adiabatic midpoint rule for the dispersion managed nonlinear schrodinger equation the dispersion managed nonlinear schrodinger equation contains a small parameter varepsilon a rapidly changing piecewise constant coefficient function and a cubic nonlinearity typical solutions are highly oscillatory and have a discontinuous time derivative and hence solving this equation numerically is a challenging task we present and analyze a tailor made time integrator which attains the desired accuracy with a significantly larger step size than traditional methods the construction of this method is based on a favorable transformation to an equivalent problem and the explicit computation of certain integrals over highly oscillatory phases the error analysis requires the thorough investigation of various cancellation effects which result in improved accuracy for special step sizes

quantification of the variance of estimated transfer functions in the presence of undermodeling we study the effect of undermodeling on the parameter variance for prediction error time domain identification in open loop we consider linear time invariant discrete time single input single output systems with known noise model we examine asymptotic expressions for the variance for large number of data this quantity depends in general on the fourth order statistical properties of the applied input however we establish a sufficient condition under which the asymptotic variance depends on the input power spectrum only for this case we deliver exact expressions for a stochastic input the undermodeling contributes to the parameter variance due to the correlation between the prediction errors and its gradients while for a deterministic input it has no influence as an additional contribution we investigate the parameter variance under the assumptions of the stochastic embedding procedure

stability of runge kutta methods in the numerical solution of equation u t au t a 0 u t this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a0u t the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

on the time fractional schrodinger equation theoretical analysis and numerical solution by matrix mittag leffler functions abstract this paper presents a deep analysis of a time dependent schrodinger equation with fractional time derivative after the discretization of the spatial operator the equation is reformulated in terms of a special system of fractional differential equations it occurs that the eigenvalues of the coefficient matrix lay on the boundary of the stability region of fractional differential equations the main difficulties in solving this system are hence related to the simultaneous presence of persisting oscillations possibly with high frequency as it is typical with schrodinger equations and a persisting memory as a consequence of the fractional order moreover an accurate spatial discretization gives rise to systems of large to very large size involving a noteworthy computational complexity by means of a theoretical analysis the exact solution is split into two or three terms depending on the order of the fractional derivative thus to face the numerical computation by different and suitably selected methods direct evaluation of matrix functions for the terms characterized by smooth behaviour but with persistent memory and a step by step strategy in conjunction with matrix function for the oscillating term in both cases krylov subspace methods are employed for the computation of matrix functions and convergence results are presented

parametric and nonparametric identification of linear systems in the presence of nonlinear distortions a frequency domain approach this paper studies the asymptotic behavior of nonparametric and parametric frequency domain identification methods to model linear dynamic systems in the presence of nonlinear distortions under some general conditions for random multisine excitations in the first part a related linear dynamic system rlds approximation to the nonlinear system nls is defined and it is shown that the differences between the nls and the rlds can be modeled as stochastic variables with known properties in the second part a parametric model for the rlds is identified convergence in probability of this model to the rlds is proven a function of dependency is defined to detect and separate the presence of unmodeled dynamics and nonlinear distortions and to bound the bias error on the transfer function estimate

on the perturbation of unbounded linear operators with topologically complemented ranges let x and y be normed spaces and let t d t x y be a linear transformation having a finite codimensional restriction with a continuous inverse equivalently let t be a operator for example if t is bounded below suppose that r t or r t is topologically complemented in y conditions are obtained under which r t s resp r t s is topologically complemented whenever s belongs to the class of precompact operators or to some wider class

mollified impulse methods for highly oscillatory differential equations we introduce a family of impulselike methods for the integration of highly oscillatory second order differential equations whose forces can be split into a fast part and a slow part methods of this family are specified by two weight functions phi psi one is used to average positions and the other to mollify the force when the fast forces are conservative and phi psi the methods here coincide with the mollified impulse methods introduced by garcia archilla sanz serna and skeel on the other hand the methods here extend to nonlinear situations a well known class of exponential integrators introduced by hairer and lubich for cases of linear fast forces a convergence analysis is presented that provides insight into the role played by the processes of mollification and averaging in avoiding order reduction a simple condition on the weight functions is shown to be both necessary and sufficient to avoid order reduction

variational integrators for electric circuits in this contribution we develop a variational integrator for the simulation of stochastic and multiscale electric circuits when considering the dynamics of an electric circuit one is faced with three special situations 1 the system involves external control forcing through external controlled voltage sources and resistors 2 the system is constrained via the kirchhoff current kcl and voltage laws kvl 3 the lagrangian is degenerate based on a geometric setting an appropriate variational formulation is presented to model the circuit from which the equations of motion are derived a time discrete variational formulation provides an iteration scheme for the simulation of the electric circuit dependent on the discretization the intrinsic degeneracy of the system can be canceled for the discrete variational scheme in this way a variational integrator is constructed that gains several advantages compared to standard integration tools for circuits in particular a comparison to bdf methods which are usually the method of choice for the simulation of electric circuits shows that even for simple lcr circuits a better energy behavior and frequency spectrum preservation can be observed using the developed variational integrator

a trigonometric method for the linear stochastic wave equation a fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented a standard finite element method is used for the spatial discretization and a stochastic trigonometric scheme for the temporal approximation this explicit time integrator allows for error bounds independent of the space discretization and thus does not have a step size restriction as in the often used stormer verlet leap frog scheme moreover it enjoys a trace formula as does the exact solution of our problem these favorable properties are demonstrated with numerical experiments

points d evaluation pour les operateurs cycliques ayant la propriete de bishop β abstract let t be a linear bounded cyclic operator in a separable complex hilbert space h let b t and ba t denote respectively the set of bounded point evaluation and the set of analytic point evaluation of t we show that if t has the bishop property then ba t b t ap t where ap t is the approximate spectrum of t in the particular case when t is an operator of multiplication by z in a hardy space this was proved by trent pacific j math 80 1979 279 on the other hand using the generalized and the local spectral theory we obtain sufficient conditions on ba t under which the spectrum of t and the local spectrum of t at any y 0 in h coincide at the end results involving the spectral picture of quasi similar cyclic operators are given

a uniformly accurate ua multiscale time integrator fourier pseudospectral method for the klein gordon schrodinger equations in the nonrelativistic limit regime a multiscale time integrator fourier pseudospectral mti fp method is proposed and analyzed for solving the klein gordon schrodinger kgs equations in the nonrelativistic limit regime with a dimensionless parameter 0 varepsilon le 1 0 1 which is inversely proportional to the speed of light in fact the solution of the kgs equations propagates waves with wavelength at o varepsilon 2 o 2 and o 1 in time and space respectively when 0 varepsilon ll 1 0 1 which brings significantly numerical burden in practical computation the mti fp method is designed by adapting a multiscale decomposition by frequency of the solution at each time step and applying the fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives respectively we rigorously establish two independent error bounds for the mti fp at o tau 2 varepsilon 2 h m 0 o 2 2 hm0 and o varepsilon 2 h m 0 o 2 hm0 for varepsilon in 0 1 0 1 with tau time step size h mesh size and m 0 ge 4 m0 4 an integer depending on the regularity of the solution which imply that the mti fp converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau o for varepsilon in 0 1 0 1 in addition the mti fp method converges optimally with quadratic convergence rate at o tau 2 o 2 in the regime when 0 tau lesssim varepsilon 2 0 2 and the error is at o varepsilon 2 o 2 independent of tau in the regime when 0 varepsilon lesssim tau 1 2 0 1 2 thus the meshing strategy requirement or varepsilon scalability of the mti fp is tau o 1 o 1 and h o 1 h o 1 for 0 varepsilon ll 1 0 1 which is significantly better than that of classical methods numerical results demonstrate that our error bounds are optimal and sharp finally the mti fp method is applied to study numerically convergence rates of the kgs equations to its limiting models in the nonrelativistic limit regime

quantifying model errors of identified transfer functions in this paper a method is presented to quantify the uncertainty on an estimated transfer function model due to disturbing process noise and model errors the model errors are modeled with a stochastic model from which parameters are estimated from the residuals the identification is performed in the frequency domain

multivalued fredholm type operators with abstract generalised inverses multivalued semi fredholm type operators with topologically complemented kernels and ranges are investigated in normed linear spaces it is shown that regularisers generalised inverses can be constructed for the classes given and that the operators considered can be characterised in terms of their regularisers continuity of the inverses is discussed and dual properties of adjoints are given

an averaging technique for highly oscillatory hamiltonian problems in this paper we are concerned with the numerical solution of highly oscillatory hamiltonian systems with a stiff linear part we construct an averaged system whose solution remains close to the exact one over bounded time intervals possesses the same adiabatic and hamiltonian invariants as the original system and is nonstiff we then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time averages developed in e cances et al numer math 100 2005 pp 211 232 eventually we demonstrate the efficiency of our approach on two test problems with one or several frequencies

stability analysis of runge kutta methods for unbounded retarded differential equations with piecewise continuous arguments abstract this paper deals with the asymptotical stability of the analytic solutions for the unbounded retarded differential equations with piecewise continuous arguments epca u t f t u t u t n t 0 u 0 u 0 where n z a sufficient condition that the differential equations are asymptotically stable is derived this paper is also concerned with the stability analysis of the runge kutta methods for equation u t au t bu t n t 0 u 0 u 0 where n z the conditions that the numerical solutions preserve the stability of the analytic solutions are obtained and some numerical experiments are given

a gautschi type method for oscillatory second order differential equations we study a numerical method for second order differential equations in which high frequency oscillations are generated by a linear part for example semilinear wave equations are of this type the numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly we prove that the method admits second order error bounds which are independent of the product of the step size with the frequencies our analysis also provides new insight into the m ollified impulse method of garcia archilla sanz serna and skeel we include results of numerical experiments with the sine gordon equation

perturbation of normally solvable linear relations in paracomplete spaces in this paper we investigate the stability of the index the nullity and the deficiency of normally solvable linear relations in paracomplete spaces under perturbation by strictly singular and t strictly singular linear relations this study led us to generalize some well known results for operators and extend some results of small perturbation of normally solvable linear relations given by t alvarez 2012 3

robust parametric transfer function estimation using complex logarithmic frequency response data the statistical properties of a logarithmic least squares frequency domain estimator are analyzed in an error in variables framework it is shown that contrary to most frequency domain estimators the logarithmic least squares estimator remains practically consistent when the variances of the frequency domain data are not a priori known if available the variance at each spectral line of the logarithmic frequency response data can be used to weight the logarithmic least squares estimator resulting in parameter estimates with a smaller variability an estimate of these variances can be derived from the residual errors at the excited spectral lines besides its robustness to lack of prior noise information it is demonstrated that the logarithmic estimator behaves remarkably well in the presence of outliers

on the generalized resolvent in banach spaces abstract we construct for a closed densely defined operator a on a banach space x a generalized resolvent on the domain of fredholm regularity of a meromorphic in the fredholm domain a of a

a parallel implementation of a two dimensional fluid laser plasma integrator for stratified plasma vacuum systems a two dimensional fluid laser plasma integrator for stratified plasma vacuum systems is presented inside a plasma a laser pulse can be longitudinally compressed from ten or more wave lengths to one or two cycles however for physically realistic simulations transversal effects have to be included because transversal instabilities can destroy the pulse and transversal compression in the plasma as well as focusing in vacuum allows much higher intensities to be reached in contrast to the one dimensional case where a two step implementation of the gautschi type exponential integrator with constant step size turned out to be sufficient it is essential to enable changes of the time step size for the two dimensional case the use of a one step version of the gautschi type integrator being accurate of second order independent of the highest frequencies arising in the system is proposed in vacuum this allows to take arbitrarily large time steps to optimize runtime and memory requirements within the plasma a splitting of the laplacian is suggested this splitting allows to evaluate the matrix functions arising in the gautschi type method by one dimensional fourier transforms it is also demonstrated how the different variants of the scheme can be parallelized numerical experiments illustrate the superior performance and accuracy of the integrator compared to the standard leap frog method finally we discuss the simulation of a layered plasma vacuum structure using the new method

stability of runge kutta methods in the numerical solution of equation u t au t a0u t a1u t 1 this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a 0u t a 1u t 1 the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

the improved linear multistep methods for differential equations with piecewise continuous arguments abstract this paper deals with the convergence of the linear multistep methods for the equation x t ax t a 0 x t numerical experiments demonstrate that the 2 step adams bashforth method is only of order p 0 when applied to the given equation an improved linear multistep methods is constructed it is proved that these methods preserve their original convergence order for ordinary differential equations odes and some numerical experiments are given

on error bounds for the gautschi type exponential integrator applied to oscillatory second order differential equations this paper studies a numerical method for second order oscillatory differential equations in which high frequency oscillations are generated by a linear time and or solution dependent part for constant linear part it is known that the method allows second order error bounds independent of the product of the step size with the frequencies and is therefore a long time step method most real world problems are not of that kind and it is important to study more general equations the analysis in this paper shows that one obtains second order error bounds even in the case of a time and or solution dependent linear part if the matrix is evaluated at averaged positions

long time energy conservation of numerical methods for oscillatory differential equations we consider second order differential systems where high frequency oscillations are generated by a linear part we present a frequency expansion of the solution and we discuss two invariants of the system that determine the coefficients of the frequency expansion these invariants are related to the total energy and the oscillatory harmonic energy of the original system for the numerical solution we study a class of symmetric methods that discretize the linear part without error we are interested in the case where the product of the step size with the highest frequency can be large in the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system this allows us to prove the near conservation of the total and the oscillatory energy over very long time intervals

a note on the gautschi type method for oscillatory second order differential equations the gautschi type method has been proposed by hochbruck and lubich for oscillatory second order differential equations they conjecture that this method allows for a uniform error bound independent of the size of the system the conjecture is proved in this note

quantifying the error in estimated transfer functions with application to model order selection previous results on estimating errors or error bounds on identified transfer functions have relied on prior assumptions about the noise and the unmodeled dynamics this prior information took the form of parameterized bounding functions or parameterized probability density functions in the time or frequency domain with known parameters it is shown that the parameters that quantify this prior information can themselves be estimated from the data using a maximum likelihood technique this significantly reduces the prior information required to estimate transfer function error bounds the authors illustrate the usefulness of the method with a number of simulation examples how the obtained error bounds can be used for intelligent model order selection that takes into account both measurement noise and under modeling is shown another simulation study compares the method to akaike s well known fpe and aic criteria

exponential collocation methods for conservative or dissipative systems abstract in this paper we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods it is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or lyapunov functions we also consider order estimates of the new methods furthermore we explore and discuss the application of our methods in important stiff gradient systems and it turns out that our methods are unconditionally energy diminishing and strongly damped even for very stiff gradient systems practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear schrodinger equation as a byproduct of this paper arbitrary order trigonometric rkn collocation methods are also presented and analysed for second order highly oscillatory general systems the paper is accompanied by numerical results that demonstrate the great potential of this work

comparison results for parabolic differential equations at resonance the objective of this paper is to develop systematically general comparison techniques for semilinear parabolic equations with periodic and homogeneous neumann boundary conditions these results are expected to play a significant role in the existence theory for parabolic boundary value problems at resonance

stability of differential equations with piecewise constant arguments of generalized type abstract in this paper we continue to consider differential equations with piecewise constant argument of generalized type epcag m u akhmet integral manifolds of differential equations with piecewise constant argument of generalized type nonlinear anal tma 66 2007 367 383 a deviating function of a new form is introduced the linear and quasilinear systems are under discussion the structure of the sets of solutions is specified necessary and sufficient conditions for stability of the zero solution are obtained our approach can be fruitfully applied to the investigation of stability oscillations controllability and many other problems of epcag some of the results were announced at the international conference on hybrid systems and applications university of louisiana lafayette 2006

error analysis of exponential integrators for oscillatory second order differential equations in this paper we analyse a family of exponential integrators for second order differential equations in which high frequency oscillations in the solution are generated by a linear part conditions are given which guarantee that the integrators allow second order error bounds independent of the product of the step size with the frequencies our convergence analysis generalizes

weak second order multirevolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise we introduce a class of numerical methods for highly oscillatory systems of stochastic differential equations with general noncommutative noise we prove global weak error bounds of order two uniformly with respect to the stiffness of the oscillations which permits to use large time steps the approach is based on the micro macro framework of multi revolution composition methods recently introduced for deterministic problems and inherits its geometric features in particular to design integrators preserving exactly quadratic first integral numerical experiments including the stochastic nonlinear schrodinger equation with space time multiplicative noise illustrate the performance and versatility of the approach

on the initial growth of interfaces in reaction diffusion equations with strong absorption we study the initial growth of the interfaces of non negative local solutions of the equation u t u m xx lambdau q when m greater than or equal to 1 and 0 c0 for some explicit c0 c0 lambda m q then the free boundary zeta t sup x u x t 0 is a heating front more precisely zeta t greater than or equal to at m q 2 1 q for any t small enough and for some a 0 if on the contrary u x 0 less than or equal to c x 2 m q with c 0 applications to solutions of the associated cauchy and dirichlet problems are also given

on oscillation properties of delay differential equations with positive and negative coefficients abstract for a scalar delay differential equation x t a t x h t b t x g t 0 a t 0 b t 0 h t t g t t a connection between the following properties is established nonoscillation of the differential equation and the corresponding differential inequalities positiveness of the fundamental function and existence of a nonnegative solution for a certain explicitly constructed nonlinear integral inequality a comparison theorem and explicit nonoscillation and oscillation results are presented

the stability radius of fredholm linear pencils abstract let t and s be two bounded linear operators from banach spaces x into y and suppose that t is fredholm and dim n t s is constant in a neighborhood of 0 let d t s be the supremum of all r 0 such that dim n t s and codim r t s are constant for all with r it is a consequence of more general results due to h bart and d c lay 1980 studia math 66 307 320 that d t s lim n n t s 1 n where n t s are some non negative extended real numbers for x y and s i the identity operator we have n t s t n where is the reduced minimum modulus a different representation of the stability radius d t s is obtained here in terms of the spectral radii of generalized inverses of t the existence of generalized resolvents for fredholm linear pencils is also considered

adiabatic filon type methods for highly oscillatory second order ordinary differential equations in this paper we study efficient numerical integrators for linear and nonlinear systems of highly oscillatory second order ordinary differential equations the systems are reformulated as a first order system which is then transformed to adiabatic variables the solution of the transformed system is a smoother function which is more accessible to numerical approximation than the original system we develop filon type methods for linear systems by approximating the integral as a linear combination of function values and derivatives we then present a special combination of filon type methods and waveform relaxation methods for nonlinear systems both types of methods can be used with far larger step sizes than those required by traditional schemes and their performance drastically improves as frequency grows as are illustrated by numerical experiments

convergence analysis of trigonometric methods for stiff second order stochastic differential equations we study a class of numerical methods for a system of second order sde driven by a linear fast force generating high frequency oscillatory solutions the proposed schemes permit the use of large step sizes have uniform global error bounds in the position i e independent of the large frequencies present in the sde and offer various additional properties this new family of numerical integrators for sde can be viewed as a stochastic generalisation of the trigonometric integrators for highly oscillatory deterministic problems

long time step methods for oscillatory differential equations considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present it is assumed that one can do an explicit separation of the rhs forces into fast forces and slow forces such that i the fast forces contain the high frequency part of the solution ii the fast forces are conservative and iii the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem the fast forces are allowed to have low frequency components particular applications for which the schemes are intended include n body problems for which most of the forces are slow and nonlinear wave phenomena for which the fast forces can be propagated by spectral methods the assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces a long time step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system an existing method that comes close to qualifying is the impulse method also known as verlet i and r respa it is shown that it might fail though for a couple of reasons first it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode and second it is unstable if the step size is near a multiple of half the period of a normal mode proposed in this paper is a mollified impulse method having an error bound that is independent of the frequency of the fast forces it is also shown to possess superior stability properties theoretical results are supplemented by numerical experiments the method is efficient and reasonably easy to implement

essential spectra of linear relations abstract five essential spectra of linear relations are defined in terms of semi fredholm properties and the index basic properties of these sets are established and the perturbation theory for semi fredholm relations is then applied to verify a generalisation of weyl s theorem for single valued operators we conclude with a mobius transform spectral mapping theorem

wold type decomposition for some regular operators abstract this paper is concerned with wold type decomposition for regular operators whose orbits under any vector satisfy some growth conditions several results on left invertible operators close to isometries are extended we also give numerous results on the moore penrose inverse for regular operators in this particular setting

instabilities and inaccuracies in the integration of highly oscillatory problems by means of a two frequency test problem we analyze the instabilities and inaccuracies that may impair the performance of multiple time steps split operator integrators in highly oscillatory situations such as those encountered in molecular dynamics astrophysics or partial differential equations describing waves considered are the impulse verlet i r respa method the mollified impulse method and the reversible averaging integrator the analysis covers errors in positions momenta and energy

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type this paper is concerned with the numerical properties of q methods for the solution of alternately advanced and retarded differential equations with piecewise continuous arguments using two q methods namely the one leg q method and the linear q method the necessary and sufficient conditions under which the analytic stability region is contained in the numerical stability region are obtained and the conditions of oscillations for the q methods are also obtained it is proved that oscillations of the analytic solution are preserved by the q methods furthermore the relationships between stability and oscillations are revealed some numerical experiments are presented to illustrate our results

normed linear relations domain decomposability adjoint subspaces and selections abstract this paper is a study of normed linear relations with emphasis on results and properties that do not necessarily hold or have no counterparts for arbitrary normed convex processes sufficient conditions are given under which the norm of a linear relation m is equal to the norm of some operator part i e a single valued linear selection of m generalizations of the closed graph theorem and the open mapping principle for linear relations are established in terms of linear selections the norm of the adjoint subspace of a closed linear relation m is shown to be equal to the norm of m without any restrictions on the domain of m the rich structure of normed linear relations is uncovered via four fundamental concepts and tools algebraic topological and proximinal operator parts and the adjoint subspace of a linear relation notions of algebraic topological and orthogonal domain decomposability of linear relations in banach spaces are studied and the relationships between these notions of domain decomposability and the corresponding notions of operator parts are determined

a perturbation result for q open quotient morphisms in normed spaces and applications to linear relations using some techniques of perturbation theory for quotient morphisms between banach spaces we obtain necessary and sufficient conditions for the stability of the topological index of a q open quotient morphism under small with respect to the gap topology perturbations with quotient morphisms as an application we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation with closed multivalued part between normed spaces under small perturbations with linear relations