transmitter optimization for the multi antenna downlink with per antenna power constraints this paper considers the transmitter optimization problem for a multiuser downlink channel with multiple transmit antennas at the base station in contrast to the conventional sum power constraint on the transmit antennas this paper adopts a more realistic per antenna power constraint because in practical implementations each antenna is equipped with its own power amplifier and is limited individually by the linearity of the amplifier assuming perfect channel knowledge at the transmitter this paper investigates two different transmission schemes under the per antenna power constraint a minimum power beamforming design for downlink channels with a single antenna at each remote user and a capacity achieving transmitter design for downlink channels with multiple antennas at each remote user it is shown that in both cases the per antenna downlink transmitter optimization problem may be transformed into a dual uplink problem with an uncertain noise this generalizes previous uplink downlink duality results and transforms the per antenna transmitter optimization problem into an equivalent minimax optimization problem further it is shown that various notions of uplink downlink duality may be unified under a lagrangian duality framework this new interpretation of duality gives rise to efficient numerical optimization techniques for solving the downlink per antenna transmitter optimization problem

spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

a hybrid element formulation for the three dimensional penalty finite element analysis of incompressible fluids this work presents a hybrid element formulation for the three dimensional penalty finite element analysis of incompressible newtonian fluids the formulation is based on a mixed variational statement in which velocity and stresses are treated as independent field variables the main advantage of this formulation is that it bypasses the use of ad hoc techniques such as selective reduced integration that are commonly used in penalty based finite element formulations and directly yields high accuracy for the velocity and stress fields without the need to carry out smoothing in addition since the stress degrees of freedom are condensed out at an element level the cost of solving for the global degrees of freedom is the same as in a standard penalty finite element method although the gain in accuracy for both the velocity and stress including the pressure fields is quite significant

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

on the computation of ground state and dynamics of schrodinger poisson slater system in this paper we deal with the computation of ground state and dynamics of the schrodinger poisson slater sps system to this end backward euler and time splitting pseudospectral methods are proposed for the nonlinear schrodinger equation with the nonlocal hartree potential approximated by solving a poisson equation the approximation approaches for the hartree potential include fast convolution algorithms which are accelerated by using fft in 1d and fast multipole method fmm in 2d and 3d and sine fourier pseudospectral methods the inconsistency in 0 mode in fourier pseudospectral approach is pointed out which results in a significant loss of high order of accuracy as expected for spectral methods numerical comparisons show that in 1d the fast convolution and sine pseudospectral approaches are compatible while in 3d the fast convolution approach based on fmm is second order accurate and the fourier pseudospectral approach is better than it from both efficiency and accuracy point of view among all these approaches the sine pseudospectral one is the best candidate in the numerics of the sps system finally we apply the backward euler and time splitting sine pseudospectral methods to study the ground state and dynamics of 3d sps system in different setups

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

multigrid methods to accelerate convergence of element by element solution algorithms for viscous incompressible flows abstract the use of iterative solvers for large systems of equations results in a slow convergence rate after the small wave length error components are resolved for a fixed convergence tolerance limit the error resolution capability of an iterative solver is a function of the mesh density two iterative solvers gmres and orthomin which utilize the element by element ebe data structure of the finite element mesh are studied to demonstrate this behavior a significant improvement in the accuracy of the solution and the rate of convergence is achieved for these iterative solvers by using a successive mesh refinement scheme like in a multigrid method this approach is found to better resolve both the large and the small scale flow phenomenon for a fine mesh while satisfying the convergence criterion in a fraction of the cpu time required by the standard iterative methods an iterative penalty finite element model is used to solve the governing equations for laminar incompressible isothermal fluid flows the solution accuracy and cpu time savings are demonstrated for two sample problems

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

an implicit algorithm within the arbitrary lagrangian eulerian formulation for solving incompressible fluid flow with large boundary motions abstract the objective of the paper is to present an implicit algorithm for incompressible fluid flow solution using the arbitrary lagrangian eulerian ale formulation and to investigate solution accuracy and stability of the algorithm the governing equations of the implicit procedure are derived using isoparametric interpolations for the fluid velocities and pressure the details suitable for general use are presented in our derivations of the fundamental equations and of the basic finite element balance equations the penalty method is utilized to eliminate the pressure on the element level accuracy and stability of the solutions are demonstrated in three examples for which the analytical solutions are known in the first example the burger s equation analogue to 1 d fluid flows is solved without and with fe mesh motion to show that the mesh motion practically does not affect the solutions in the second example the solitary wave motion with large displacements of the free boundary is solved as a benchmark problem in the third example the fluid flow in an infinite contractible and expandable pipe with prescribed large radial wall motion is solved this example is specifically attractive for biological flows as in blood vessels and lung airways all solutions presented show that the proposed algorithm is sufficiently accurate and stable since the algorithm is implicit high accuracy of results can be achieved with a relatively large time step

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

an improved fast multipole algorithm for potential fields a new version of the fast multipole method fmm for potential fields is presented we introduce a new representation of potentials in which most translation operators are diagonal as a result for double precision calculations in two dimensions we obtain an improvement of a factor of two to four in speed compared to previously published algorithms the improvement is expected to be much greater in three dimensions the performance of the method is illustrated with several numerical examples

downlink mmse transceiver optimization for multiuser mimo systems duality and sum mse minimization we address the problem of minimum mean square error mmse transceiver design for point to multipoint transmission in multiuser multiple input multiple output mimo systems we focus on the problem of minimizing the downlink sum mse under a sum power constraint it is shown that this problem can be solved efficiently by exploiting a duality between the downlink and uplink mse feasible regions we propose two different optimization frameworks for downlink mmse transceiver design the first one solves an equivalent uplink problem then the result is transferred to the original downlink problem duality ensures that any uplink mmse scheme e g linear mmse reception or mmse decision feedback equalization dfe has a downlink counterpart we propose two globally optimum algorithms based on convex optimization the basic idea of the second framework is to perform optimization in an alternating manner by switching between the virtual uplink and downlink channels this strategy exploits that the same mse can be achieved in both links for a given choice of transmit and receive filters this iteration is proven to be convergent

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

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

linear transceiver design for downlink multiuser mimo systems downlink interference duality approach this paper considers linear transceiver design for downlink multiuser multiple input multiple output mimo systems we examine different transceiver design problems we focus on two groups of design problems the first group is the weighted sum mean square error wsmse i e symbol wise or user wise wsmse minimization problems and the second group is the minimization of the maximum weighted mean square error wmse symbol wise or user wise wmse problems the problems are examined for the practically relevant scenario where the power constraint is a combination of per base station bs antenna and per symbol user and the noise vector of each mobile station is a zero mean circularly symmetric complex gaussian random variable with arbitrary covariance matrix for each of these problems we propose a novel downlink interference duality based iterative solution each of these problems is solved as follows first we establish a new mean square error mse downlink interference duality second we formulate the power allocation part of the problem in the downlink channel as a geometric program gp third using the duality result and the solution of gp we utilize alternating optimization technique to solve the original downlink problem for the first group of problems we have established symbol wise and user wise wsmse downlink interference duality these duality are established by formulating the noise covariance matrices of the interference channels as fixed point functions on the other hand for the second group of problems we have established symbol wise and user wise mse downlink interference duality these duality are established by formulating the noise covariance matrices of the interference channels as marginally stable convergent discrete time switched systems the proposed duality based iterative solutions can be extended straightforwardly to solve many other linear transceiver design problems we also show that our mse downlink interference duality unify all existing mse duality in our simulation results we have observed that the proposed duality based iterative algorithms utilize less total bs power than that of the existing algorithms

penalty finite element analysis of 3 d navier stokes equations abstract the steady state navier stokes equations in three dimensions are solved by a penalty finite element method for the problems of wall driven cavity flow in a cubical box and natural convection in a cubical cavity subjected to differential side heating the present solutions are compared with recent finite difference solutions of the wall driven cavity problem for reynolds numbersre 100 and 400 the agreement is found to be very good

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

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

coulomb interactions on planar structures inverting the square root of the laplacian we present an adaptive fast multipole method for inverting the square root of the laplacian in two dimensions solving this problem is the dominant computational cost in many applications arising in electrical engineering geophysical fluid dynamics and the study of thin films it corresponds to the evaluation of the field induced by a planar distribution of charge or vorticity our algorithm is direct and assumes only that the source distribution is discretized using an adaptive quad tree the amount of work grows linearly with the number of mesh points

downlink mmse transceiver optimization for multiuser mimo systems mmse balancing we study the problem of mean square error mse transceiver design for point to multipoint transmission in multiuser multiple input multiple output mimo systems we focus on four optimization problems minimizing the maximum weighted layer wise or user wise mse under a total power constraint minimizing the total transmit power subject to a set of layer wise or user wise mse requirements the non convexity of these problems makes the derivation of globally optimal algorithms very difficult for transceiver design new optimal power allocation strategies are derived for the problems considered here based on the power allocation we study how this can be used for iterative transceiver optimization we propose iterative algorithms and prove the monotonic convergence of the algorithms the major advantages of the proposed algorithms are their low complexity and fast convergence behavior during the first few iterations the proposed algorithms are suitable not only for downlink transmission but also for uplink transmission with a total power constraint

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

time averaging for the strongly confined nonlinear schrodinger equation using almost periodicity abstract we study the limiting behavior of a nonlinear schrodinger equation describing a 3 dimensional gas that is strongly confined along the vertical z direction the confinement induces fast oscillations in time that need to be averaged out since the hamiltonian in the z direction is merely assumed confining without any further specification the associated spectrum is discrete but arbitrary and the fast oscillations induced by the nonlinear equation entail countably many frequencies that are arbitrarily distributed for that reason averaging cannot rely on small denominator estimates or like to overcome these difficulties we prove that the fast oscillations are almost periodic in time with values in a sobolev like space that we completely identify we then exploit the existence of long time averages for almost periodic functions to perform the necessary averaging procedure in our nonlinear problem

fast calculation of energy and mass preserving solutions of schrodinger poisson systems on unbounded domains this paper deals with the numerical solution of the time dependent schrodinger poisson system in the spherically symmetric case since the problem is posed on an unbounded domain one has to introduce artificial boundary conditions to confine the computational domain the main topic of this work is the construction of a so called discrete transparent boundary condition tbc for a crank nicolson type predictor corrector scheme for solving the schrodinger poisson system this scheme has the property of mass and energy conservation exactly on the discrete level we propose different strategies for the discrete tbc and present an efficient implementation finally a numerical example illustrate the findings and shows the comparison results between the different approaches

an improved fast multipole algorithm for potential fields on the line a new version of the fast multipole method fmm for the evaluation of potential fields on the line is introduced it uses a new representation of potential fields based on generalized gaussian quadratures j ma v rokhlin and s wandzura siam j numer anal 33 1996 pp 971 996 n yarvin and v rokhlin siam j sci comput 20 1999 pp 699 718 in this representation most translation operators are diagonal to efficiently incorporate this representation into the fmm an apparatus is introduced for transforming between different types of expansions this apparatus is highly general and is based on formulae for the least squares approximation of linear operators the performance of the method is illustrated with several numerical examples it is roughly twice as fast as previously published algorithms

finite element navier stokes calculation of three dimensional turbulent flow near a propeller les equations de navier stokes sont resolues par une methode d elements finis a fonction de penalite l helice est modelisee par un disque actuateur et le transport turbulent par une equation d energie cinetique turbulente

generalized linear precoder and decoder design for mimo channels using the weighted mmse criterion we address the problem of designing jointly optimum linear precoder and decoder for a mimo channel possibly with delay spread using a weighted minimum mean squared error mmse criterion subject to a transmit power constraint we show that the optimum linear precoder and decoder diagonalize the mimo channel into eigen subchannels for any set of error weights furthermore we derive the optimum linear precoder and decoder as functions of the error weights and consider specialized designs based on specific choices of error weights we show how to obtain 1 the maximum information rate design 2 qos based design we show how to achieve any set of relative snrs across the subchannels and 3 the unweighted mmse and equal error design for fixed rate systems

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

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

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

derivation of the gross pitaevskii equation for the dynamics of bose einstein condensate consider a system of n bosons in three dimensions interacting via a repulsive short range pair potential n 2 v n xi xj where x x1 xn denotes the positions of the particles let hn denote the hamiltonian of the system and let n t be the solution to the schrodinger equation suppose that the initial data n 0 satisfies the energy condition h n 0 h k n n 0i c k n k for k 1 2 we also assume that the k particle density matrices of the initial state are asymptotically factorized as n we prove that the k particle density matrices of n t are also asymptotically factorized and the one particle orbital wave function solves the gross pitaevskii equation a cubic non linear schrodinger equation with the coupling constant given by the scattering length of the potential v we also prove the same conclusion if the energy condition holds only for k 1 but the factorization of n 0 is assumed in a stronger sense ams subject classification number 81v70 81t18 35q55

unconditional stability and convergence of crank nicolson galerkin fems for a nonlinear schrodinger helmholtz system the paper is concerned with the unconditional stability and optimal l 2 error estimates of linearized crank nicolson galerkin fems for a nonlinear schrodinger helmholtz system in mathbb r d d 2 3 by introducing a corresponding time discrete system we separate the error into two parts i e the temporal error and the spatial error since the latter is tau independent the uniform boundedness of numerical solutions in l infty norm follows an inverse inequality immediately without any restrictions on time stepsize then optimal error estimates are obtained in a routine way numerical examples in both two and three dimensional spaces are given to illustrate our theoretical results

the strongly confined schrodinger poisson system for the transport of electrons in a nanowire we study the limit of the three dimensional schrodinger poisson system with a singular perturbation to model a quantum electron gas that is strongly confined near an axis for well prepared data which are polarized on the ground space of the transversal hamiltonian the resulting model is the cubic defocusing nonlinear schrodinger equation our main tool is a refined analysis of the poisson kernel when acting on strongly confined densities in that direction an appropriate scaling of the initial data is required to avoid divergent integrals when the gas concentrates on the axis

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

accurate and efficient computation of nonlocal potentials based on gaussian sum approximation abstract we introduce an accurate and efficient method for the numerical evaluation of nonlocal potentials including the 3d 2d coulomb 2d poisson and 3d dipole dipole potentials our method is based on a gaussian sum approximation of the singular convolution kernel combined with a taylor expansion of the density starting from the convolution formulation of the nonlocal potential for smooth and fast decaying densities we make a full use of the fourier pseudospectral plane wave approximation of the density and a separable gaussian sum approximation of the kernel in an interval where the singularity the origin is excluded the potential is separated into a regular integral and a near field singular correction integral the first is computed with the fourier pseudospectral method while the latter is well resolved utilizing a low order taylor expansion of the density both parts are accelerated by fast fourier transforms fft the method is accurate 14 16 digits efficient o n log n complexity low in storage easily adaptable to other different kernels applicable for anisotropic densities and highly parallelizable

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

adiabatic decoupling and time dependent born oppenheimer theory we reconsider the time dependent born oppenheimer theory with the goal to carefully separate between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands band crossings are allowed and our results are local in the sense that they hold up to the first time when a band crossing is encountered the adiabatic decoupling leads to an effective schrodinger equation for the nuclei including contributions from the berry connection

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

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

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

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

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

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

finite element analysis of oscillatory flow with heat transfer inside a square cavity in this work a semi implicit projection type finite element method is applied to solve the incompressible oscillatory cavity flow with heat transfer the two dimensional time dependent navier stokes and energy equations are numerically integrated by a time split galerkin finite element method using direct matrix solvers the time accuracy of the method was tested by calculating the transient buoyancy driven flow in a square cavity with differentially heated vertical walls the oscillatory cavity flow is then studied considering the effects of heat transfer reynolds number and oscillatory stokes number to measure the stability of the scheme for such a high singularity phenomenon the proposed scheme is found to be stable and convergent for high rayleigh numbers and will be applicable to general problems involving flow and heat transfer especially in three dimensions

robust transceiver optimization for downlink multiuser mimo systems this correspondence addresses the joint transceiver design for downlink multiuser multiple input multiple output mimo systems with imperfect channel state information csi at the base station bs and mobile stations mss by incorporating antenna correlation at both ends of the channel and taking channel estimation errors into account we solve two robust design problems minimization of the weighted sum mean square error mse and minimization of the maximum weighted mse these problems are solved as follows first we establish three kinds of mse uplink downlink duality by transforming only the power allocation matrices from uplink channel to downlink channel and vice versa second in the uplink channel we formulate the power allocation part of each problem ensuring global optimality finally based on the solution of the uplink power allocation and the mse duality results for each problem we propose an iterative algorithm that performs optimization alternatively between the uplink and downlink channels computer simulations verify the robustness of the proposed design compared to the nonrobust naive design

on the mse duality of the broadcast channel and the multiple access channel we present a mean square error mse duality between the broadcast channel and the multiple access channel for multiantenna users communicating with a single base station we introduce three levels of the duality which allow for a problem specific customization with different computational complexities and resolutions the first level preserves the sum mse during the conversion from the uplink to the downlink and vice versa whereas the second level not only keeps the sum mse constant but also ensures the preservation of the individual users mses the third level involves the finest resolution and preserves the individual streams mses and the individual streams signal to interference and noise ratio sinr simultaneously in contrast with hitherto existing mse dualities the proposed sort of duality features a lower complexity since no mse computation detouring is necessary during the conversion to the dual domain and is capable of handling all combinations of active and passive transmitters and receivers moreover we show how two of these three dualities can be exploited to solve the unweighted total sum mse minimization problem and the weighted sum mse minimization in the broadcast channel in an efficient way by revealing the hidden convexity in the first case and drastically reducing the computational complexity in the latter case

efficient numerical methods for computing ground states and dynamics of dipolar bose einstein condensates new efficient and accurate numerical methods are proposed to compute ground states and dynamics of dipolar bose einstein condensates becs described by a three dimensional 3d gross pitaevskii equation gpe with a dipolar interaction potential due to the high singularity in the dipolar interaction potential it brings significant difficulties in mathematical analysis and numerical simulations of dipolar becs in this paper by decoupling the two body dipolar interaction potential into short range or local and long range interactions or repulsive and attractive interactions the gpe for dipolar becs is reformulated as a gross pitaevskii poisson type system based on this new mathematical formulation we prove rigorously existence and uniqueness as well as nonexistence of the ground states and discuss the existence of global weak solution and finite time blow up of the dynamics in different parameter regimes of dipolar becs in addition a backward euler sine pseudospectral method is presented for computing the ground states and a time splitting sine pseudospectral method is proposed for computing the dynamics of dipolar becs due to the adoption of new mathematical formulation our new numerical methods avoid evaluating integrals with high singularity and thus they are more efficient and accurate than those numerical methods currently used in the literatures for solving the problem extensive numerical examples in 3d are reported to demonstrate the efficiency and accuracy of our new numerical methods for computing the ground states and dynamics of dipolar becs

short note a short note on simplified pseudospectral methods for computing ground state and dynamics of spherically symmetric schrodinger poisson slater system in a recent paper we proposed and compared various approaches to compute the ground state and dynamics of the schrodinger poisson slater sps system for general external potential and initial condition concluding that the methods based on sine pseudospectral discretization in space are the best candidates this note is concerned with the case that the external potential and initial condition are spherically symmetric for the sps system with spherical symmetry via applying a proper change of variables into the reduced quasi 1d model we simplify the methods proposed for the general 3d case such that both the memory and computational load are significantly reduced

robust sum mse optimization for downlink multiuser mimo systems with arbitrary power constraint generalized duality approach this paper considers linear minimum mean square error mmse transceiver design problems for downlink multiuser multiple input multiple output mimo systems where imperfect channel state information is available at the base station bs and mobile stations mss we examine robust sum mean square error mse minimization problems the problems are examined for the generalized scenario where the power constraint is per bs per bs antenna per user or per symbol and the noise vector of each ms is a zero mean circularly symmetric complex gaussian random variable with arbitrary covariance matrix for each of these problems we propose a novel duality based iterative solution each of these problems is solved as follows first we establish a novel sum average mean square error amse duality second we formulate the power allocation part of the problem in the downlink channel as a geometric program gp third using the duality result and the solution of gp we utilize alternating optimization technique to solve the original downlink problem to solve robust sum mse minimization constrained with per bs antenna and per bs power problems we have established novel downlink uplink duality on the other hand to solve robust sum mse minimization constrained with per user and per symbol power problems we have established novel downlink interference duality for the total bs power constrained robust sum mse minimization problem the current duality is established by modifying the constraint function of the dual uplink channel problem and for the robust sum mse minimization with per bs antenna and per user symbol power constraint problems our duality are established by formulating the noise covariance matrices of the uplink and interference channels as fixed point functions respectively we also show that our sum amse duality are able to solve other sum mse based robust design problems computer simulations verify the robustness of the proposed robust designs compared to the nonrobust naive designs

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

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

rate optimization for multiuser mimo systems with linear processing this paper focuses on linear transceiver design for rate optimization in multiuser gaussian multple input multiple output mimo systems we focus on two design criteria 1 maximizing the weighted sum rate subject to a total power constraint 2 maximizing the minimum user rate subject to a total power constraint for these problems new power allocation strategies are derived which can be formulated as geometric programs gps involving mean square errors mses based on these solutions we propose iterative algorithms where each iteration contains the optimization of the uplink power uplink receive filters and downlink receive filters monotonic convergence of the algorithms is proved simulations show that the algorithms outperform existing linear schemes additionally we extend the results to other variations of the problems such as the problem of sum rate constrained or user rate constrained power minimization and the problem of sum rate maximization subject to user rate constraints and a total power constraint

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

a new fast multipole accelerated poisson solver in two dimensions we present an adaptive fast multipole method for solving the poisson equation in two dimensions the algorithm is direct assumes that the source distribution is discretized using an adaptive quad tree and allows for dirichlet neumann periodic and free space conditions to be imposed on the boundary of a square the amount of work per grid point is comparable to that of classical fast solvers even for highly nonuniform grids

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

on the roles of minimization and linearization in least squares finite element models of nonlinear boundary value problems in this paper we examine the roles of minimization and linearization in the least squares finite element formulations of nonlinear boundary values problems the least squares principle is based upon the minimization of the least squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations in the present case we consider l 2 norms since the least squares method is independent of the discretization procedure and the solution scheme the least squares principle suggests that minimization should be performed prior to linearization where linearization is employed in the context of either the picard or newton iterative solution procedures however in the least squares finite element analysis of nonlinear boundary value problems it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations the main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged the assessment is performed mathematically through an examination of the variational setting for the least squares formulation of an abstract nonlinear boundary value problem and also computationally through the numerical simulation of the least squares finite element solutions of both a nonlinear form of the poisson equation and also the incompressible navier stokes equations the assessment suggests that although the least squares principle indicates that minimization should be performed prior to linearization such an approach is often impractical and not necessary

weighted sum rate optimization for downlink multiuser mimo coordinated base station systems centralized and distributed algorithms this paper considers the joint linear transceiver design problem for the downlink multiuser multiple input multiple output mimo systems with coordinated base stations bss we consider maximization of the weighted sum rate with per bs antenna power constraint problem we propose novel centralized and computationally efficient distributed iterative algorithms that achieve local optimum to the latter problem these algorithms are described as follows first by introducing additional optimization variables we reformulate the original problem into a new problem second for the given precoder matrices of all users the optimal receivers are computed using minimum mean square error mmse method and the optimal introduced variables are obtained in closed form expressions third by keeping the introduced variables and receivers constant the precoder matrices of all users are optimized by using second order cone programming socp and matrix fractional minimization approaches for the centralized and distributed algorithms respectively finally the second and third steps are repeated until these algorithms converge we have shown that the proposed algorithms are guaranteed to converge we also show that the proposed algorithms require less computational cost than that of the existing linear algorithm all simulation results demonstrate that our distributed algorithm achieves the same performance as that of the centralized algorithm moreover the proposed algorithms outperform the existing linear algorithm in particular when each of the users has single antenna we have observed that the proposed algorithms achieve the global optimum

existence of multiple positive solutions to nonhomogeneous schrodinger poisson system in this paper we consider the existence of multiple solutions to the following nonhomogeneous generalized schrodinger poisson system u ku q f u g u h x in r 3 2 qf u in r 3 where q 0 is a parameter 0 h x h x l 2 r 3 and g is asymptotically linear or superliner at infinity we show that there exists q 0 0 such that the system has at least two positive radial solutions for q 0 q 0

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 note on marginal stability of switched systems in this note we present criteria for marginal stability and marginal instability of switched systems for switched nonlinear systems we prove that uniform stability is equivalent to the existence of a common weak lyapunov function cwlf that is generally not continuous for switched linear systems we present a unified treatment for marginal stability and marginal instability for both continuous time and discrete time switched systems in particular we prove that any marginally stable system admits a norm as a cwlf by exploiting the largest invariant set contained in a polyhedron several insightful algebraic characteristics are revealed for marginal stability and marginal instability

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

adiabatic approximation of the schrodinger poisson system with a partial confinement the stationary case asymptotic quantum transport models of a two dimensional gas are presented the models are the stationary versions of those introduced in a previous paper by ben abdallah mehats pinaud the starting point is a singular perturbation of the three dimensional stationary schrodinger poisson system posed on bounded domain the electron injection in the device is modeled thanks to open boundary conditions under a small density assumption the asymptotics lead to a full two dimensional first order approximation of the initial model an intermediate model called the 2 5d adiabatic model in ben abdallah mehats pinaud is then introduced it shares the same structure as the limit but is shown to be a second order approximation of the three dimensional model

computing the ground state and dynamics of the nonlinear schrodinger equation with nonlocal interactions via the nonuniform fft we present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear schrodinger equation nlse with nonlocal interactions based on a fast and accurate evaluation of the long range interactions via the nonuniform fast fourier transform nufft we begin with a review of the fast and accurate nufft based method in 29 for nonlocal interactions where the singularity of the fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates we then extend the method to compute other nonlocal interactions whose fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform many of these interactions do not decay at infinity in the physical space which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain the performance of our method against other existing methods is illustrated numerically with particular attention on the effect of the size of the computational domain in the physical space finally to study the ground state and dynamics of the nlse we propose efficient and accurate numerical methods by combining the nufft method for potential evaluation with the normalized gradient flow using backward euler fourier pseudospectral discretization and time splitting fourier pseudospectral method respectively extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the nlse with various nonlocal interactions numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency

a time dependent born oppenheimer approximation with exponentially small error estimates we present the construction of an exponentially accurate time dependent born oppenheimer approximation for molecular quantum mechanics

efficient and spectrally accurate numerical methods for computing ground and first excited states in bose einstein condensates in this paper we present two efficient and spectrally accurate numerical methods for computing the ground and first excited states in bose einstein condensates becs we begin with a review on the gradient flow with discrete normalization gfdn for computing stationary states of a nonconvex minimization problem and show how to choose initial data effectively for the gfdn for discretizing the gradient flow we use sine pseudospectral method for spatial derivatives and either backward euler scheme besp or backward forward euler schemes for linear nonlinear terms bfsp for temporal derivatives both besp and bfsp are spectral order accurate for computing the ground and first excited states in bec of course they have their own advantages i for linear case besp is energy diminishing for any time step size where bfsp is energy diminishing under a constraint on the time step size ii at every time step the linear system in bfsp can be solved directly via fast sine transform fst and thus it is extremely efficient and in besp it needs to be solved iteratively via fst by introducing a stabilization term and thus it could be efficient too comparisons between besp and bfsp as well as other existing numerical methods are reported in terms of accuracy and total computational time our numerical results show that both besp and bfsp are much more accurate and efficient than those existing numerical methods in the literature finally our new numerical methods are applied to compute the ground and first excited states in bec in one dimension 1d 2d and 3d with a combined harmonic and optical lattice potential for demonstrating their efficiency and high resolution

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