The Study Of A Class Of Modified Local Minimax Algorithms For Computing Excited States Of BEC

This paper discussed the calculation of a class of excited states in Bose-Einstein condensate(BEC)equation.A modified local minimax(LMM)algorithm for constraint saddle point of unit sphere was proposed and the convergence of the algorithm was analyzed.Meanwhile,the preconditioner strategy was introduced to improve the efficiency of the algorithm.First,we reviewed the Gross-Pitaevskii equation of the BEC,gradient flow with discrete normalization for computing the ground state solution in BEC,and the basic idea of the unconstrained LMM for saddle point.Next,a modified LMM algorithm for the saddle point problem with the constraint of unit sphere was proposed by defining constrained local peak selection and using projection gradient descent direction in order.Then the algorithm was applied to the simulation of the excited states in BEC and abundant numerical results were obtained.Then we proved the strong energy attenuation of the Arimijo step size criterion.Furthermore,the global convergence of the modified LMM algorithm was confirmed under the constrained PS conditions and other standard assumptions.Finally,since the computational efficiency of the modified LMM algorithm based on projection gradient was not approving in calculation of excited state with high index,three kinds of preconditioners were introduced to optimize the algorithm.The precision and efficiency were compared with the un-preconditioner algorithm and more simulation results of BEC excited state were given.