homogenization limits and wigner transforms we present a theory for carrying out homogenization limits for quadratic functions called energy densities of solutions of initial value problems ivps with anti self adjoint spatial pseudo differential operators pdos the approach is based on the introduction of phase space wigner matrix measures that are calculated by solving kinetic equations involving the spectral properties of the pdo the weak limits of the energy densities are then obtained by taking moments of the wigner measure the very general theory is illustrated by typical examples like semi classical limits of schrodinger equations with or without a periodic potential the homogenization limit of the acoustic equation in a periodic medium and the classical limit of the dirac equation 1997 john wiley sons inc

improving performance of fxlms algorithm for active noise control of impulsive noise this paper concerns active noise control anc of impulsive noise modeled using non gaussian stable processes the most famous filtered x least mean square fxlms algorithm for anc systems is based on minimization of variance of error signal for the impulse noise the fxlms algorithm becomes unstable as second order moments do not exist for non gaussian processes among the existing algorithms for anc of impulsive noise one is based on minimizing least mean p power lmp of the error signal resulting in fxlmp algorithm the other is based on modifying on the basis of statistical properties the reference signal in update of fxlms algorithm in this paper the proposed algorithm is an extension of the later approach extensive simulations are carried out which demonstrate the effectiveness of the proposed algorithm it achieves the best performance among the existing algorithms and at the same computational complexity as that of fxlms algorithm

a conservative fourier pseudo spectral method for the nonlinear schrodinger equation a fourier pseudo spectral method that conserves mass and energy is developed for a two dimensional nonlinear schrodinger equation by establishing the equivalence between the semi norm in the fourier pseudo spectral method and that in the finite difference method we are able to extend the result in ref 56 to prove that the optimal rate of convergence of the new method is in the order of o n r 2 in the discrete l 2 norm without any restrictions on the grid ratio where n is the number of modes used in the spectral method and is the time step size a fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method numerical examples are presented to show the efficiency and accuracy of the new method

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice i homogeneous problems we present a computational approach for the wkb approximation of the wave function of an electron moving in a periodic one dimensional crystal lattice we derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the bloch spectrum of the schrodinger operator relying on the framework of the multibranch entropy solutions introduced by brenier and corrias we compute efficiently multiphase solutions using well adapted and simple numerical schemes in this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method

numerical approximation of the schrodinger equation with the electromagnetic field by the hagedorn wave packets in this paper we approximate the semi classical schrodinger equation in the presence of electromagnetic field by the hagedorn wave packets approach by operator splitting the hamiltonian is divided into the modified part and the residual part the modified hamiltonian which is the main new idea of this paper is chosen by the fact that hagedorn wave packets are localized both in space and momentum so that a crucial correction term is added to the truncated hamiltonian and is treated by evolving the parameters associated with the hagedorn wave packets the residual part is treated by a galerkin approximation we prove that with the modified hamiltonian only the hagedorn wave packets dynamics give the asymptotic solution with error o e1 2 where e is the scaled planck constant we also prove that the galerkin approximation for the residual hamiltonian can reduce the approximation error to o ek 2 where k depends on the number of hagedorn wave packets added to the dynamics this approach is easy to implement and can be naturally extended to the multidimensional cases unlike the high order gaussian beam method in which the non constant cut off function is necessary and some extra error is introduced the hagedorn wave packets approach gives a practical way to improve accuracy even when e is not very small

gaussian wave packet transform based numerical scheme for the semi classical schrodinger equation with random inputs abstract in this work we study the semi classical limit of the schrodinger equation with random inputs and show that the semi classical schrodinger equation produces o e oscillations in the random variable space with the gaussian wave packet transform the original schrodinger equation is mapped to an ordinary differential equation ode system for the wave packet parameters coupled with a partial differential equation pde for the quantity w in rescaled variables further we show that the w equation does not produce e dependent oscillations and thus it is more amenable for numerical simulations we propose multi level sampling strategy in implementing the gaussian wave packet transform where in the most costly part i e simulating the w equation it is sufficient to use e independent samples we also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm and hopefully shed light on possible future directions

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

a posteriori error control and adaptivity for crank nicolson finite element approximations for the linear schrodinger equation we derive optimal order a posteriori error estimates for fully discrete approximations of linear schrodinger type equations in the l infty l 2 l l 2 norm for the discretization in time we use the crank nicolson method while for the space discretization we use finite element spaces that are allowed to change in time the derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of schrodinger equations the final estimates are obtained using energy techniques and residual type estimators various numerical experiments for the one dimensional linear schrodinger equation in the semiclassical regime verify and complement our theoretical results the numerical implementations are performed with both uniform partitions and adaptivity in time and space for adaptivity we further develop and analyze an existing time space adaptive algorithm to the cases of schrodinger equations the adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem especially for small values of the planck constant

dimension reduction of the schrodinger equation with coulomb and anisotropic confining potentials we consider dimension reduction for the three dimensional 3d schrodinger equation with the coulomb interaction and an anisotropic confining potential to lower dimensional models in the limit of infinitely strong confinement in one or two space dimensions and obtain formally the surface adiabatic model sam or surface density model sdm in two dimensions 2d and the line adiabatic model lam in one dimension 1d efficient and accurate numerical methods for computing ground states and dynamics of the sam sdm and lam models are presented based on efficient and accurate numerical schemes for evaluating the effective potential in lower dimensional models they are applied to find numerically convergence and convergence rates for the dimension reduction from 3d to 2d and 3d to 1d in terms of ground state and dynamics which confirm some existing analytical results for the dimension reduction in the literature in particular we explain and demonstrate that the standard schrodinger poisson system in 2

high order time splitting hermite and fourier spectral methods in this paper we are concerned with the numerical solution of the time dependent gross pitaevskii equation gpe involving a quasi harmonic potential primarily we consider discretisations that are based on spectral methods in space and higher order exponential operator splitting methods in time the resulting methods are favourable in view of accuracy and efficiency moreover geometric properties of the equation such as particle number and energy conservation are well captured regarding the spatial discretisation of the gpe we consider two approaches in the unbounded domain we employ a spectral decomposition of the solution into hermite basis functions on the other hand restricting the equation to a sufficiently large bounded domain fourier techniques are applicable for the time integration of the gpe we study various exponential operator splitting methods of convergence orders two four and six our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting fourier and hermite pseudospectral methods for the time evolution of the gpe furthermore we illustrate the effectiveness of higher order time splitting methods compared to standard integrators in a long term integration

stability analysis of difference schemes for variable coefficient schro dinger type equations we consider the stability of difference schemes for the solution of the initial boundary value problem for the equation u 1 left a x t u x right x b x t u c x t u f x t where u a b c and f are complex valued functions using energy methods we establish the stability of a general two level scheme which includes euler s method crank nicolson s method and the backward euler method if the coefficient a x t is purely imaginary the explicit euler method is unconditionally unstable for this case we propose a new scheme with appropriately chosen artificial dissipation which we prove to be conditionally stable

the semiclassical limit of the defocusing nls hierarchy we establish the semiclassical limit of the one dimensional defocusing cubic nonlinear schrodinger nls equation complete integrability is exploited to obtain a global characterization of the weak limits of the entire nls hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows consequently this also establishes the zero dispersion limit of the modified korteweg de vries equation that resides in that hierarchy we have adapted and clarified the strategy introduced by lax and levermore to study the zero dispersion limit of the korteweg de vries equation expanding it to treat entire integrable hierarchies and strengthening the limits obtained a crucial role is played by the convexity of the underlying log determinant with respect to the times associated with the commuting flows c 1999 john wiley sons inc

high order exponential operator splitting methods for time dependent schrodinger equations in this paper we deduce high order error bounds for exponential operator splitting methods the employed techniques are specific to linear differential equations of the form u t a u t b u t t geq 0 involving an unbounded operator a in particular evolutionary schrodinger equations with sufficiently regular initial values are included in the analysis

fast and accurate evaluation of nonlocal coulomb and dipole dipole interactions via the nonuniform fft we present a fast and accurate algorithm for the evaluation of nonlocal long range coulomb and dipole dipole interactions in free space the governing potential is simply the convolution of an interaction kernel u mathbf x and a density function rho mathbf x psi mathbf x 2 for some complex valued wave function psi mathbf x permitting the formal use of fourier methods these are hampered by the fact that the fourier transform of the interaction kernel widehat u mathbf k has a singularity and or widehat rho mathbf k ne0 at the origin mathbf k bf 0 in fourier phase space thus accuracy is lost when using a uniform cartesian grid in mathbf k which would otherwise permit the use of the fft for evaluating the convolution here we make use of a high order discretization of the fourier integral accelerated by the nonuniform fast fourier transform nufft by adopting spherical and polar phase space discretizations in three and two dimensions respectively

numerical study of time splitting and space time adaptive wavelet scheme for schrodinger equations in this paper we present a time splitting and space time adaptive wavelet scheme for solving schrodinger equations with a small planck constant the scheme is stable and it can overcome high frequency pseudo oscillation of the solution the concrete steps and numerical experiments are also given

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice ii impurities confinement and bloch oscillations we present a computational approach for the wkb approximation of the wavefunction of an electron moving in a periodic one dimensional crystal lattice by means of a nonstrictly hyperbolic system whose flux function stems from the bloch spectrum of the schrodinger operator this second part focuses on the handling of the source terms which originate from adding a slowly varying exterior potential physically relevant examples are the occurrence of bloch oscillations in case it is linear a quadratic one modelling a confining field and the harmonic coulomb term resulting from the inclusion of a donor impurity inside an otherwise perfectly homogeneous lattice

nonlinear system identification in impulsive environments nonlinear system identification has been studied under the assumption that the noise has finite second and higher order statistics in many practical applications impulsive measurement noise severely weakens the effectiveness of conventional methods in this paper spl alpha stable noise is used as a noise model in such case the minimum mean square error mmse criterion is no longer an appropriate metric for estimation error due to the lack of finite second order statistics of the noise therefore we adopt minimum dispersion criterion which in turn leads to the adaptive least mean pth power lmp algorithm it is shown that the lmp algorithm under the spl alpha stable noise model converges as long as the step size satisfies certain conditions the effect of p on the performance is also investigated compared with conventional methods the proposed method is more robust to impulsive noise and has better performance

a new robust variable step size nlms algorithm a new framework for designing robust adaptive filters is introduced it is based on the optimization of a certain cost function subject to a time dependent constraint on the norm of the filter update particularly we present a robust variable step size nlms algorithm which optimizes the square of the a posteriori error we also show the link between the proposed algorithm and another one derived using a robust statistics approach in addition a theoretical model for predicting the transient and steady state behavior and a proof of almost sure filter convergence are provided the algorithm is then tested in different environments for system identification and acoustic echo cancelation applications

convergence of a semiclassical wavepacket based time splitting for the schrodinger equation we propose a new algorithm for solving the semiclassical time dependent schrodinger equation the algorithm is based on semiclassical wavepackets the focus of the analysis is only on the time discretization convergence is proved to be quadratic in the time step and linear in the semiclassical parameter varepsilon

mass preserving discontinuous galerkin methods for schrodinger equations we construct analyze and numerically validate a class of mass preserving direct discontinuous galerkin ddg schemes for schrodinger equations subject to both linear and nonlinear potentials up to round off error these schemes preserve the discrete version of the mass of the continuous solution for time discretization we use the crank nicolson for linear schrodinger equations and the strang splitting for nonlinear schrodinger equations so that numerical mass is still preserved at each time step the ddg method when applied to linear schrodinger equations is shown to have the optimal k 1 th order of accuracy for polynomial elements of degree k the numerical tests demonstrate both accuracy and capacity of these methods

numerical computations for n coupled nonlinear schrodinger equations by split step spectral methods in this paper various split step spectral sssp schemes are proposed for n coupled nonlinear schrodinger n cnls equations especially for the systems with n 3 for which numerical studies are few these schemes are spectrally accurate in space and sth s 1 2 4 6 8 order in time they are proved to be conservative and to admit the exact plane wave solution extensive numerical experiments are carried out for the 3 and 4 cnls systems to confirm the theoretical analysis half of the schemes are shown to possess wonderful ability of capturing high frequency waves however the eighth order schemes which seems to be optimal fail in this test accuracy and efficiency of the schemes are compared with each other and the high order schemes exhibit better since the eighth order schemes are not better than the sixth order ones it is believed that constructing too much high order schemes are unnecessary moreover interactions of two soliton solutions are also well simulated by the fourth order and the sixth order schemes so the two methods are sufficient for use

the separation of zeros for entire functions of exponential type soit f une fonction entiere a valeur reelle sur l axe reel r de type exponentiel de carre integrable sur r si f a une infinite de zeros sur cet axe alors il existe au moins une paire de zeros consecutifs dont la distance est au moins

bloch decomposition based gaussian beam method for the schrodinger equation with periodic potentials the linear schrodinger equation with periodic potentials is an important model in solid state physics the most efficient direct simulation using a bloch decomposition based time splitting spectral method 18 requires the mesh size to be o e where e is the scaled semiclassical parameter in this paper we generalize the gaussian beam method introduced in jin et al 23 to solve this problem asymptotically we combine the technique of bloch decomposition and the eulerian gaussian beam method to arrive at an eulerian computational method that requires mesh size of o e the accuracy of this method is demonstrated via several numerical examples

spectral splitting method for nonlinear schrodinger equations with singular potential we consider the time dependent one dimensional nonlinear schrodinger equation with pointwise singular potential by means of spectral splitting methods we prove that the evolution operator is approximated by the lie evolution operator where the kernel of the lie evolution operator is explicitly written this result yields a numerical procedure which is much less computationally expensive than multi grid methods previously used furthermore we apply the lie approximation in order to make some numerical experiments concerning the splitting of a soliton interaction among solitons and blow up phenomenon

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

convergence analysis of high order time splitting pseudospectral methods for nonlinear schrodinger equations in this work the issue of favorable numerical methods for the space and time discretization of low dimensional nonlinear schrodinger equations is addressed the objective is to provide a stability and error analysis of high accuracy discretizations that rely on spectral and splitting methods as a model problem the time dependent gross pitaevskii equation arising in the description of bose einstein condensates is considered for the space discretization pseudospectral methods collocated at the associated quadrature nodes are analyzed for the time integration high order exponential operator splitting methods are studied where the decomposition of the function defining the partial differential equation is chosen in accordance with the underlying spectral method the convergence analysis relies on a general framework of abstract nonlinear evolution equations and fractional power spaces defined by the principal linear part essential tools in the derivation of a temporal global error estimate are furthe

gaussian beam methods for the schrodinger equation with discontinuous potentials we propose eulerian and lagrangian gaussian beam methods for the schrodinger equation with discontinuous potentials at the quantum barriers where the potential is discontinuous we derive suitable interface conditions to account for quantum scattering information these scattering interface conditions are then built into the numerical fluxes in the eulerian level set formulation of the gaussian beam methods and are also used in the lagrangian formulation including an interface condition for the hessian matrix we carry both 1d and 2d numerical examples to verify the accuracy of the method

practical symplectic partitioned runge kutta and runge kutta nystrom methods we present new symmetric fourth and sixth order symplectic partitioned runge kutta and runge kutta nystrom methods we studied compositions using several extra stages optimising the efficiency an effective error ef is defined and an extensive search is carried out using the extra parameters the new methods have smaller values of ef than other methods found in the literature when applied to several examples they perform up to two orders of magnitude better than previously known method which is in very good agreement with the values of ef

the gaussian wave packet transform efficient computation of the semi classical limit of the schrodinger equation part 1 formulation and the one dimensional case in this paper an efficient and accurate method for simulating the propagation of a localized solution of the schrodinger equation with a smooth potential near the semiclassical limit is presented we are interested computing arbitrarily accurate solutions when the non dimensional planck s constant e is small but not negligible the method is based on a time dependent transformation of the independent variables closely related to gaussian wave packets a rescaled wave function w satisfies a new schrodinger equation with a time dependent potential which is a perturbation of the harmonic oscillator the perturbation being o e so that all stiffness in space and time is greatly reduced in fact for integration in a fixed time interval the number of modes required to fully resolve the problem decreases when e is decreased the original wave function may be reconstructed by fourier interpolation although expectation values of the observables can be computed directly from the function w itself if the initial condition is a gaussian wave packet very few modes are necessary to fully resolve the w variable so for short time very accurate solutions can be obtained at low computational cost initial conditions other than gaussians wave packets can also be used in this paper the gaussian wave packet transform is carefully outlined and applied to the schrodinger equation in one dimension detailed numerical tests show the efficiency and accuracy of the approach in the sequel of this paper this approach has been extended to the multidimensional case

numerical study of time splitting spectral discretizations of nonlinear schrodinger equations in the semiclassical regimes in this paper we study the performance of time splitting spectral approximations for general nonlinear schrodinger equations nls in the semiclassical regimes where the planck constant varepsilon is small the time splitting spectral approximation under study is explicit unconditionally stable and conserves the position density in l 1 moreover it is time transverse invariant and time reversible when the corresponding nls is extensive numerical tests are presented for weak strong focusing defocusing nonlinearities for the gross pitaevskii equation and for current relaxed quantum hydrodynamics the tests are geared towards the understanding of admissible meshing strategies for obtaining correct physical observables in the semiclassical regimes furthermore comparisons between the solutions of the nls and its hydrodynamic semiclassical limit are presented

fourier splitting methods for the dynamics of rotating bose einstein condensates abstract we present a new method to propagate rotating bose einstein condensates subject to explicitly time dependent trapping potentials using algebraic techniques we combine magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the hamiltonian that can be computed using only fourier transforms at the cost of solving a small system of polynomial equations the resulting scheme solves the challenging component of the nonlinear hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating bose einstein condensates the method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates e g for small nonlinearities since it retains the near integrable structure of the problem for large nonlinearities the method remains highly efficient if higher order p 2 is sought furthermore we show how it can be adapted to the presence of dissipation terms numerical examples illustrate the performance of the scheme

symmetrized local error estimators for time reversible one step methods in nonlinear evolution equations abstract prior work on computable defect based local error estimators for linear time reversible integrators is extended to nonlinear and nonautonomous evolution equations we prove that the asymptotic results from the linear case auzinger and koch 2018 remain valid i e the modified estimators yield an improved asymptotic order as the step size goes to zero typically the computational effort is only slightly higher than for conventional defect based estimators and it may even be lower in some cases we illustrate this by some examples and present numerical results for evolution equations of schrodinger type solved by either time splitting or magnus type integrators finally we demonstrate that adaptive time stepping schemes can be successfully based on our local error estimators

adaptive volterra filter with continuous lp norm using a logarithmic cost for nonlinear active noise control abstract the filtered x least mean lp norm fxlmp algorithm is proven to be useful for nonlinear active noise control nanc systems however its performance deteriorates when the impulsive noises are presented in nanc systems to surmount this shortcoming a new nonlinear adaptive algorithm based on volterra expansion model vfxloglmp is developed in this paper which is derived by minimizing the lp norm of logarithmic cost it is found that the fxlmp and vfxloglmp require to select an appropriate value of p according to the prior information on noise characteristics which prohibit their practical applications based on vfxloglmp algorithm we proposed a continuous lp norm algorithm with logarithmic cost vfxlogclmp which does not need the parameter selection and thresholds estimation benefiting from the various error norms for 1 p 2 it remains the robustness of vfxloglmp moreover the convergence behavior of vfxlogclmp for moving average secondary paths and stochastic input signals is performed compared to the existing algorithms two versions of the proposed algorithms have much better convergence and stability in impulsive noise environments

strang splitting for a semilinear schrodinger equation with damping and forcing abstract we propose and analyze a strang splitting method for a cubic semilinear schrodinger equation with forcing and damping terms and subject to periodic boundary conditions the nonlinear part is solved analytically whereas the linear part space derivatives damping and forcing is approximated by the exponential trapezoidal rule the necessary operator exponentials and functions can be computed efficiently by fast fourier transforms if space is discretized by spectral collocation under natural regularity assumptions we first show global existence of the problem in h 4 t and establish global bounds depending on properties of the forcing the main result of our error analysis is first order convergence in h 1 t and second order convergence in l 2 t on bounded time intervals

convergence analysis of high order time splitting pseudo spectral methods for rotational gross pitaevskii equations a convergence analysis of time splitting pseudo spectral methods adapted for time dependent gross pitaevskii equations with additional rotation term is given for the time integration high order exponential operator splitting methods are studied and the space discretization relies on the generalized laguerre fourier spectral method with respect to the x y x y variables as well as the hermite spectral method in the z z direction essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations fractional power spaces defined by the principal linear part a sobolev type inequality in a curved rectangle and results on the asymptotical distribution of the nodes and weights associated with gau laguerre quadrature the obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained provided that the problem data satisfy suitable regularity requirements a numerical example confirms the theoretical convergence estimate

numerical methods for computing ground states and dynamics of nonlinear relativistic hartree equation for boson stars efficient and accurate numerical methods are presented for computing ground states and dynamics of the three dimensional 3d nonlinear relativistic hartree equation both without and with an external potential this equation was derived recently for describing the mean field dynamics of boson stars in its numerics due to the appearance of pseudodifferential operator which is defined in phase space via symbol spectral method is more suitable for the discretization in space than other numerical methods such as finite difference method etc for computing ground states a backward euler sine pseudospectral besp method is proposed based on a gradient flow with discrete normalization and respectively for computing dynamics a time splitting sine pseudospectral tssp method is presented based on a splitting technique to decouple the nonlinearity both besp and tssp are efficient in computation via discrete sine transform and are of spectral accuracy in spatial discretization tssp is of second order accuracy in temporal discretization and conserves the normalization in discretized level in addition when the external potential and initial data for dynamics are spherically symmetric the original 3d problem collapses to a quasi 1d problem for which both besp and tssp methods are extended successfully with a proper change of variables finally extensive numerical results are reported to demonstrate the spectral accuracy of the methods and to show very interesting and complicated phenomena in the mean field dynamics of boson stars

the backward phase flow and fbi transform based eulerian gaussian beams for the schrodinger equation we propose the backward phase flow method to implement the fourier bros iagolnitzer fbi transform based eulerian gaussian beam method for solving the schrodinger equation in the semi classical regime the idea of eulerian gaussian beams has been first proposed in 12 in this paper we aim at two crucial computational issues of the eulerian gaussian beam method how to carry out long time beam propagation and how to compute beam ingredients rapidly in phase space by virtue of the fbi transform we address the first issue by introducing the reinitialization strategy into the eulerian gaussian beam framework essentially we reinitialize beam propagation by applying the fbi transform to wavefields at intermediate time steps when the beams become too wide to address the second issue inspired by the original phase flow method we propose the backward phase flow method which allows us to compute beam ingredients rapidly numerical examples demonstrate the efficiency and accuracy of the proposed algorithms

active noise control with on line estimation of non gaussian noise characteristics abstract active noise control anc is a methodology for attenuating noise based on adaptive signal processing algorithms anc is well assessed for the attenuation of gaussian noise but the rejection of non gaussian impulsive noise signals represents a much more critical task that may even impair algorithm convergence to overcome this problem the adaptive filter weight update process must be modified by discarding or discounting samples associated with impulsive noise this can be done either by modeling the impulsive noise with a non gaussian distribution such as the symmetric stable s s distribution or by applying an outlier detection method with both approaches the accuracy in the noise description appears to be crucial for effective noise reduction this paper proposes two novel approaches for the attenuation of impulsive noise both for invariant and time varying noise distributions the first one is based on the on line estimation of an s s model of the noise probabilistic description the second relies on a simple on line recursive procedure that reliably estimates amplitude thresholds for outlier detection both methods compare favorably with competitor approaches while maintaining a sufficiently low algorithm complexity several examples are shown to demonstrate the algorithms effectiveness

a wigner measure analysis of the dufort frankel scheme for the schrodinger equation we apply wigner transform techniques to the analysis of the dufort frankel difference scheme for the schrodinger equation and to the continuous analogue of the scheme in the case of a small scaled planck constant semiclassical regime 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 or not caustics develop numerical test examples are presented to help interpret the theory and to compare the dufort frankel scheme to other difference schemes for the schr odinger equation

defect based local error estimators for splitting methods with application to schrodinger equations part ii higher order methods for linear problems in this work defect based local error estimators for higher order exponential operator splitting methods are constructed and analyzed in the context of time dependent linear schrodinger equations the technically involved procedure is carried out in detail for a general three stage third order splitting method and then extended to the higher order case asymptotical correctness of the a posteriori local error estimator is proven under natural commutator bounds for the involved operators and along the way the known non stiff order conditions and a priori convergence bounds are recovered the theoretical error estimates for higher order splitting methods are confirmed by numerical examples for a test problem of schrodinger type further numerical experiments for a test problem of parabolic type complement the investigations

a generalized laguerre fourier hermite pseudospectral method for computing the dynamics of rotating bose einstein condensates a time splitting generalized laguerre fourier hermite pseudospectral method is proposed for computing the dynamics of rotating bose einstein condensates becs in two and three dimensions the new numerical method is based on the following i the use of a time splitting technique for decoupling the nonlinearity ii the adoption of polar coordinates in two dimensions and cylindrical coordinates in three dimensions such that the angular rotation term becomes constant coefficient and iii the construction of eigenfunctions for the linear operator by properly scaling the generalized laguerre fourier and hermite functions the new method enjoys the following properties i it is explicit time reversible and time transverse invariant ii it conserves the position density and is spectrally accurate in space and second order or fourth order accurate in time and iii it solves the problem in the original whole space instead of in a truncated artificial computational domain the method is also extend

a numerical study of adaptive space and time discretisations for gross pitaevskii equations as a basic principle benefits of adaptive discretisations are an improved balance between required accuracy and efficiency as well as an enhancement of the reliability of numerical computations in this work the capacity of locally adaptive space and time discretisations for the numerical solution of low dimensional nonlinear schrodinger equations is investigated the considered model equation is related to the time dependent gross pitaevskii equation arising in the description of bose einstein condensates in dilute gases the performance of the fourier pseudo spectral method constrained to uniform meshes versus the locally adaptive finite element method and of higher order exponential operator splitting methods with variable time stepsizes is studied numerical experiments confirm that a local time stepsize control based on a posteriori local error estimators or embedded splitting pairs respectively is effective in different situations with an enhancement either in efficiency or reliability as expected adaptive time splitting schemes combined with fast fourier transform techniques are favourable regarding accuracy and efficiency when applied to gross pitaevskii equations with a defocusing nonlinearity and a mildly varying regular solution however the numerical solution of nonlinear schrodinger equations in the semi classical regime becomes a demanding task due to the highly oscillatory and nonlinear nature of the problem the spatial mesh size and the time increments need to be of the size of the decisive parameter 0 e 1 especially when it is desired to capture correctly the quantitative behaviour of the wave function itself the required high resolution in space constricts the feasibility of numerical computations for both the fourier pseudo spectral and the finite element method nevertheless for smaller parameter values locally adaptive time discretisations facilitate to determine the time stepsizes sufficiently small in order that the numerical approximation captures correctly the behaviour of the analytical solution further illustrations for gross pitaevskii equations with a focusing nonlinearity or a sharp gaussian as initial condition respectively complement the numerical study

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

numerical study of fractional nonlinear schrodinger equations using a fourier spectral method we provide a detailed numerical investigation of dispersive schrodinger type equations involving a fractional laplacian in an one dimensional case by an appropriate choice of the dispersive exponent both mass and energy sub and supercritical regimes can be identified this allows us to study the possibility of finite time blow up versus global existence the nature of the blow up the stability and instability of nonlinear ground states and the long time dynamics of solutions the latter is also studied in a semiclassical setting moreover we numerically construct ground state solutions of the fractional nonlinear schrodinger equation

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

defect based local error estimators for splitting methods with application to schrodinger equations part iii the present work is concerned with the efficient time integration of nonlinear evolution equations by exponential operator splitting methods defect based local error estimators serving as a reliable basis for adaptive stepsize control are constructed and analyzed in the context of time dependent nonlinear schrodinger equations asymptotical correctness of the local error estimators associated with the first order lie trotter and second order strang splitting methods is proven numerical examples confirm the theoretical results and illustrate the performance of adaptive stepsize control

a new sobolev gradient method for direct minimization of the gross pitaevskii energy with rotation in this paper we improve traditional steepest descent methods for the direct minimization of the gross pitaevskii gp energy with rotation at two levels we first define a new inner product to equip the sobolev space h 1 and derive the corresponding gradient second for the treatment of the mass conservation constraint we use a projection method that avoids more complicated approaches based on modified energy functionals or traditional normalization methods the descent method with these two new ingredients is studied theoretically in a hilbert space setting and we give a proof of the global existence and convergence in the asymptotic limit to a minimizer of the gp energy the new method is implemented in both finite difference and finite element two dimensional settings and is used to compute various complex configurations with vortices of rotating bose einstein condensates the new sobolev gradient method shows better numerical performances compared to classical l 2 or h 1 gradient methods especially when high rotation rates are considered

optimized high order splitting methods for some classes of parabolic equations we are concerned with the numerical solution obtained by splitting methodsof certain parabolic partial differential equations splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients it has been demonstrated that this se cond order barrier can be overcome by using splitting methods with complex valued coefficients with positive real parts in this way method s of orders 3 to 14 by using the suzuki yoshida triple and quadruple jump composition procedure have been explicitly built here we reconsider this technique an d show that it is inherently bounded to order 14 and clearly sub optimal with respect to error constants as an alternative we solve directly the algebraic equations arising from the order conditions and construct methods of orders 6 and 8 that are the most accurate ones available at present time even when low accuracies are desired we also show that in the general case 14 is not an order barrier for splitting method s with complex coefficients with positive real part by building explicitly a method of order 16 as a composition of methods of order 8

improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm convergence analysis of stochastic gradient adaptive filters using the sign algorithm is presented in this paper the methods of analysis currently available in literature assume that the input signals to the filter are white this restriction is removed for gaussian signals in our analysis expressions for the second moment of the coefficient vector and the steady state error power are also derived simulation results are presented and the theoretical and empirical curves show a very good match

defect based local error estimators for splitting methods with application to schrodinger equations part i the linear case we introduce a defect correction principle for exponential operator splitting methods applied to time dependent linear schrodinger equations and construct a posteriori local error estimators for the lie trotter and strang splitting methods under natural commutator bounds on the involved operators we prove asymptotical correctness of the local error estimators and along the way recover the known a priori convergence bounds numerical examples illustrate the theoretical local and global error estimates

an efficient and stable numerical method for the maxwell dirac system in this paper we present an explicit unconditionally stable and accurate numerical method for the maxwell dirac system md and use it to study dynamics of md as preparatory steps we take the three dimensional 3d maxwell dirac system scale it to obtain a two parameter model and review plane wave solution of free md then we present a time splitting spectral method tssp for md the key point in the numerical method is based on a time splitting discretization of the dirac system and to discretize nonlinear wave type equations by pseudospectral method for spatial derivatives and then solving the ordinary differential equations odes in phase space analytically under appropriate chosen transmission conditions between different time intervals the method is explicit unconditionally stable time reversible time transverse invariant and of spectral order accuracy in space and second order accuracy in time moreover it conserves the particle density exactly in discretized level and gives exact results for plane wave solution of free md extensive numerical tests are presented to confirm the above properties of the numerical method furthermore the tests also suggest the following meshing strategy or e resolution is admissible in the nonrelativistic limit regime 0 e 1 spatial mesh size h o e and time step t o e2 where the parameter e is inversely proportional to the speed of light

analysis of a leap frog pseudospectral scheme for the schrodinger equation the numerical properties of a leap frog pseudospectral scheme for the schrodinger equation are analyzed stability second order accuracy in time and spectral accuracy in space are discussed considering the linear schrodinger equation with potential in a periodic setting further issues regarding phase error gauge invariance conservation properties and commutation relations are addressed results of numerical experiments are reported to demonstrate the validity and limitations of the theoretical findings and for comparison with the well known crank nicholson finite difference scheme

adaptive volterra filters for active control of nonlinear noise processes this paper presents a volterra filtered x least mean square lms algorithm for feedforward active noise control the research has demonstrated that linear active noise control anc systems can be successfully applied to reduce the broadband noise and narrowband noise specifically such linear anc systems are very efficient in reduction of low frequency noise however in some situations the noise that comes from a dynamic system may he a nonlinear and deterministic noise process rather than a stochastic white or tonal noise process and the primary noise at the canceling point may exhibit nonlinear distortion furthermore the secondary path estimate in the anc system which denotes the transfer function between the secondary source secondary speaker and the error microphone may have nonminimum phase and hence the causality constraint is violated if such situations exist the linear anc system will suffer performance degradation an implementation of a volterra filtered x lms vfxlms algorithm based on a multichannel structure is described for feedforward active noise control numerical simulation results show that the developed algorithm achieves performance improvement over the standard filtered x lms algorithm for the following two situations 1 the reference noise is a nonlinear noise process and at the same time the secondary path estimate is of nonminimum phase 2 the primary path exhibits the nonlinear behavior in addition the developed vfxlms algorithm can also be employed as an alternative in the case where the standard filtered x lms algorithm does not perform well

numerical computations for long wave short wave interaction equations in semi classical limit this paper presents and compares various numerical techniques for the long wave short wave interaction equations in addition to the standard explicit implicit schemes and the spectral methods a novel scheme srk which is based on a time splitting approach combined with the runge kutta method is presented we demonstrate that not only the srk scheme is efficient compared to the split step spectral methods but it can apply directly to problems with general boundary conditions the conservation properties of the numerical schemes are discussed numerical simulations are reported for case studies with different types of initial data the present study enhances our understanding of the behavior of nonlinear dispersive waves in the semi classical limit

frozen gaussian approximation based domain decomposition methods for the linear schrodinger equation beyond the semi classical regime the paper is devoted to develop efficient domain decomposition methods for the linear schrodinger equation beyond the semiclassical regime which does not carry a small enough rescaled planck constant for asymptotic methods e g geometric optics to produce a good accuracy but which is too computationally expensive if direct methods e g finite difference are applied this belongs to the category of computing middle frequency wave propagation where neither asymptotic nor direct methods can be directly used with both efficiency and accuracy motivated by recent works of the authors on absorbing boundary conditions antoine et al 2014 13 and yang and zhang 2014 43 we introduce semiclassical schwarz waveform relaxation methods sswr which are seamless integrations of semiclassical approximation to schwarz waveform relaxation methods two versions are proposed respectively based on herman kluk propagation and geometric optics and we prove the convergence and provide numerical evidence of efficiency and accuracy of these methods

the automatic solution of partial differential equations using a global spectral method a spectral method for solving linear partial differential equations pdes with variable coefficients and general boundary conditions defined on rectangular domains is described based on separable representations of partial differential operators and the one dimensional ultraspherical spectral method if a partial differential operator is of splitting rank 2 such as the operator associated with poisson or helmholtz the corresponding pde is solved via a generalized sylvester matrix equation and a bivariate polynomial approximation of the solution of degree n x n y is computed in o n x n y 3 2 operations partial differential operators of splitting rank 3 are solved via a linear system involving a block banded matrix in o min n x 3 n y n x n y 3 operations numerical examples demonstrate the applicability of our 2d spectral method to a broad class of pdes which includes elliptic and dispersive time evolution equations the resulting pde solver is written in matlab and is publicly available as part of chebfun it can resolve solutions requiring over a million degrees of freedom in under 60 seconds an experimental implementation in the julia language can currently perform the same solve in 10 seconds

a fourth order time splitting laguerre hermite pseudospectral method for bose einstein condensates a fourth order time splitting laguerre hermite pseudospectral method is introduced for bose einstein condensates becs in three dimensions with cylindrical symmetry the method is explicit time reversible and time transverse invariant it conserves the position density and is spectral accurate in space and fourth order accurate in time moreover the new method has two other important advantages i it reduces a three dimensional 3 d problem with cylindrical symmetry to an effective two dimensional 2 d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain the method is applied to vector gross pitaevskii equations vgpes for multicomponent becs extensive numerical tests are presented for the one dimensional 1 d gpe the 2 d gpe with radial symmetry the 3 d gpe with cylindrical symmetry as well as 3 d vgpes for two component becs to show the efficiency and accuracy of the new numerical method

stable explicit schemes for equations of the schro dinger type most conventional explicit finite difference schemes e g euler s scheme for solving the parabolic equation of schrodinger type u t iu xx are unconditionally unstable this difficulty can be overcome by introducing a dissipative term to the conventional explicit schemes based on this approach we derive a class of new explicit finite difference schemes which are conditionally stable spans two time levels and are o k h 2 accurate we also determine the schemes from this class that have the least restrictive stability requirements it is interesting to note that the analog of the lax wendroff scheme is unstable

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

two completely explicit and unconditionally convergent fourier pseudo spectral methods for solving the nonlinear schrodinger equation abstract this paper aims to construct and analyze two new fourier pseudo spectral fps methods for the general nonlinear schrodinger nls equation the two fps methods have two merits unconditional convergence and complete explicitness in the practical computation further more by introducing a modified mass functional and a modified energy functional the two fps methods are proved to preserve the total mass and energy in the discrete sense besides the standard energy method the key techniques used in our numerical analysis are a mathematical induction argument and a lifting technique without any restriction on the grid ratio and initial value we establish the optimal error estimate of the two fps methods for solving the general nls equation while previous work just is valid for the cubic nls equation and requires small initial value for the focusing case these two fps methods are proved to be spectrally accurate in space and second order accurate in time respectively the analysis framework can be used to prove the unconditional convergence of many other fourier pseudo spectral methods for solving the nls type equations we investigate the effect of the nonlinear term on the progression simulation of the plane wave the conservation of the invariants and the effect of initial data on the blow up solution via different parameters numerical results are reported to show the accuracy and efficiency of the proposed methods

a finite element method with mesh adaptivity for computing vortex states in fast rotating bose einstein condensates numerical computations of stationary states of fast rotating bose einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices in this paper we propose a low order finite element method with mesh adaptivity by metric control as an alternative approach to the commonly used high order finite difference or spectral approximation methods the mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states an imaginary time propagation method and a sobolev gradient descent method we first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that refinement using simultaneously the real and imaginary part of the solution is successful mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state considerable computational time saving is obtained compared to uniform mesh computations the new method is applied to compute difficult cases relevant for physical experiments large nonlinear interaction constant and high rotation rates

semi implicit operator splitting pade method for higher order nonlinear schrodinger equations abstract we propose a semi implicit finite difference operator splitting pade ospd method for solving the higher order nonlinear schrodinger equation which describes the optical soliton wave propagation in fibers the method achieves fourth order of accuracy in space and has been proven to be stable by linear stability analysis numerical experiments and comparisons are investigated to show the advantages and effectivity of the ospd method and some interesting collision behaviors are also observed

dispersion relation equation preserving fdtd method for nonlinear cubic schrodinger equation in this study we aim to solve the cubic nonlinear schrodinger cnls equation by the method of fractional steps over a time step from t n to t n 1 the linear part of the schrodinger equation is solved firstly through four time integration steps in this part of the simulation the explicit symplectic scheme of fourth order accuracy is adopted to approximate the time derivative term the second order spatial derivative term in the linear schrodinger equation is approximated by centered scheme the resulting symplectic and space centered difference scheme renders an optimized numerical dispersion relation equation in the second part of the simulation the solution of the nonlinear equation is computed exactly thanks to the embedded invariant nature within each time increment the proposed semi discretized difference scheme underlying the modified equation analysis of second kind and the method of dispersion error minimization has been assessed in terms of the spatial modified wavenumber or the temporal angular frequency resolution several problems have been solved to show that application of this new finite difference scheme for the calculation of one and two dimensional schrodinger equations can deemed conserve hamiltonian quantities and preserve dispersion relation equation dre

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice iii from ab initio models to wkb for schrodinger poisson this work is concerned with the semiclassical approximation of the schrodinger poisson equation modeling ballistic transport in a 1d periodic potential by means of wkb techniques it is derived by considering the mean field limit of a n body quantum problem then k multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for pauli s exclusion principle numerical experiments display the behaviour of self consistent wave packets and screening effects

efficient methods for linear schrodinger equation in the semiclassical regime with time dependent potential we build efficient and unitary hence stable methods for the solution of the linear time dependent schrodinger equation with explicitly time dependent potentials in a semiclassical regime the magnus zassenhaus schemes presented here are based on a combination of the zassenhaus decomposition bader et al 2014 found comput math 14 689 720 doi 10 1007 s10208 013 9182 8 1 with the magnus expansion of the time dependent hamiltonian we conclude with numerical experiments 1 http dx doi org 10 1007 s10208 013 9182 8

numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size

a minimisation approach for computing the ground state of gross pitaevskii systems in this paper we present a minimisation method for computing the ground state of systems of coupled gross pitaevskii equations our approach relies on a spectral decomposition of the solution into hermite basis functions inserting the spectral representation into the energy functional yields a constrained nonlinear minimisation problem for the coefficients for its numerical solution we employ a newton like method with an approximate line search strategy we analyse this method and prove global convergence appropriate starting values for the minimisation process are determined by a standard continuation strategy numerical examples with two and three component two dimensional condensates are included these experiments demonstrate the reliability of our method and nicely illustrate the effect of phase segregation