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

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

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

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

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

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

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

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

a markov chain that models genetic algorithms in noisy environments abstract in this study we construct a markov chain that models genetic algorithms gas in noisy environments fitness functions are assumed to be perturbed by additive or multiplicative random noise that takes on finitely many values we first compute the transition probabilities of this markov chain exactly we then focus on the environment where fitness functions are additively disturbed and further analyze the chain in order to investigate the transition and convergence properties of gas in this noisy environment the markov chain has only one positive recurrent communication class and it follows immediately from this property that gas eventually find at least one globally optimal solution with probability 1 furthermore our analysis shows that the chain has a stationary distribution that is also its steady state distribution using this property and the transition probabilities of the chain we derive an upper bound for the number of iterations sufficient to ensure with at least a specified probability that a ga selects a globally optimal solution upon termination

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

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

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

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

measurement methods comparisions and linear statistical relationship the linear statistical relationship model can be used to compare two measurement procedures crude methods are now typically used in practice because the statistical concepts are often not understood by practitioners to clarify concepts we define three types of equivalency based on accuracy and precision of the measurement methods and show that the confidence interval for slope depends on sample size and information size these insights lead to obtaining bounded confidence intervals and equivalence tests the derived results provide the basis for designing and analyzing a comparison experiment an example from practice is used to illustrate the methodology

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

optimal stabilization of families of linear stochastic differntial equations with jump corfficients and multiplicative noise stabilization of families of finite dimensional linear systems by constant linear feedback is studied by introducing a sequence of finite horizon parameter optimization problems if the set of stabilizing constant linear feedback controllers is not empty and a regularity condition holds a subsequence of solutions to the optimization problems converges to a constant stabilizing controller

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

convergence analysis of canonical genetic algorithms this paper analyzes the convergence properties of the canonical genetic algorithm cga with mutation crossover and proportional reproduction applied to static optimization problems it is proved by means of homogeneous finite markov chain analysis that a cga will never converge to the global optimum regardless of the initialization crossover operator and objective function but variants of cga s that always maintain the best solution in the population either before or after selection are shown to converge to the global optimum due to the irreducibility property of the underlying original nonconvergent cga these results are discussed with respect to the schema theorem