracket operator we show that standard runge kutta schemes fail in recovering the main qualitative feature of these flows that is isospectrality since they cannot recover arbitrary cubic conservation laws this failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

a theory for learning by weight flow on stiefel grassman manifold recently we introduced the concept of neural network learning on stiefel grassman manifold for multilayer perceptron like networks contributions of other authors have also appeared in the scientific literature about this topic this article presents a general theory for it and illustrates how existing theories may be explained within the general framework proposed here

on quadratic stability of systems with structured uncertainty this paper considers the problem of stability robustness with respect to a class of nonlinear time varying perturbations which are bounded in a component wise rather than aggregated manner a family of robustness bounds is parameterized in terms of a non singular symmetric matrix it is shown that the problem of computing the largest robustness bound over the set of non singular symmetric matrices can be approximated by a smooth minimization problem over a compact set a convergent algorithm for computing an optimal robustness bound is proposed in the form of a gradient flow

global convergence of nonlinear cascade flows with morse bott zero dynamics abstract we derive a sufficient condition that a flow captures the dynamics on an invariant submanifold this leads to a refinement of the lasalle invariance principle as a consequence we generalize a well known global asymptotic stability result of nonlinear cascade systems to show global convergence to a compact invariant set this includes the case where a globally asymptotically stable system is coupled to a morse bott flow

explicit magnus expansions for nonlinear equations in this paper we develop and analyse new explicit magnus expansions for the nonlinear equation y a t y y defined in a matrix lie group in particular integration methods up to order four are presented in terms of integrals which can be either evaluated exactly or replaced by conveniently adapted quadrature rules the structure of the algorithm allows us to change the step size and even the order during the integration process thus improving its efficiency several examples are considered including isospectral flows and highly oscillatory nonlinear differential equations

in memory of kenneth g wilson kenneth wilson had an enormous impact on the renormalization group and field theories in general i had the great pleasure to work in three fields to which he contributed essentially critical phenomena gauge invariance in duality and confinement and flow equations and similarity renormalization

isospectral flows on a class of finite dimensional jacobi matrices abstract we present a new matrix valued isospectral ordinary differential equation that asymptotically block diagonalizes n n zero diagonal jacobi matrices employed as its initial condition this o d e features a right hand side with a nested commutator of matrices and structurally resembles the double bracket o d e studied by r w brockett in 1991 we prove that its solutions converge asymptotically that the limit is block diagonal and above all that the limit matrix is defined uniquely as follows for n even a block diagonal matrix containing 2 2 blocks such that the super diagonal entries are sorted by strictly increasing absolute value furthermore the off diagonal entries in these 2 2 blocks have the same sign as the respective entries in the matrix employed as the initial condition for n odd there is one additional 1 1 block containing a zero that is the top left entry of the limit matrix the results presented here extend some early work of kac and van moerbeke

2d time lapse seismic tomography using an active time constraint atc approach a 2d seismic time lapse inversion approach can image the evolution of seismic velocities over time and space the forward modeling is based on solving the eikonal equation using a second order fast marching method wave paths are represented by fresnel volumes rather than by conventional rays this approach accounts for complex velocity models and has the advantage of considering the effects of wave frequency on velocity resolution the aim of time lapse inversion is to find changes in velocities of each cell in the model as a function of time each model along the time axis is called a reference space model this approach can be simplified into an inverse problem that seeks the optimum of several reference space models taken together using the approximation that the change in seismic velocity varies linearly in time between two subsequent reference models the method is demonstrated on a synthetic example that includes regularization in time in the cost function and reduces inversion artifacts associated with noise in the data by comparison with independent inversions at each time

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

lie algebraic aspects of the finite nonperiodic toda flows in this paper we review various aspects of nonperiodic full and tridiagonal toda flow theory in particular we consider gradient structures in the dynamics and geometry of these systems and we compare and contrast a number of formulations of the nonperiodic toda equations we describe the structure of these systems on general semisimple lie algebras in the case of the full kostant asymmetric toda flow we describe some of the background behind its integrability and explain the role of noncommutative integrability in its qualitative behavior we describe the relationship between the asymmetric toda flows and the symmetric and indefinite toda flows and show how one may conjugate from the full kostant toda flows to the full symmetric toda flows via a poisson map we describe briefly related toda systems such as nonabelian toda and the peakon flows

an ode based method for computing the distance of coprime polynomials to common divisibility the problem of computing the distance of two real coprime polynomials to the set of polynomials with a nontrivial greatest common divisor gcd appears in computer algebra signal processing and control theory it has been studied in the literature under the names approximate common divisor varepsilon gcd and distance to uncontrollability existing solution methods use different types of local optimization methods and require a user defined initial approximation in this paper we propose a new method that allows us to include constraints on the coefficients of the polynomials moreover the method proposed in the paper is more robust to the initial approximation than the newton type optimization methods available in the literature our approach consists of two steps 1 reformulate the problem as a problem of determining the structured distance to singularity of an associated sylvester matrix and 2 integrate a system of odes which describes the gradient associated to the functional to be minim

moyal implementation of flow equations a non perturbative approach to quantum many body systems we show how wegner s flow equations can be reformulated as ordinary differential equations through the use of the moyal bracket in finite dimensional hilbert spaces the introduction of the moyal bracket leads naturally to the identification of a small expansion parameter namely the inverse of the dimensionality of the space this expansion corresponds to a non perturbative treatment of the coupling constant in the case of infinite dimensional spaces plays the role of the small parameter and the moyal formulation then allows for a semi classical treatment of the flow equation we demonstrate these statements for the lipkin and dicke models as well as for the symmetric x4 and double well potentials

erratum addendum differential equations for roaming pseudospectra paths to extremal points and boundary tracking we correct an error that incurred in our paper published in siam j numer anal 49 2011 pp 1194 1209 and give new algorithms for tracking the boundary of the pseudospectrum which are suitable for large sparse matrices

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

statistical physics algorithms that converge in recent years there has been significant interest in adapting techniques from statistical physics in particular mean field theory to provide deterministic heuristic algorithms for obtaining app

asymptotic hamiltonian dynamics the toda lattice the three wave interaction and the non holonomic chaplygin sleigh abstract in this paper we discuss asymptotic stability in energy preserving systems which have an almost poisson structure in particular we consider a class of poisson systems which includes the toda lattice in standard hamiltonian systems one of course does not expect asymptotic stability the key here is the structure of the phase space of the poisson or almost poisson systems and the nature of the equilibria our systems are in some sense generalizations of the integrable toda lattice system but are not integrable in general as a particular example we point out an interesting connection between three mechanical two degree of freedom systems that exhibit asymptotic stability two of them are classical hamiltonian systems while the third is a non holonomic system non holonomic systems are generally energy preserving but not hamiltonian but the system analyzed here turns out to have a phase space which is the union of hamiltonian ones we also discuss various higher dimensional examples

universal computation and other capabilities of hybrid and continuous dynamical systems we explore the simulation and computational capabilities of hybrid and continuous dynamical systems the continuous dynamical systems considered are ordinary differential equations odes for hybrid systems we concentrate on models that combine odes and discrete dynamics e g finite automata we review and compare four such models from the literature notions of simulation of a discrete dynamical system by a continuous one are developed we show that hybrid systems whose equations can describe a precise binary timing pulse exact clock can simulate arbitrary reversible discrete dynamical systems defined on closed subsets of r n the simulations require continuous odes in ir2n with the exact clock as input all four hybrid systems models studied here can implement exact clocks we also prove that any discrete dynamical system in rn can be simulated by continuous odes in rt2n 1 we use this to show that smooth odes in ri3 can simulate arbitrary turing machines and hence possess the power of universal computation we use the famous asynchronous arbiter problem to distinguish between hybrid and continuous dynamical systems we prove that one cannot build an arbiter with devices described by a system of lipschitz odes on the other hand all four hybrid systems models considered can implement arbiters even if their odes are lipschitz

a dynamical systems approach to weighted graph matching graph matching is a fundamental problem that arises frequently in the areas of distributed control computer vision and facility allocation in this paper we consider the optimal graph matching problem for weighted graphs which is computationally challenging due the combinatorial nature of the set of permutations contrary to optimization based relaxations to this problem in this paper we develop a novel relaxation by constructing dynamical systems on the manifold of orthogonal matrices in particular since permutation matrices are orthogonal matrices with nonnegative elements we define two gradient flows in the space of orthogonal matrices the first minimizes the cost of weighted graph matching over orthogonal matrices whereas the second minimizes the distance of an orthogonal matrix from the finite set of all permutations the combination of the two dynamical systems converges to a permutation matrix which provides a suboptimal solution to the weighted graph matching problem finally our approach is shown to be promising by illustrating it on nontrivial problems

on explicitly solvable gradient systems of moser karmarkar type abstract a generalization and a classification of explicitly solvable gradient systems which appear in linear programming and neurodynamics are studied it is shown that nonlinear dynamical systems of moser karmarkar type generically take both a lax representation and a gradient form a related optimization problem on a simplex is discussed an integration procedure and a complete description of the invariant manifolds and the phase portrait of the dynamical systems of moser karmarkar type are also investigated

on computation of the stability radius for nonlinearly structured perturbations this paper considers the problem of finding a perturbation matrix with the least spectral norm such that a matrix valued function becomes singular where the dependence of the function on the perturbation is allowed to be nonlinear it is proved that such a problem can be approximated by a smooth unconstrained minimization problem with compact sublevel sets a computational procedure proposed based on this result is demonstrated to be effective in both linear and nonlinear cases

steepest descent algorithms for optimization under unitary matrix constraint in many engineering applications we deal with constrained optimization problems with respect to complex valued matrices this paper proposes a riemannian geometry approach for optimization of a real valued cost function t of complex valued matrix argument w under the constraint that w is an n times n unitary matrix we derive steepest descent sd algorithms on the lie group of unitary matrices u n the proposed algorithms move towards the optimum along the geodesics but other alternatives are also considered we also address the computational complexity and the numerical stability issues considering both the geodesic and the nongeodesic sd algorithms armijo step size 1 adaptation rule is used similarly to 2 but with reduced complexity the theoretical results are validated by computer simulations the proposed algorithms are applied to blind source separation in mimo systems by using the joint diagonalization approach 3 we show that the proposed algorithms outperform other widely used algorithms

an iterative method for computing robustness of polynomial stability we propose a method for computing the distance of a stable polynomial to the set of unstable ones both in the hurwitz and in the schur case the method is based on the reformulation of the problem as the structured distance to instability of a companion matrix associated to a polynomial we first introduce the structured e pseudospectrum of a companion matrix and write a system of ordinary differential equations which maximize the real part or the absolute value of elements of the structured e pseudospectrum and then exploit the knowledge of the derivative of the maximizers with respect to e to devise a quadratically convergent iteration furthermore we use a variant of the same odes to compute the boundary of structured pseudospectra and compare them to unstructured ones an extension to constrained perturbations is also considered

3 d traveltime computation using the fast marching method we present a fast algorithm for solving the eikonal equation in three dimensions based on the fast marching method the algorithm is of the order o n log n where n is the total number of grid points in the computational domain the algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain the method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity in addition the method resolves any overturning propagation wavefronts we begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details we then show examples of traveltime propagation through the seg eage salt model using point source and planewave initial conditions and analyze the error in constant velocity media the algorithm allows for any shape of the initial wavefront while a point source is the most commonly used initial condition initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another the algorithm presented here is designed for computing first arrival traveltimes nonetheless since it exploits the fast marching method for solving the eikonal equation we believe it is the fastest of all possible consistent schemes to compute first arrivals

on solving an isospectral flow in this paper we expand the solution of the matrix ordinary differential system originally due to bloch and iserles of the form x n x 2 t 0 x 0 x 0 sym n n s o n where sym n denotes the space of real n n symmetric matrices and s o n denotes the lie algebra of real n n skew symmetric matrices the flow is solved using explicit magnus expansion which respects the isospectrality of the system we represent the terms of expansion as binary rooted trees and deduce an explicit formalism to construct the trees recursively

hamiltonian structure of dynamical systems which solve linear programming problems abstract a new class of completely integrable hamiltonian systems which solve linear programming problems is introduced the relations with reduced toda flows double bracket equations interior point methods are established an interpretation in terms of total actions on kahler manifolds is given a complete description of the phase portrait including the explicit description of action angle variables is provided

stabilization of nonholonomic systems using isospectral flows in this paper we derive and analyze a discontinuous stabilizing feedback for a lie algebraic generalization of a class of kinematic nonholonomic systems introduced by brockett new directions in applied mathematics p hilton and g young eds springer verlag new york 1982 pp 11 27 the algorithm involves discrete switching between isospectral and norm decreasing flows we include a rigorous analysis of the convergence

multistencils fast marching methods a highly accurate solution to the eikonal equation on cartesian domains a wide range of computer vision applications require an accurate solution of a particular hamilton jacobi hj equation known as the eikonal equation in this paper we propose an improved version of the fast marching method fmm that is highly accurate for both 2d and 3d cartesian domains the new method is called multistencils fast marching msfm which computes the solution at each grid point by solving the eikonal equation along several stencils and then picks the solution that satisfies the upwind condition the stencils are centered at each grid point and cover its entire nearest neighbors in 2d space two stencils cover 8 neighbors of the point whereas in 3d space six stencils cover its 26 neighbors for those stencils that are not aligned with the natural coordinate system the eikonal equation is derived using directional derivatives and then solved using higher order finite difference schemes the accuracy of the proposed method over the state of the art fmm based techniques has been demonstrated through comprehensive numerical experiments

runge kutta smoother for suppression of computational mode instability of leapfrog scheme abstract the runge kutta smoother is applied to suppress nonlinear numerical instabilities in the leapfrog scheme for time integration of the korteweg de vries equation the accuracy of integration is compared with that by the use of the second order smoother the runge kutta smoother enables us to make long time integration of the korteweg de vries equation for large amplitudes

optimal control and geodesic flows in this paper we analyze and generalize from the point of view of the maximum principle a class of nonlinear optimal control problems originally introduced in brockett 1994 the optimal control problems discussed here lie on the adjoint orbits of compact lie groups or on grassmann manifolds and the structure of the equations that arise depends on the metric used the so called normal metric we consider some special cases of the general problem and their meaning from the point of view of riemannian geometry and hamiltonian mechanics

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

functional dissipation microarrays for classification in this article we describe a new method of extracting information from signals called functional dissipation that proves to be very effective for enhancing classification of high resolution texture rich data our algorithm bypasses to some extent the need to have very specialized feature extraction techniques and can potentially be used as an intermediate feature enhancement step in any classification scheme functional dissipation is based on signal transforms but uses the transforms recursively to uncover new features we generate a variety of masking functions and extract features with several generalized matching pursuit iterations in each iteration the recursive process modifies several coefficients of the transformed signal with the largest absolute values according to the specific masking function in this way the greedy pursuit is turned into a slow controlled dissipation of the structure of the signal that for some masking functions enhances separation among classes our case study in this paper is the classification of crystallization patterns of amino acids solutions affected by the addition of small quantities of proteins

analog computation with dynamical systems a b s t r a c t physical systems exhibit various levels of complexity their long term dynamics may converge to fixed points or exhibit complex chaotic behavior this paper presents a theory that enables to interpret natural processes as special purpose analog computers since physical systems are naturally described in continuous time a definition of computational complexity for continuous time systems is required in analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems discrete or continuous in time on the basis of an intrinsic time scale of the system dissipative dynamical systems are classified into the computational complexity classes pd co rpd npd and exp t corresponding to their standard counterparts according to the complexity of their long term behavior the complexity of chaotic attractors relative to regular ones leads to the conjecture pa fi npj continuous time flows have been proven useful in solving various practical problems our theory provides the tools for an algorithmic analysis of such flows as an example we analyze the continuous hopfield network 1998 elsevier science b v all rights reserved

an iterative method for computing the pseudospectral abscissa for a class of nonlinear eigenvalue problems we consider the following class of nonlinear eigenvalue problems left sum i 1 m a i p i lambda right v 0 where a 1 ldots a m are given n times n matrices and the functions p 1 ldots p m are assumed to be entire this does not only include polynomial eigenvalue problems but also eigenvalue problems arising from systems of delay differential equations our aim is to compute the epsilon pseudospectral abscissa i e the supremum of the real parts of the points in the epsilon pseudospectrum which is the complex set obtained by joining all solutions of the eigenvalue problem under perturbations delta a i i 1 m of norm at most epsilon of the matrices a i i 1 m under mild assumptions guaranteeing the existence of a globally rightmost point of the epsilon pseudospectrum we prove that it is sufficient to restrict the analysis to rank one perturbations of the form delta a i beta i u v where u in mathbb c n and v in mathbb c n wit

reaching an optimal consensus dynamical systems that compute intersections of convex sets in this paper multi agent systems minimizing a sum of objective functions where each component is only known to a particular node is considered for continuous time dynamics with time varying interconnection topologies assuming that each node can observe a convex solution set of its optimization component and the intersection of all such sets is nonempty the considered optimization problem is converted to an intersection computation problem by a simple distributed control rule the considered multi agent system with continuous time dynamics achieves not only a consensus but also an optimal agreement within the optimal solution set of the overall optimization objective directed and bidirectional communications are studied respectively and connectivity conditions are given to ensure a global optimal consensus in this way the corresponding intersection computation problem is solved by the proposed decentralized continuous time algorithm we establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non smooth analysis

flow equations and normal ordering a survey first we give an introduction to the method of diagonalizing or block diagonalizing continuously a hamiltonian and explain how this procedure can be used to analyse the two dimensional hubbard model then we give a short survey on applications of this flow equation on other models finally we outline how symmetry breaking can be introduced by means of a symmetry breaking of the normal ordering not of the hamiltonian

experimental computation of real numbers by newtonian machines following a methodology we have proposed for analysing the nature of experimental computation we prove that there is a 3 dimensional newtonian machine which given any point x 2 0 1 can generate an infinite sequence pn qn for n 1 2 of rational number interval approximations that converges to x as n 1 the machine is a system for scattering and collecting particles the theorem implies that every point x 2 0 1 is computable by a simple newtonian kinematic system that is bounded in space and mass and for which the calculation of the n th approximation of x takes place in o n time with o n energy we describe a set of variants of the scatter machine which explain why our machine is non deterministic

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

on the suppression of numerical oscillations using a non linear filter the idea of using a non linear filtering algorithm to eliminate numerically generated oscillations is investigated a detailed study is conducted to follow the development of numerical oscillations and their interaction with the filter a relaxation procedure is also proposed to enhance the effectiveness of the filter three model problems a 2d steady state scalar convection diffusion equation a 1d unsteady gas dynamics flow with shock and a 1d linear wave equation have been designed to test the performance of the filtering algorithm the effectiveness of the filter is assessed for convection schemes of different dispersive and diffusive characteristics demonstrating that it is effective in eliminating oscillations with short wavelength but oscillations of longer wavelengths are virtually unaffected it is concluded that a proper combination of non linear filter and dispersive numerical scheme is a viable alternative to dissipative schemes in resolving flows with sharp gradients and discontinuities

numerical integration methods for the double bracket flow in this paper new methods up to order four based on the magnus expansion are proposed for the numerical integration of the double bracket equation the magnus series is constructed term by term by means of recurrences and a bound on the convergence domain is also provided the new integrators preserve the most salient qualitative features of the flow and are computationally more efficient than other standard lie group solvers such as the runge kutta munthe kaas class of algorithms