We formulate a damped oscillating particle method to solve the stationary nonlinear Schrödinger equation (NLSE). The ground state solutions are found by a converging damped oscillating evolution equation that can be discretized with symplectic numerical techniques. The method is demonstrated for three different cases: for the single-component NLSE with an attractive self-interaction, for the single-component NLSE with a repulsive self interaction and a constraint on the angular momentum, and for the two-component NLSE with a constraint on the total angular momentum. We reproduce the so called yrast curve for the single-component case, described in [A. D. Jackson et al., Europhys. Lett. 95, 30002 (2011)], and produce for the first time an analogous curve for the two-component NLSE. The numerical results are compared with analytic solutions and competing numerical methods. Our method is well suited to handle a large class of equations and can easily be adapted to further constraints and components.
Abstract.We present an algorithm for mixed precision iterative refinement on the constrained and weighted linear least squares problem, the CWLSQ problem. The approximate solution is obtained by solving the CWLSQ problem with the weighted QR factorization [6]. With backward errors for the weighted QR decomposition together with perturbation bounds for the CWLSQ problem we analyze the convergence behaviour of the iterative refinement procedure.In the unweighted case the initial convergence rate of the error of the iteratively refined solution is determined essentially by the condition number. For the CWLSQ problem the initial convergence behaviour is more complicated, The analysis shows that the initial convergence is dependent both on the condition of the problem related to the solution, x, and the vector X = Wr, where W is the weight matrix and r is the residual.We test our algorithm on two examples where the solution is known and the condition number of the problem can be varied. The computational test confirms the theoretical results and verifies that mixed precision iterative refinement, using the system matrix and the weighted QR decomposition, is an effective way of improving an approximate solution to the CWLSQ problem.
Motivated by numerous experiments on Bose-Einstein condensed atoms which have been performed in tight trapping potentials of various geometries (elongated and/or toroidal/annular), we develop a general method which allows us to reduce the corresponding three-dimensional GrossPitaevskii equation for the order parameter into an effectively one-dimensional equation, taking into account the interactions (i.e., treating the width of the transverse profile variationally) and the curvature of the trapping potential. As an application of our model we consider atoms which rotate in a toroidal trapping potential. We evaluate the state of lowest energy for a fixed value of the angular momentum within various approximations of the effectively one-dimensional model and compare our results with the full solution of the three-dimensional problem, thus getting evidence for the accuracy of our model.
A hybrid algorithm consisting of a Gauss-Newton method and a second order method for solving constrained and weighted nonlinear least squares problems is developed, analyzed and tested. One of the advantages of the algorithm is that arbitrary large weights can be handled and that the weights in the merit function do not get unnecessary large when the iterates diverge from a saddle point. The local convergence properties for the Gauss-Newton method is thoroughly analyzed and simple ways of estimating and calculating the local convergence rate for the Gauss-Newton method are given. Under the assumption that the constrained and weighted linear least squares subproblems attained in the Gauss-Newton method are not too ill-conditioned, global convergence towards a rst order KKT point is proved.
.A framework and an algorithm for using modified Gram-Schmidt for constrained and weighted linear least squares problems is presented . It is shown that a direct implementation of a weighted modified Gram-Schmidt algorithm is unstable for heavily weighted problems . It is shown that, in most cases it is possible to get a stable algorithm by a simple modification free from any extra computational costs . In particular, it is not necessary to perform reorthogonalization .Solving the weighted and constrained linear least squares problem with the presented weighted modified Gram-Schmidt algorithm is seen to be numerically equivalent to an algorithm based on a weighted Householder-like QR factorization applied to a slightly larger problem. This equivalence is used to explain the instability of the weighted modified Gram-Schmidt algorithm . If orthogonality, with respect to a weighted inner product, of the columns in Q is important then reorthogonalization can be used . One way of performing such reorthogonalization is described .Computational tests are given to show the main features of the algorithm .
The present work is concerned with new ideas of potential value for solving differential equations. First, a brief introduction to particle methods in mechanics is made by revisiting the vibrating string. The full case of'nonlinear motion is studied and the corresponding nonlinear differential equations are derived. It is suggested that the particle origin of these equations is of more general interest than usually considered. A novel possibility to develop particle methods for solving differential equations in a direct way is investigated. The dynamical functional particle method (DFPM) is developed as a solution method for boundary value problems. DFPM is based on the concept of an interaction functional as a dynamical force field acting on quasi particles. The approach is not limited to linear equations. We e.xemptify by applying DFPM to several linear Schrodinger type of problems as welt as a nonlinear case. It is seen that DFPM performs veiy well in comparison with some standard numerical libraries. In all cases, the convergence rates are exponential in time.
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