two grid method for the two dimensional time dependent schrodinger equation by the finite element method abstract in this paper we construct a backward euler full discrete two grid finite element scheme for the two dimensional time dependent schrodinger equation with this method the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two poisson equations on the same fine grid we analyze the error estimate of the standard finite element solution and the two grid solution in the h 1 norm it is shown that the two grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h k k 1 finally a numerical experiment indicates that our two grid algorithm is more efficient than the standard finite element method

time consistent equilibrium reinsurance investment strategy for n competitive insurers under a new interaction mechanism and a general investment framework abstract we propose a new interaction mechanism for n 2 competitive insurers under which we investigate the time consistent equilibrium reinsurance investment strategy for n insurers each insurer can purchase reinsurance for reducing the claim risk and invest in the financial market for increasing his wealth we consider a general investment framework which includes typical diversified and concentrated investment patterns in the literature the objective of each insurer is to find a time consistent equilibrium reinsurance investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth by using the stochastic control technique we obtain the explicit time consistent equilibrium reinsurance investment strategies and corresponding equilibrium value functions through analyzing the general investment strategy we find the criterion of whether the insurer should invest in a specific risky asset furthermore we examine theoretically and numerically how the price parameters of risky assets affect the concrete investment pattern we find that when the appreciation rate of one risky asset exceeds some times as those of other risky assets the insurer will only invest in this asset otherwise the insurer will adopt the diversified investment pattern we also numerically examine the influences of the number of insurers and the surplus jump on the time consistent equilibrium reinsurance strategy

a random matrix theoretic approach to handling singular covariance estimates in many practical situations we would like to estimate the covariance matrix of a set of variables from an insufficient amount of data more specifically if we have a set of n independent identically distributed measurements of an m dimensional random vector the maximum likelihood estimate is the sample covariance matrix here we consider the case where n m such that this estimate is singular noninvertible and therefore fundamentally bad we present a radically new approach to deal with this situation based on the idea of dimensionality reduction through an ensemble of isotropically random unitary matrices we obtain two estimates cov and invcov which are estimates for the covariance matrix and the inverse covariance matrix respectively both estimates retain the original eigenvectors while altering the eigenvalues we have a closed form analytical expression for cov and invcov in terms of the eigenvector eigenvalue decomposition of the sample covariance we motivate the use of invcov through applications to linear estimation supervised learning and high resolution spectral estimation we also compare the performance of these estimators with other more conventional methods

the cramer rao bound for estimating a sparse parameter vector the goal of this contribution is to characterize the best achievable mean squared error mse in estimating a sparse deterministic parameter from measurements corrupted by gaussian noise to this end an appropriate definition of bias in the sparse setting is developed and the constrained cramer rao bound crb is obtained this bound is shown to equal the crb of an estimator with knowledge of the support set for almost all feasible parameter values consequently in the unbiased case our bound is identical to the mse of the oracle estimator combined with the fact that the crb is achieved at high signal to noise ratios signal to noise ratio snrs by the maximum likelihood technique our result provides a new interpretation for the common practice of using the oracle estimator as a gold standard against which practical approaches are compared

the finite element method for the time dependent gross pitaevskii equation with angular momentum rotation we consider the time dependent gross pitaevskii equation describing the dynamics of rotating bose einstein condensates and its discretization with the finite element method we analyze a mass conserving crank nicolson type discretization and prove corresponding a priori error estimates with respect to the maximum norm in time and the l2 and energy norm in space the estimates show that we obtain optimal convergence rates under the assumption of additional regularity for the solution to the gross pitaevskii equation we demonstrate the performance of the method in numerical experiments by siam 2017

symplectic methods for the nonlinear schrodinger equation abstract in this paper we show that the spatial discretization of the nonlinear schrodinger equation leads to a hamiltonian system which can be simulated with symplectic numerical schemes in particular we apply two symplectic integrators to the nonlinear schrodinger equation and we demonstrate that they are able to produce accurate results and to preserve very well the invariants of the original system such as the energy and charge

compressed sensing with basis mismatch performance bounds and sparse based estimator compressed sensing cs is a promising emerging domain that outperforms the classical limit of the shannon sampling theory if the measurement vector can be approximated as the linear combination of few basis vectors extracted from a redundant dictionary matrix unfortunately in realistic scenario the knowledge of this basis or equivalently of the entire dictionary is often uncertain i e corrupted by a basis mismatch bm error the consequence of the bm problem is that the estimation accuracy in terms of bayesian mean square error bmse of popular sparse based estimators collapses even if the support is perfectly estimated and in the high signal to noise ratio snr regime this saturation effect considerably limits the effective viability of these estimation schemes in the first part of this work the bayesian cramer rao bound bcrb is derived for cs model with unstructured bm we show that the bcrb foresees the saturation effect of the estimation accuracy of standard sparse based estimators as for instance the omp cosamp or the bp in addition we provide an approximation of this bmse threshold in the second part and in the context of the structured bm model a new estimation scheme called bias correction estimator bice is proposed and its statistical properties are studied the bice acts as a post processing estimation layer for any sparse based estimator and mitigates considerably the bm degradation finally the bice i is a blind algorithm i e is unaware of the uncorrupted dictionary matrix ii is generic since it can be associated to any sparse based estimator iii is fast i e the additional computational cost remains low and iv has good statistical properties to illustrate our results and propositions the bice is applied in the challenging context of the compressive sampling of non bandlimited impulsive signals

derivation of the bias of the normalized sample covariance matrix in a heterogeneous noise with application to low rank stap filter in a previous work we have developed a low rank lr spatio temporal adaptive processing stap filter when the disturbance is modeled as the sum of a low rank spherically invariant random vector sirv clutter and a zero mean white gaussian noise this lr stap filter is built from the normalized sample covariance matrix nscm and exhibits good robustness properties to secondary data contamination by target components in this correspondence we derive the bias of the nscm with this noise model we show that the eigenvectors estimated from the nscm are unbiased the new expressions of the expectation of nscm eigenvalues are also given from these results we also show that the estimate of the clutter subspace projector based on the nscm used in our lr stap is a consistent estimate of the true one results on numerical data validates the theoretical approach

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

time consistent non zero sum stochastic differential reinsurance and investment game under default and volatility risks abstract this paper investigates a non zero sum stochastic differential game between two mean variance insurers these two insurers are concerned about their terminal wealth and the relative performance compared with each other we assume that they can buy proportional reinsurance and invest in a financial market consisting of a risk free asset a stock and a defaultable bond the price process of stock is driven by the constant elasticity of variance cev model and the defaultable bond recovers a proportion of value at default so these two insurers are faced with insurance risk volatility risk and default risk the non zero sum goal of these insurers is to maximize the mean variance utility of a weighted value of their terminal and relative wealths we solve the mean variance problem in the time consistent case and establish the extended hamilton jacobi bellman systems for the post default case and the pre default case respectively furthermore we derive the closed form solutions of the nash equilibrium reinsurance and investment strategies for these two insurers in the end of this paper we calibrate the parameters based on real data and several numerical examples are provided to illustrate the effects of economic parameters on the equilibrium strategies

superconvergence analysis of finite element method for the time dependent schrodinger equation in this paper we consider the two dimensional time dependent schrodinger equation firstly we use the rectangular lagrange type finite element of order p to get a semi discrete scheme of the equation and discuss the superconvergence error estimate in the h 1 norm secondly we use the crank nicolson method in time to get a fully discrete scheme of the equation and the superconvergence estimate in the h 1 norm can be obtained in this scheme finally a numerical example with the order p 1 is provided to verify our theoretical results

fully discrete methods for the nonlinear schrodinger equation abstract in this paper we construct and analyze fully discrete methods for the nonlinear schrodinger equation these methods are based on a galerkin finite element formulation in space and use time stepping techniques which are modifications of implicit runge kutta methods the resulting schemes are both efficient and circumvent the well known order reduction phenomenon affecting runge kutta methods numerical experiments are also presented

the solution of nonlinear schrodinger equations using orthogonal spline collocation abstract an orthogonal spline collocation semidiscretization previously applied in the numerical solution of the cubic schrodinger equation is extended to the solution of schrodinger equations which possess general power nonlinearities and nonlinear terms that contain derivatives the accuracy of the method is examined as well as its ability to conserve approximations to certain invariants associated with the theoretical solution the results of numerical experiments are presented in which the integration in time is performed using a routine from a software library particular attention is paid to the value and limitations of conservation as an indicator of accuracy

an energy conservative difference scheme for the nonlinear fractional schrodinger equations abstract in this paper an energy conservative crank nicolson difference scheme for nonlinear riesz space fractional schrodinger equations is studied we give a rigorous analysis of the conservation properties including mass conservation and energy conservation in the discrete sense based on brouwer fixed point theorem the existence of the difference solution is proved by virtue of the energy method the difference solution is shown to be unique and convergent at the order of o 2 h 2 in the l 2 norm with time step and mesh size h finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results

nonconforming finite element method for a generalized nonlinear schrodinger equation abstract in this paper an efficient nonconforming finite element method fem is studied with e q 1 r o t element for a generalized nonlinear schrodinger equation first we prove a novel result of the consistency error estimate with order o h2 for e q 1 r o t element which leads to the superconvergent error estimate in broken h1 norm for semi discrete scheme while previous work only derive convergent results for this element second a linearized backward euler scheme is established and a time discrete system is introduced to split the error into two parts the temporal error and the spatial error by using a rigorous analysis for the regularity of the time discrete system and the proved characters of e q 1 r o t element the optimal l2 error estimate is obtained without any time step restrictions which leads to the numerical solution can be bounded in l norm by an inverse inequality unconditionally then the supercloseness estimate is arrived at with the above achievements third global superconvergence results are deduced through interpolated postprocessing technique at last numerical examples are provided to confirm the theoretical analysis here h is the subdivision parameter and is the time step

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

asymptotic achievability of the cramer rao bound for noisy compressive sampling we consider a model of the form y ax n where x isin cm is sparse with at most l nonzero coefficients in unknown locations y isin cn is the observation vector a isin cn times m is the measurement matrix and n isin cn is the gaussian noise we develop a cramer rao bound on the mean squared estimation error of the nonzero elements of x corresponding to the genie aided estimator gae which is provided with the locations of the nonzero elements of x intuitively the mean squared estimation error of any estimator without the knowledge of the locations of the nonzero elements of x is no less than that of the gae assuming that l n is fixed we establish the existence of an estimator that asymptotically achieves the cramer rao bound without any knowledge of the locations of the nonzero elements of x as n rarr infin for a a random gaussian matrix whose elements are drawn i i d according to n 0 1

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

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

online sparse system identification and signal reconstruction using projections onto weighted ell_ 1 balls this paper presents a novel projection based adaptive algorithm for sparse signal and system identification the sequentially observed data are used to generate an equivalent sequence of closed convex sets namely hyperslabs each hyperslab is the geometric equivalent of a cost criterion that quantifies data mismatch sparsity is imposed by the introduction of appropriately designed weighted l1 balls and the related projection operator is also derived the algorithm develops around projections onto the sequence of the generated hyperslabs as well as the weighted l1 balls the resulting scheme exhibits linear dependence with respect to the unknown system s order on the number of multiplications additions and an o llog2l dependence on sorting operations where l is the length of the system signal to be estimated numerical results are also given to validate the performance of the proposed method against the least absolute shrinkage and selection operator lasso algorithm and two very recently developed adaptive sparse schemes that fuse arguments from the lms rls adaptation mechanisms with those imposed by the lasso rational

numerical simulation of nonlinear schro dinger systems a new conservative scheme we present a new numerical scheme for nonlinear schrodinger type equations the scheme conserves the energy and charge of the systems and it is linearly implicit numerical experiments for several physical problems show that our scheme is stable accurate and efficient

stability analysis of finite difference schemes for quantum mechanical equations of motion for a pdf involving both space and time variables stability criteria are presently shown to change drastically when the equation contains i as in the quantum mechanical equations of motion it is further noted that the stability of finite difference schemes for quantum mechanical equations of motion depends on both spatial and temporal zoning it is possible to compare a free particle green s function to the solution of a simple diffusion equation and the quantum mechanical motion of a free particle to fresnel diffraction in optics

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

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

time consistent investment reinsurance strategy for mean variance insurers with a defaultable security abstract this paper considers an optimal investment and reinsurance problem involving a defaultable security for an insurer under the mean variance criterion in a jump diffusion risk model the insurer can purchase proportional reinsurance or acquire new insurance business and invest in a financial market consisting of a risk free asset a stock and a defaultable bond in particular the correlation between the insurance risk model and the financial market is also considered from a game theoretic perspective the extended hamilton jacobi bellman systems of equations are established for the post default case and the pre default case respectively in both cases closed form expressions for the optimal time consistent investment reinsurance strategies and the corresponding value functions are derived moreover some properties of optimal strategies value functions and efficient frontiers are discussed either analytically or numerically

superconvergence of a finite element approximation to the solution of a sobolev equation in a single space variable a standard galerkin method for a quasilinear equation of sobolev type using continuous piecewise polynomial spaces is presented and analyzed optimal order error estimates are established in various norms and nodal superconvergence is demonstrated discretization in time by explicit single step methods is discussed

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

finite element approximation for 2nd order elliptic eigenvalue problems with nonlocal boundary or transition conditions in this paper we use two symplectic schemes to simulate the ablowitz ladik model associated to the cubic nonlinear schrodinger equation and we compare them with nonsymplectic methods

a quadratic b spline finite element method for solving nonlinear schrodinger equation a numerical solution of the nonlinear schrodinger nls equation is obtained by using the quadratic b spline finite element method test problems concerning motion birth and bound states of solitons of the nlg equation are studied to evaluate the method results are compared with various numerical methods

comparisons between sine gordon and perturbed nonlinear schrodinger equations for modeling light bullets beyond critical collapse abstract the sine gordon sg equation and perturbed nonlinear schrodinger nls equations are studied numerically for modeling the propagation of two space dimensional 2d localized pulses the so called light bullets in nonlinear dispersive optical media we begin with the 2 1 sg equation obtained as an asymptotic reduction in the two level dissipationless maxwell bloch system followed by the review on the perturbed nls equation in 2d for sg pulse envelopes which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical cubic focusing nls the perturbed nls is approximated by truncating the nonlinearity into finite higher order terms undergoing focusing defocusing cycles efficient semi implicit sine pseudospectral discretizations for sg and perturbed nls are proposed with rigorous error estimates numerical comparison results between light bullet solutions of sg and perturbed nls as well as critical nls are reported which validate that the solution of the perturbed nls as well as its finite term truncations are in qualitative and quantitative agreement with the solution of sg for the light bullets propagation even after the critical collapse of cubic focusing nls in contrast standard critical nls is in qualitative agreement with sg only before its collapse as a benefit of such observations pulse propagations are studied via solving the perturbed nls truncated by reasonably many nonlinear terms which is a much cheaper task than solving sg equation directly

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

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

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

existence and stability of ground states for fully discrete approximations of the nonlinear schrodinger equation in this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear schrodinger equation on the real line more precisely we consider a symplectic time splitting integrator applied to a discrete nonlinear schrodinger equation with additional dirichlet boundary conditions on a large interval we give conditions ensuring the existence of a numerical ground state which is close in energy norm to the continuous ground state such result is valid under a cfl condition of the form tau h 2 le c where tau and h denote the time and space step size respectively furthermore we prove that if the initial datum is symmetric and close to the continuous ground state eta then the associated numerical solution remains close to the orbit of eta gamma cup alpha e i alpha eta for very long times

optimal investment policies for a firm with a random risk process exponential utility and minimizing the probability of ruin we consider a firm that is faced with an uncontrollable stochastic cash flow or random risk process there is one investment opportunity a risky stock and we study the optimal investment decision for such firms there is a fundamental incompleteness in the market in that the risk to the investor of going bankrupt cannot be eliminated under any investment strategy since the random risk process ensures that there is always a positive probability of ruin bankruptcy we therefore focus on obtaining investment strategies which are optimal in the sense of minimizing the risk of ruin in particular we solve for the strategy that maximizes the probability of achieving a given upper wealth level before hitting a given lower level this policy also minimizes the probability of ruin we prove that when there is no risk free interest rate this policy is equivalent to the policy that maximizes utility from terminal wealth for a fixed terminal time when the firm has an exponential utility function this validates a longstanding conjecture about the relation between minimizing probability of ruin and exponential utility when there is a positive risk free interest rate the conjecture is shown to be false we also solve for the optimal policy for the related objective of minimizing the expected discounted penalty paid upon ruin

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

derivation of the discrete conservation laws for a family of finite difference schemes abstract we present conservative schemes in finite differences applied to nonlinear odes and nonlinear klein gordon schrodinger and dirac field equations a systematic approach is used and an explicit derivation of the corresponding conservation laws is given

robust optimal investment and reinsurance of an insurer under variance premium principle and default risk abstract this paper considers a robust optimal investment and reinsurance problem under model ambiguity and default risk for an insurer who can trade in a saving account a stock and a defaultable bond and aims to maximize the minimal expected cara utility the surplus process of the insurer is assumed to follow the cramer lundberg model in particular both the insurance and reinsurance premium are assumed to be calculated via the variance premium principle by using the dynamic programming approach we study the pre default case and post default case respectively then closed form expressions for the optimal strategies and the corresponding value function are derived finally numerical examples are given to illustrate our main results and we discuss relevant economic insights obtained from these results

linearly implicit methods for the nonlinear schrodinger equation in nonhomogeneous media the nonlinear schrodinger equation in one dimensional cartesian coordinates is studied numerically by means of three linearly implicit finite difference methods that result in block tridiagonal matrices at each time level as a function of the damping pumping and changes in the refraction index the numerical experiments performed to determine the effects of the time step spatial step size and implicitness parameters of the dispersion and nonlinear terms show that the most accurate results are obtained when both the dispersion and nonlinear terms are treated by means of a second order accurate trapezoidal discretization and full linearization of the nonlinear terms it is also shown that if either the dispersion or the nonlinear terms are treated with a first order accurate approximation substantial errors are found and these errors are largest when the nonlinear terms are discretized by means of a first order difference and do not decrease substantially as the number of grid points is increased it has been observed that when solitons move from a medium to another one characterized by a higher refraction index the amplitude and speed of the soliton increase whereas its width decreases upon approaching the interface the amplitude of the soliton was found to increase downstream while some radiation was observed upstream both the increase in amplitude and radiation were found to increase with the change in the refraction index the effect of the distance where the index of refraction changes was found to be small if this distance is smaller than the width of the soliton for solitons propagating from a medium to another one of lower refraction index the amplitude and speed of the soliton decrease whereas its width increases the amplitude of the soliton increases downstream and spreads towards the boundary where it is reflected when the index of refraction decreases below a threshold it was found that the soliton may become trapped at the interface the results presented in this paper indicate that theories based on treating solitons as particles perturbation methods inverse scattering and variational formulations can provide accurate results only when the change in the refraction index is sufficiently small

continuous finite element methods which preserve energy properties for nonlinear problems abstract continuous finite elements are used to develop numerical methods which preserve energy properties for nonlinear ordinary and partial differential equations schemes are constructed that give high orders of accuracy time discretization algorithms for the cahn hilliard klein gordon korteweg de vries and schrodinger equations are presented comparisons are made with existing schemes

conservative super convergent and hybrid discontinuous galerkin methods applied to nonlinear schrodinger equations abstract using a unified framework the formulation of a super convergent discontinuous galerkin sdg method and a hybridized discontinuous galerkin hdg version both applied to a general nonlinear schrodinger equation is presented conservation of the mass and the energy is studied theoretically for the semi discrete formulation and for the fully discrete method using the modified crank nicolson time scheme conservation of both quantities is numerically validated on two dimensional problems and high order approximations a numerical study of convergence illustrates the advantages of the new formulations over the traditional local discontinuous galerkin ldg method numerical experiments show that the approximation of the initial discrete energy converges with order 2 k 1 which is better than that obtained by the standard continuous finite element which is only of order 2k when polynomials of degree k are used

semi global approach for propagation of the time dependent schrdinger equation for time dependent and nonlinear problems a detailed exposition of highly efficient and accurate method for the propagation of the time dependent schrdinger equation 50 is presented the method is readily generalized to solve an arbitrary set of ode s the propagation is based on a global approach in which large time intervals are treated as a whole replacing the local considerations of the common propagators the new method is suitable for various classes of problems including problems with a time dependent hamiltonian nonlinear problems non hermitian problems and problems with an inhomogeneous source term in this paper a thorough presentation of the basic principles of the propagator is given we give also a special emphasis on the details of the numerical implementation of the method for the first time we present the application for a non hermitian problem by a numerical example of a one dimensional atom under the influence of an intense laser field the efficiency of the method is demonstrated by a comparison with the common rungekutta approach

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

optimal error analysis of cranknicolson schemes for a coupled nonlinear schrdinger system in 3d the paper is concerned with the time step condition of the commonly used semi implicit cranknicolson finite difference schemes for a coupled nonlinear schrdinger system in three dimensional space we present the optimal l2 error estimate without any restriction on time step while all previous works require certain time step conditions our approach is based on a rigorous analysis in both real and imaginary parts of the energy estimate inequality of the error function numerical examples for both two dimensional and three dimensional models are investigated and numerical results illustrate our theoretical analysis

a stochastic portfolio optimization model with bounded memory this paper considers a portfolio management problem of merton s type in which the risky asset return is related to the return history the problem is modeled by a stochastic system with delay the investor s goal is to choose the investment control as well as the consumption control to maximize his total expected discounted utility under certain situations we derive the explicit solutions in a finite dimensional space

cramer rao type bounds for sparse bayesian learning in this paper we derive hybrid bayesian and marginalized cramer rao lower bounds hcrb bcrb and mcrb for the single and multiple measurement vector sparse bayesian learning sbl problem of estimating compressible vectors and their prior distribution parameters we assume the unknown vector to be drawn from a compressible student t prior distribution we derive crbs that encompass the deterministic or random nature of the unknown parameters of the prior distribution and the regression noise variance we extend the mcrb to the case where the compressible vector is distributed according to a general compressible prior distribution of which the generalized pareto distribution is a special case we use the derived bounds to uncover the relationship between the compressibility and mean square error mse in the estimates further we illustrate the tightness and utility of the bounds through simulations by comparing them with the mse performance of two popular sbl based estimators we find that the mcrb is generally the tightest among the bounds derived and that the mse performance of the expectation maximization em algorithm coincides with the mcrb for the compressible vector we also illustrate the dependence of the mse performance of sbl based estimators on the compressibility of the vector for several values of the number of observations and at different signal powers

split step spectral schemes for nonlinear dirac systems abstract the paper considers split step spectral schemes for the numerical integration of nonlinear dirac systems in 1 1 dimensions proofs of stability and convergence are given along with numerical experiments which clearly show the superiority of the suggested methods over standard and split step finite difference algorithms

corrections to asymptotic achievability of the cramer rao bound for noisy compressive sampling given n noisy measurements denoted by mathbf y and an overcomplete gaussian dictionary mathbf a the authors of 1 establish the existence and the asymptotic statistical efficiency of an unbiased estimator unaware of the locations of the nonzero entries collected in set mathcal i in the deterministic l sparse signal mathbf x more precisely there exists an estimator hat mathbf x mathbf y mathbf a unaware of set mathcal i with a variance reaching the oracle cramer rao bound in the asymptotic scenario i e for n l rightarrow infty and l n rightarrow alpha in 0 1 as was noted in the paper fundamental limits and constructive methods for estimation and sensing of sparse signals by b babadi the existence proof remains true even though lemma 3 5 and 20 the paper asymptotic achievability of the cramer rao bound for noisy compressive sampling are inexact in this note the exact closed form expression of the variance of the estimator hat mathbf x mathbf y mathbf a is provided and its practical usefulness is numerically illustrated with the orthogonal matching pursuit estimator

convergence of an energy preserving scheme for the zakharov equations in one space dimension an energy preserving linearly implicit finite difference scheme is presented for approximating solutions to the periodic cauchy problem for the one dimensional zakharov system of two nonlinear partial differential equations first order convergence estimates are obtained in a standard energy norm in terms of the initial errors and the usual discretization errors

compact finite difference schemes with high accuracy for one dimensional nonlinear schrodinger equation abstract in this paper two compact finite difference schemes are presented for the numerical solution of the one dimensional nonlinear schrodinger equation the discrete l 2 norm error estimates show that convergence rates of the present schemes are of order o h 4 2 numerical experiments on some model problems show that the present schemes preserve the conservation laws of charge and energy and are of high accuracy

solving the generalized nonlinear schrodinger equation via quartic spline approximation this paper is concerned with a new conservative finite difference method for solving the generalized nonlinear schrodinger gnls equation iut uxx f u 2 u 0 the numerical scheme is constructed through the semidiscretization and an application of the quartic spline approximation central difference and extrapolation formulae are used for approximating the neumann boundary conditions introduced both continuous and discrete energy conservation and the stability property are investigated the numerical method provides an efficient and reliable way for computing long time solitary solutions given by the gnls equation numerical examples are given to demonstrate our conclusions

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

dissipative or conservative finite difference schemes for complex valued nonlinear partial differential equations abstract we propose a new procedure for designing finite difference schemes that inherit energy conservation or dissipation property from complex valued nonlinear partial differential equations pdes such as the nonlinear schrodinger equation the ginzburg landau equation and the newell whitehead equation the procedure is a complex version of the procedure that furihata has recently presented for real valued nonlinear pdes furthermore we show that the proposed procedure can be modified for designing linearly implicit finite difference schemes that inherit energy conservation or dissipation property

new schemes for the nonlinear shrodinger equation in this paper a new scheme is derived from a multisymplectic integrator of the nonlinear shrodinger equation nls we can also call the new scheme multisymplectic scheme since multisymplectic integrators can be constructed by using symplectic schemes in both time and space directions we can use the composition method to get higher order integrators correspondingly the higher order composition schemes of the new scheme can be obtained

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

analysis of fisher information and the cramer rao bound for nonlinear parameter estimation after random compression in this paper we analyze the impact of compression with complex random matrices on fisher information and the cramer rao bound crb for estimating unknown parameters in the mean value function of a complex multivariate normal distribution we consider the class of random compression matrices whose distribution is right unitarily invariant the compression matrix whose elements are i i d standard complex normal random variables is one such matrix we show that for all such compression matrices the fisher information matrix has a complex matrix beta distribution we also derive the distribution of crb these distributions can be used to quantify the loss in crb as a function of the fisher information of the noncompressed data in our numerical examples we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in crb

maximum norm error bound of a linearized difference scheme for a coupled nonlinear schrodinger equations in this article a linearized conservative difference scheme for a coupled nonlinear schrodinger equations is studied the discrete energy method and an useful technique are used to analyze the difference scheme it is shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm numerical experiments are presented to support the theoretical results

a multisymplectic integrator for the periodic nonlinear schrodinger equation a multisymplectic integrator for the periodic nonlinear schrodinger equation is presented in this paper its accuracy is proved by introducing a norm we investigate its nonlinear stability we also discuss the relationship between this multisymplectic integrator and two variational integrators which are derived by using the discrete multisymplectic field theory and the finite element method

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

a differential quadrature algorithm for simulations of nonlinear schrodinger equation numerical simulations of nonlinear schrodinger equation are studied using differential quadrature method based on cosine expansion propogation of a soliton interaction of two solitons birth of standing and mobile solitons and bound state solutions are simulated the accuracy of the method dq is measured using maximum error norm the results are compared with some earlier works the lowest two conserved quantities are computed numerically for all cases

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

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

lyapunov type operators for numerical solutions of pdes in the present paper numerical methods are developed to approximate the solutions of some evolutionary nonlinear problems the continuous problems are transformed into some lyapunov type equations and then analysed for existence uniqueness convergence stability and error estimates the main idea consists in applying fourier analysis and von neumann criterion acting translation and scaling parameter methods to obtain contractive operators leading to fixed point theory

a semi discrete scheme for the stochastic nonlinear schrodinger equation we study the convergence of a semi discretized version of a numerical scheme for a stochastic nonlinear schrodinger equation the nonlinear term is a power law and the noise is multiplicative with a stratonovich product our scheme is implicit in the deterministic part of the equation as is usual for conservative equations we also use an implicit discretization of the noise which is better suited to stratonovich products we consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate however in the semi discrete case no ito formula is available and we have to use a discrete form of this tool also in the course of the proof we need to introduce a cut off of the diffusion coefficient which allows to treat the nonlinearity then we prove convergence by a compactness argument due to the presence of noise and to the implicit discretization of the noise this is rather complicated and technical we finally obtain convergence of the discrete solutions in various topologies

optimal time consistent reinsurance investment strategy with delay for an insurer under a defaultable market abstract in this paper we consider a reinsurance investment problem with delay for an insurer under the mean variance criterion in a defaultable market the financial market consists of a risk free bond a stock and a defaultable bond the insurer s surplus process is described by a jump diffusion risk model and the price process of the stock is assumed to follow a constant elasticity of variance cev model in particular we take the delay of feedback time for strategy into account applying stochastic control approach we derive the time consistent reinsurance investment strategy in post default case and pre default case explicitly via a game theoretic framework respectively finally numerical examples and sensitivity analyses are provided to show the impact of financial parameters on the optimal strategies