squared wavefunctions approach to periodic solutions of vector nonlinear schrodinger equation we develop a method of squared wavefunctions for the vector nonlinear schrodinger equation the squared wavefunctions of the octet representation of su 3 group give periodic solutions in terms of weierstrass elliptic functions specific limits of the obtained solution are the plane wave the soliton and cnoidal waves which were previously obtained using the ansatz of stationary motion

model reduction of nonstationary lpv systems this paper focuses on the model reduction of nonstationary linear parameter varying nslpv systems we provide a generalization of the balanced truncation procedure for the model reduction of stable nslpv systems along with a priori error bounds then for illustration purposes this method is applied to reduce the model of a two mass translational system furthermore we give an approach for the model reduction of stabilizable and detectable systems which requires the development and use of coprime factorizations for nslpv models for the general class of eventually periodic lpv systems which includes periodic and finite horizon systems as special cases our results can be explicitly computed using semidefinite programming

multidimensional logical transforms this correspondence reviews the logical transform and pre sents new theoretical and empirical developments novel extensions of the logical transform into spaces of three and four dimensions are de scribed using the hexahedral and octahedral tessellations these new tessellations are important in that they yield sufficiently compact neighborhoods to permit multidimensional cellular logic transforms to be carried out in real time by table lookup applications of logical transforms in these tessellations to data and image analysis are provided

an error bound for a discrete reduced order model of a linear multivariable system the design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction in order for the design based on the order reduction to include a guarantee of stability it is sufficient to have a bound on the model error previous work has provided such a bound for continuous time systems for algorithms based on balancing in this note an linftybound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing

general relaxation methods for initial value problems with application to multistep schemes recently an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial value problems using runge kutta methods we generalize this approach to multistep methods including all general linear methods of order two or higher and many other classes of schemes we prove the existence of a valid relaxation parameter and high order accuracy of the resulting method in the context of general equations including but not limited to conservative or dissipative systems the theory is illustrated with several numerical examples

a method for the integration in time of certain partial differential equations abstract a method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations the method can be a useful alternative to the usual leapfrog technique in that it does not suffer from the occurrence of blowup phenomena numerical examples concerning the korteweg de vries equation and the nonlinear schrodinger equation are given

on morphological operators based on rank filters rank order morphological operators have been designed to better cope with shape variations and noise than the corresponding operators with plain structuring elements in this paper starting from a brief historical overview of rank order filters and erosions dilations in digital image processing rank order based morphological operators are surveyed we also highlight the relations between these operators and show that many are similar if not equivalent an extensive bibliography is provided

two classes of linearly implicit local energy preserving approach for general multi symplectic hamiltonian pdes abstract two classes of efficient and robust schemes are proposed for the general multi symplectic hamiltonian systems using the invariant energy quadratization ieq approach the schemes are linear second order accurate local energy preserving and preserve the global energy they are not restricted to specific forms of the nonlinear part of the state function and only require solving linear equations at each time step we applied the new schemes to various multi symplectic hamiltonian pdes to demonstrate their effectiveness computational efficiency and accuracy

solutions and laws of conservation for coupled nonlinear schrodinger equations lie group analysis a set of two coupled nonlinear schr odinger equations is systematically analyzed by means of lie group technique the physical situations under consideration include nonlinear propagation in strongly birefringent and multimode optical fibers the most general lie group of point symmetries its lie algebra and a group of adjoint representations that correspond to the lie algebra are identified as a result a complete list of group invariant exact solutions is obtained and compared with earlier results the corresponding laws of conservation are derived employing noether s theorem

ξ filters a new family of nonlinear digital convolution filters is described having unusual and valuable properties the response is constant phase over the entire input frequency range and in some forms no signal whatsoever is passed beyond the cutoff frequency and side lobes are identically zero by simply iterating the transform upon which a filter is based without changing the size of the convolution kernel the cutoff frequency may be varied use of filters is analyzed and demonstrated as applied to ultrahigh speed image processing using table lookup computation

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

a galerkin method for a nonlinear integro differential wave system a galerkin method is introduced for the numerical solution of a system equivalent to the equation tt 1 0 x2dx xx which models the vibrations of a string when the increase in tension due to extension is taken into account we prove the convergence of the semidiscrete approximations and present some numerical tests

an hr adaptive method for the cubic nonlinear schrodinger equation abstract the nonlinear schrodinger equation nlse is one of the most important equations in quantum mechanics and appears in a wide range of applications including optical fibre communications plasma physics and biomolecule dynamics it is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients adaptive moving mesh methods r adaptive attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features this approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest adaptive mesh refinement h adaptive where the mesh is locally coarsened or refined is an alternative adaptive strategy which is popular for time independent problems in this paper we consider the effectiveness of a combined method h r adaptive to solve the nlse in one space dimension simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches the method is also shown to control the spatial error based on the user s input error tolerance evidence is also presented indicating second order spatial convergence using a novel monitor function to generate the adaptive moving mesh

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

balanced truncation of linear time varying systems in this paper balanced truncation of linear time varying systems is studied in discrete and continuous time based on relatively basic calculations with time varying lyapunov equations inequalities we are able to derive both upper and lower error bounds for the truncated models these results generalize well known time invariant formulas the case of time varying state dimension is considered input output stability of all truncated balanced realizations is also proven the method is finally successfully applied to a high order model

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

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

a study of the recursion yn 1 yn tymn we provide a detailed study of the recursion y 0 1 y n 1 y n y m n n 0 1 m 1 which arises either as a model discretization of a nonlinear ode or in the use of the energy method sharp bounds and asymptotic estimates are given for the size of the iterates y n

hamiltonian conserving discrete canonical equations based on variational difference quotients abstract new discrete mechanics based on the assumption of the discrete time is proposed the discrete mechanics does not contain any continuous differentiation but contains only difference quotients resulting discrete hamiltonian s canonical equations are single time step difference equations and exactly conserve the hamiltonian the canonical equations give the numerical results more accurately than the heun scheme and the 4th order runge kutta scheme

pseudo spectral solution of nonlinear schro dinger equations abstract we compare four pseudo spectral split step methods for solving a class of nonlinear schrodinger nls equations the importance of observing the l 2 invariance of the continuous problem is demonstrated through numerical experiments the best performance is obtained by transforming the given equation to an nls equation where two of the coefficients satisfy a simple algebraic relationship the problem can be solved efficiently in terms of the new variables and the l 2 norm of the computed solution is time invariant

quasi periodic and periodic solutions for coupled nonlinear schrodinger equations of manakov type we consider travelling periodic and quasi periodic wave solutions in coupled nonlinear schrodinger equations in fibre optics these equations can be used to model single mode fibres with strong birefringence and two mode optical fibres recently these equations appear as a model describing pulse pulse interactions in wavelength division multiplexed channels of optical fibre transmission systems in some cases this model reduces to the integrable manakov system ims two phase quasi periodic solutions for the ims are given in terms of two dimensional kleinian functions the reduction of quasi periodic solutions to elliptic functions is discussed new solutions are found in terms of generalized hermite polynomials which are associated with two gap treibich verdier potentials

quasiperiodic and periodic solutions for vector nonlinear schrodinger equations we consider quasiperiodic and periodic cnoidal wave solutions of a set of n component vector nonlinear schrodinger equations vnlses in a biased photorefractive crystal with a drift mechanism of nonlinear response and kerr type nonlinearity n component nonlinear schrodinger equations can be used to model self trapped mutually incoherent wave packets these equations also model pulse pulse interactions in wavelength division multiplexed channels of optical fiber transmission systems quasiperiodic wave solutions for the vnlses in terms of n dimensional kleinian functions are presented periodic solutions in terms of hermite polynomials and generalized hermite polynomials for n component nonlinear schrodinger equations are found

a tau method based on non uniform space time elements for the numerical simulation of solitons abstract in this paper we discuss the numerical simulation of soliton solutions of schrodnger s system of nonlinear partial differential equations the technique used is based on a novel approach to the tau method based on space time elements the original operator is not discretized in either space or time the perturbation term which in the tau method is a by product of the computation is used to measure the error in the equation or defect and therefore to control the accuracy of the computation in an adaptive way this leads to an accurate and economical numerical procedure which even for approximations of a very low degree can simulate surfaces with very sharp gradients in the direction of both space and time the tau method formulation presented in this paper preserves very accurately the principle of conservation of energy however our numerical examples support the idea that the latter is a necessary but not a sufficient condition to guarantee an efficient approximation of soliton solutions of schrodinger s equation by means of a method of numerical simulation

petrov galerkin methods for nonlinear dispersive waves abstract petrov galerkin methods based on piecewise linear interpolants for the korteweg de vries and related equations are studied both accuracy and stability are analysed for the linearised case it is shown that in the nonlinear case the order of accuracy of the standard galerkin procedure is reduced and an alternative technique is therefore proposed which retains the fourth order accuracy in space this is found to perform well when compared with finite difference or other finite element schemes

convergence of the lambert mcleod trajectory solver and of the celf method a trajectory problem is an initial value problemd y dt f y y 0 where the interest lies in obtaining the curve traced by the solution the trajectory rather than in finding the actual correspondanc between values of the parametert and points on that curve we prove the convergence of the lambert mcleod scheme for the numerical integration of trajectory problems we also study the celf method an explicit procedure for the integration in time of semidiscretizations of pdes which has some useful conservation properties the proofs rely on the concept of restricted stability introduced by stetter in order to show the convergence of the methods an idea of strang is also employed whereby the numerical solution is compared with a suitable perturbation of the theoretical solution rather than with the theoretical solution itself

the multisymplectic diamond scheme we introduce a class of general purpose linear multisymplectic integrators for hamiltonian wave equations based on a diamond shaped mesh on each diamond the pde is discretized by a symplectic runge kutta method the scheme advances in time by filling in each diamond locally leading to greater efficiency and parallelization and easier treatment of boundary conditions compared to methods based on rectangular meshes

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

some new structure preserving algorithms for general multi symplectic formulations of hamiltonian pdes many partial differential equations pdes can be written as a multi symplectic hamiltonian system which has three local conservation laws namely multi symplectic conservation law local energy conservation law and local momentum conservation law in this paper we give several systematic methods for discretizing general multi symplectic formulations of hamiltonian pdes including a local energy preserving algorithm a class of global energy preserving methods and a local momentum preserving algorithm the methods are illustrated by the nonlinear schrodinger equation and the korteweg de vries equation numerical experiments are presented to demonstrate the conservative properties of the proposed numerical methods

methods for the numerical solution of the nonlinear schroedinger equation optimal l2 rates of convergence are established for several fully discrete schemes for the numerical solution of the nonlinear schroedinger equation both finite differences and finite elements are considered for the discretization in space while the integration in time is treated either by the leap frog technique or by a modified crank nicolson procedure which generalizes the one suggested by delfour fortin and payne and possesses two useful conserved quantities

high order compact splitting multisymplectic method for the coupled nonlinear schrodinger equations in this paper we develop a new kind of multisymplectic integrator for the coupled nonlinear schrodinger cnls equations the cnls equations are cast into multisymplectic formulation then it is split into a linear multisymplectic formulation and a nonlinear hamiltonian system the space of the linear subproblem is approximated by a high order compact hoc method which is new in multisymplectic context the nonlinear subproblem is integrated exactly for splitting and approximation we utilize an hoc sms integrator its stability and conservation laws are investigated in theory numerical results are presented to demonstrate the accuracy conservation laws and to simulate various solitons as well for the hoc sms integrator they are consistent with our theoretical analysis

model reduction of multidimensional and uncertain systems model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure these methods involve a complete generalization of balanced realizations balanced gramians and balanced truncation model reduction with guaranteed error bounds based on solutions to a pair of linear matrix inequalities which generalize lyapunov equations the resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems

stochastic conformal schemes for damped stochastic klein gordon equation with additive noise abstract in this article stochastic conformal schemes of the damped stochastic klein gordon equation with additive noise are studied it is shown that this equation possesses the stochastic conformal multi symplectic conservation law under appropriate boundary conditions the global momentum evolution law and the global energy evolution law are proposed we chiefly develop the stochastic conformal preissman scheme the stochastic conformal discrete gradient scheme and the stochastic conformal euler box scheme to preserve the geometric structures of the original system specifically we make theoretical discussions on the three proposed schemes to obtain corresponding discrete conservation laws or discrete evolution laws then the damped stochastic linear klein gordon equation and the damped stochastic nonlinear klein gordon equation with cubic nonlinearity are taken as examples to demonstrate the validity of the proposed schemes through numerical experiments and comparisons the superiorities of the proposed schemes are fully shown which are consistent with our theoretical analysis moreover the mean square convergence orders of the three stochastic conformal schemes in time direction and space direction are tested numerically

a least squares approach to reduce stable discrete linear systems preserving their stability abstract a new stability preserving model reduction algorithm for discrete linear siso systems based on a least squares approach is proposed similar to the pade approximation an equation system for the markov parameters involving a high dimensional hankel matrix is considered it is proved that approximate solutions computed via the moore penrose pseudo inverse give rise to a stability preserving reduction scheme furthermore the proposed algorithm is compared to the balanced truncation method showing comparable performance of the reduced systems

balanced truncation for discrete time markov jump linear systems this technical note investigates the model reduction problem for mean square stable discrete time markov jump linear systems for this class of systems a balanced truncation algorithm is developed the reduced order model is suboptimal however the approximation error which is captured by means of the stochastic gain is bounded from above by twice the sum of singular numbers associated to the truncated states of each mode such a result allows rigorous simplification of the dynamics of each mode in an independent manner with respect to a metric which is relevant from a robust control point of view

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes

an adaptive method of lines solution of the korteweg de vries equation abstract following a method of lines formulation the korteweg de vries equation is solved using a static spatial remeshing algorithm based on the equidistribution principle which allows the number of nodes to be significantly reduced as compared to a fixed grid solution several finite difference schemes including direct and stagewise procedures are compared and the results of a large number of computational experiments are presented which demonstrate that the selection of a spatial approximation scheme for the third order derivative term is the primary determinant of solution accuracy

dissipativity based asynchronous filtering for periodic markov jump systems abstract this paper is devoted to developing a periodic and asynchronous filter for periodic markov jump systems a periodic and asynchronous filter means that the filter parameters are s periodic for a given operating mode and its operating modes are unnecessary to be identical but form a periodic hidden markov model together with the modes of the plant a sufficient condition is proposed to guarantee that the filtering error system is exponentially mean square stable and strictly q s r dissipative to realize this filter a slack matrix is introduced to deal with the nonlinearity and finsler s lemma is applied such that the decision variables are reduced finally a numerical simulation provides verification for the effectiveness of our designed filter

b spline finite element studies of the non linear schrodinger equation abstract the non linear schrodinger equation is solved numerically by a b spline finite element method the approach used is based on collocation of cubic b splines over spatial finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range time integration of the resulting system of ordinary differential equations is effected using a crank nicolson approximation standard problems are used to validate the algorithm comparisons are made with published numerical and analytical solutions the proposed method exhibits good conservation properties and performs well the development of temporally recurrent states from spatially periodic initial conditions is examined through computer experiments the birth of mobile as well as static solitons out of maxwellian initial data is observed

numerical experience with the nonlinear schrodinger equation abstract increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear schrodinger equation without changing the initial condition u x 0 sech x leads to bound states of an increasing number of solitons this results in very steep gradients in space and time and so provides a more severe test of numerical methods than before in particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples

studies in numerical nonlinear instability iii augmented hamiltonian systems leap frog explicit mid point discretizations of hamiltonian systems are considered it is proved that the discrete evolution preserves the symplectic structure of the phase space the kolmogorov arnold moser kam theory is applied to guarantee the boundedness of the computed points under suitable restrictions of the time step the general ideas introduced are employed to provide an extensive analysis of the leap frog discretization of the complex equation iz t z 2 z 0 which describes the nonlinear selfinteraction of a fourier mode the paper emphasizes the relations between numerical analysis and classical mechanics dynamical systems

an explicit finite difference scheme with exact conservation properties abstract a finite difference scheme for the numerical study of the korteweg de vries equation is constructed it is explicit and yet conserves exactly the energy of the computed solutions the underlying idea can also be applied to more general equations or systems numerical experiments are included

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

analysis of a galerkin approach applied to a system of coupled schrodinger equations abstract we study error and convergence rate of a fully discrete fourier galerkin scheme to approximate the solutions of a system consisting of two coupled cubic schrodinger equations with cross modulation which is a model for the propagation of a nonlinear pulse in a linearly birefringent kerr optical fiber we also establish a result on existence and uniqueness of a solution of the corresponding initial value problem

h model reduction of markovian jump linear systems abstract in this paper the h model reduction problem for linear systems that possess randomly jumping parameters is studied the development includes both the continuous and discrete cases it is shown that the reduced order models exist if a set of matrix inequalities is feasible an effective iterative algorithm involving linear matrix inequalities is suggested to solve the matrix inequalities characterizing the model reduction solutions using the numerical solutions of the matrix inequalities the reduced order models can be obtained an example is given to illustrate the proposed model reduction method

decoupled local global energy preserving schemes for the n coupled nonlinear schrodinger equations abstract we develop two local energy preserving integrators and a global energy preserving integrator for the general multisymplectic hamiltonian system when applied to the 1d and multi dimensional n coupled nonlinear schrodinger equations the given schemes have the exact preservation of the local global conservation law and are decoupled in the components n n 1 2 n i e each of the components can be solved independently the decoupled feature is significant and helpful for overcoming the computational difficulty of the n coupled n 3 nonlinear schrodinger equations especially of the multi dimensional case the composition method is employed to improve the accuracy of the schemes in time and the discrete fast fourier transform is used to reduce the computational complexity several numerical experiments are carried out to exhibit the behaviors of the wave solutions numerical results confirm the theoretical results

multi stage symplectic schemes of two kinds of hamiltonian systems for wave equations abstract multi stage schemes for wave equation are constructed their stability conditions are discussed

general local energy preserving integrators for solving multi symplectic hamiltonian pdes in this paper we propose and investigate a general approach to constructing local energy preserving algorithms which can be of arbitrarily high order in time for solving hamiltonian pdes this approach is based on the temporal discretization using continuous runge kutta type methods and the spatial discretization using pseudospectral methods or gauss legendre collocation methods the local energy conservation law of our new schemes is analyzed in detail the effectiveness of the novel local energy preserving integrators is demonstrated by coupled nonlinear schrodinger equations and 2d nonlinear schrodinger equations with external fields our new schemes are compared with some classical multi symplectic and symplectic schemes in numerical experiments the numerical results show the remarkable long term behaviour of our new schemes

error bounds for balanced model reduction of linear time varying systems error bounds are developed for balanced truncation of linear time varying systems leading to an extension of the twice the sum of the tail formulas well known in the time invariant case the approach relies on an operator theoretic framework for analysis of linear time varying systems this provides a multivariable notion of frequency for such systems which are thus characterized by rational functions of many complex variables allowing the problem to be formulated in the linear fractional framework using a time varying version of standard necessary conditions for reduced order modeling based on convex operator inequalities we show that these error bounds for balanced truncation are related to the closest possible reduced order modeling error in a sense which parallels the time invariant case

the full positive flows of manakov hierarchy hamiltonian structures and conservation laws based on four sets of lenard recursion sequences and zero curvature equation we derive the full positive flows of the manakov hierarchy associated with a 3x3 matrix spectral problem from which some new nonlinear evolution equations are proposed with the help of the darboux transformation soliton solutions of two new nonlinear evolution equations in the manakov hierarchy are constructed as two special reductions the full positive flows of the coupled modified korteweg de vries hierarchy and the sasa satsuma hierarchy are deduced in which some new nonlinear evolution equations are included and then we construct the hamiltonian structures of the manakov hierarchy and infinite conservation laws of several nonlinear evolution equations

on the balanced truncation and coprime factors reduction of markovian jump linear systems abstract the paper deals with the balanced truncation and coprime factors reduction of markovian jump linear mjl systems which can have mode varying state input and output dimensions we develop machinery for balancing mean square stable mjl system realizations using generalized gramians and strict lyapunov inequalities and provide an improved a priori upper bound on the error induced in the balanced truncation process we also generalize the coprime factors reduction method and in doing so extend the applicability of the balanced truncation technique to the class of mean square stabilizable and detectable mjl systems we provide tools to establish mean square stabilizability and detectability of the considered mjl systems in addition a notion of right coprime factorization of mjl systems and methods to construct such factorizations are given the error measure in the coprime factors reduction approach while still norm based does not directly capture the mismatch between the nominal system and the reduced order model as is the case in the balanced truncation approach where mean square stable models are considered instead the error measure is given in terms of the distance between the coprime factors realizations and thus has an interpretation in terms of robust feedback stability the paper concludes with an illustrative example which demonstrates how to apply the coprime factors model reduction approach

energy conserving methods for the nonlinear schrodinger equation in this paper we further develop recent results in the numerical solution of hamiltonian partial differential equations pdes brugnano et al 2015 by means of energy conserving methods in the class of line integral methods in particular the runge kutta methods named hamiltonian boundary value methods hbvms we shall use hbvms for solving the nonlinear schrodinger equation nlse of interest in many applications we show that the use of energy conserving methods able to conserve a discrete counterpart of the hamiltonian functional confers more robustness on the numerical solution of such a problem

an improved error estimate for reduced order models of discrete time systems the authors derive under weaker conditions a discrete time counterpart of k glover s 1984 error estimate for reduced order models following the same lines a simple proof of glover s result for continuous time systems can be given

reachability and observability reduction for linear switched systems with constrained switching we present an algorithm for reducing the number of continuous states of a discrete time linear switched system such that the reduced system has the same input output behavior as the original system for a subset of switching sequences the procedure can be interpreted as reachability and observability reduction for a linear switched system with constrained switching the proposed method is expected to be useful for abstraction based control synthesis methods for hybrid systems

balanced truncation for a class of stochastic jump linear systems and model reduction for hidden markov models this paper develops a generalization of the balanced truncation algorithm applicable to a class of discrete time stochastic jump linear systems the approximation error which is captured by means of the stochastic l 2 gain is bounded from above by twice the sum of singular numbers associated to the truncated states similar to the case of linear time invariant systems a two step model reduction algorithm for hidden markov models is also developed the first step relies on the aforementioned balanced truncation algorithm due to a topological equivalence established between hidden markov models and a subclass of stochastic jump linear systems in a second step the positivity constraints which reflect the hidden markov model structure are enforced by solving a low dimensional optimization problem

a superconvergent finite element method for the korteweg de vries equation an unconditionally stable fully discrete finite element method for the korteweg de vries equation is presented in addition to satisfying optimal order global estimates it is shown that this method is superconvergent at the nodes the algorithm is derived from the conservative method proposed by the second author by the introduction of a small time independent forcing term into the discrete equations this term is a form of the quasiprojection which was first employed in the analysis of superconvergence phenomena for parabolic problems however in the present work unlike in the parabolic case the quasiprojection is used as perturbation of the discrete equations and does not affect the choice of initial values

numerical approximation of quadratic observables of schrodinger type equations in the semi classical limit we apply wigner transform techniques to the analysis of difference methods for schrodinger type equations in the case of a small planck constant in this way we are able to obtain sharp conditions on the spatial temporal grid which guarantee convergence for average values of observables as the planck constant tends to zero the theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not numerical test examples are presented to help interpret the theory

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

a numerov galerkin technique applied to a finite element shallow water equations model with enforced conservation of integral invariants and selective lumping abstract a new two stage numerov galerkin method is presented and applied following a suggestion by cullen and morton 1 to achieve higher accuracy by combining the galerkin product with a high order compact numerov difference approximation to derivatives in the nonlinear advection operator of the shallow water equations the practical applicability of the new method is supported by computational examples involving medium 10 days and longterm 20 days integrations of the nonlinear shallow water equations small scale noise was eliminated by periodical application of a shuman filter to only one component of the velocity conservation of integral invariants is ensured by using a new augmented lagrangian multiplier penalty method in the numerov galerkin method experiments were performed using the new method with differently weighted selective lumping schemes for the mass matrix the new method exhibits a consistently higher accuracy than the single stage galerkin method and requires much less computational effort a hypothesis is offered to explain the increased accuracy obtained by selective lumping schemes in the long term numerical integrations 10 20 days as compared to the consistent mass approach in numerov galerkin schemes

some observations on a static spatial remeshing method based on equidistribution principles this paper proposes a method of lines solution procedure with time and space adaptation for one dimensional systems of partial differential equations whose solutions display steep moving fronts the spatial remeshing algorithm which is a variation of the method published by sanz serna and christie and an extension suggested by revilla is a static remeshing method based on equidistribution principles the selection of the several algorithm components i e grid placement criterion spatial discretization scheme time integrator adaptation frequency and parameter tuning are investigated and illustrated with several test examples i e the cubic schrodinger equation a model of a single step reaction with diffusion and a model of flame propagation

an engineer s guide to soliton phenomena application of the finite element method abstract the paper attempts an elementary survey of the physical and mathematical background appertaining to solitons and discusses in particular the numerical solution of three types of dispersive nonlinear partial differential equations exhibiting soliton type solutions namely the korteweg de vries equation the nonlinear schrodinger equation and the sine gordon equation throughout this study a semidiscrete galerkin method is applied using a finite element discretization in space and a step by step time integration of the resulting system of nonlinear ordinary differential equations depending upon the special type of the evolutionary equation the application of a petrov galerkin procedure may increase significantly the numerical stability accuracy and effectivity of the different approaches are demonstrated on a series of computer plots

relative error h sub infinity identification from autocorrelation data a stochastic realization method a variant of the balanced stochastic truncation bst method for approximated realization of power spectrum matrices is shown to form the basis for an identification procedure that is well suited to the task of determining relative error bounded approximate plant models for use in control design from input output cross correlation data central to the theory is a novel l sub infinity norm bound on the relative error between an exact realization of the data and bst approximate realization

the numerical treatment of the nonlinear schrodinger equation abstract the implicit midpoint rule for the integration in time of a fourth order accurate semidiscretization in space of the nonlinear schrodinger equation is studied the stability analysis for the present method is carried out by the conservation of the energy in time three increasingly difficult problems are tested by the method mentioned comparison with other methods is presented the results obtained here seem to be in better agreement with the exact solutions

approximations of stochastic hybrid systems this paper develops a notion of approximation for a class of stochastic hybrid systems that includes as special cases both jump linear stochastic systems and linear stochastic hybrid automata our approximation framework is based on the recently developed notion of the so called stochastic simulation functions these lyapunov like functions can be used to rigorously quantify the distance or error between a system and its approximate abstraction for the class of jump linear stochastic systems and linear stochastic hybrid automata we show that the computation of stochastic simulation functions can be cast as a tractable linear matrix inequality problem this enables us to compute the modeling error incurred by abstracting some of the continuous dynamics or by neglecting the influence of stochastic noise or even the influence of stochastic discrete jumps

efficient energy preserving wavelet collocation schemes for the coupled nonlinear schrodinger boussinesq system abstract second and fourth order energy preserving wavelet collocation schemes are proposed for the coupled nonlinear schrodinger boussinesq system based on the hamiltonian structure and composition technique a fast solver improves the computational efficiency of the schemes some numerical experiments are conducted to show the efficiency and accuracy of the present schemes

fully discrete explicit locally entropy stable schemes for the compressible euler and navier stokes equations abstract recently relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation here we generalize this approach to guarantee local entropy inequalities for finitely many convex functionals entropies and apply the resulting methods to the compressible euler and navier stokes equations based on the unstructured h p adaptive ssdc framework of entropy conservative or dissipative semidiscretizations using summation by parts and simultaneous approximation term operators we develop the first discretizations for compressible computational fluid dynamics that are primary conservative locally entropy stable in the fully discrete sense under a usual cfl condition explicit except for the parallelizable solution of a single scalar equation per element and arbitrarily high order accurate in space and time we demonstrate the accuracy and the robustness of the fully discrete explicit locally entropy stable solver for a set of test cases of increasing complexity

optimal filter approximations in conditionally gaussian pairwise markov switching models we consider a general triplet markov gaussian linear system bf x bf r bf y where bf x is an hidden continuous random sequence bf r is an hidden discrete markov chain bf y is an observed continuous random sequence when the triplet bf x bf r bf y is a classical conditionally gaussian linear state space model cglssm the mean square error optimal filter is not workable with a reasonable complexity and different approximate methods e g based on particle filters are used we propose two contributions the first one is to extend the cglssm to a new more general model called the conditionally gaussian pairwise markov switching model cgpmsm in which bf x is not necessarily markov given bf r the second contribution is to consider a particular case of cgpmsm in which bf r bf y is markov and in which an exact filter optimal in the sense of mean square error can be performed with linear time complexity some experiments show that the proposed method and the suited particle filter have comparable efficiency while the second one is much faster

orthogonal spline collocation methods for schr o dinger type equations in one space variable we examine the use of orthogonal spline collocation for the semi discreti za tion of the cubic schr o dinger equation and the two dimensional parabolic equation of tappert in each case an optimal order l 2 estimate of the error in the semidiscrete approximation is derived for the cubic schr o dinger equation we present the results of numerical experiments in which the integration in time is performed using a routine from a software library