In this paper we propose stochastic multi-symplectic conservation law for stochastic Hamiltonian partial differential equations,and develop a stochastic multisymplectic method for numerically solving a kind of stochastic nonlinear Schrodinger equations.It is shown that the stochasticmulti-symplecticmethod preserves themultisymplectic structure,the discrete charge conservation law,and deduces the recurrence relation of the discrete energy.Numerical experiments are performed to verify the good behaviors of the stochastic multi-symplectic method in cases of both solitary wave and collision.
Inspired by an interesting idea of Cai and Zhang,we formulate and prove the convex k-sparse decomposition of vectors that is invariant with respect to the l_(1) norm.This result fits well in discussing compressed sensing problems under the Restricted Isometry property,but we believe it also has independent interest.As an application,a simple derivation of the RIP recovery conditionδk+θk,k<1 is presented.
In this paper,we consider the positive semi-definite space tensor cone constrained convex program,its structure and algorithms.We study defining functions,defining sequences and polyhedral outer approximations for this positive semidefinite space tensor cone,give an error bound for the polyhedral outer approximation approach,and thus establish convergence of three polyhedral outer approximation algorithms for solving this problem.We then study some other approaches for solving this structured convex program.These include the conic linear programming approach,the nonsmooth convex program approach and the bi-level program approach.Some numerical examples are presented.
In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law,discrete charge conservation law and discrete energy evolution law almost surely.Numerical experiments confirm well the theoretical analysis results.Furthermore,we present a detailed numerical investigation of the optical phenomena based on the compact scheme.By numerical experiments for various amplitudes of noise,we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time.In particular,if the noise is relatively strong,the soliton will be totally destroyed.Meanwhile,we observe that the phase shift is sensibly modified by the noise.Moreover,the numerical results present inelastic interaction which is different from the deterministic case.
We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems(SHS).Three types of systems,SHS with multiplicative noise,special separable Hamiltonians and multiple additive noise,respectively,are considered in this paper.Stochastic Runge-Kutta(SRK)methods for these systems are investigated,and the corresponding conditions for SRK methods to preserve the symplectic property are given.Based on the weak/strong order and symplectic conditions,some effective schemes are derived.In particular,using the algebraic computation,we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise,and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise,respectively.The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.
We study a new trust region affine scaling method for general bound constrained optimization problems. At each iteration, we compute two trial steps. We compute one along some direction obtained by solving an appropriate quadratic model in an ellipsoidal region. This region is defined by an affine scaling technique. It depends on both the distances of current iterate to boundaries and the trust region radius. For convergence and avoiding iterations trapped around nonstationary points, an auxiliary step is defined along some newly defined approximate projected gradient. By choosing the one which achieves more reduction of the quadratic model from the two above steps as the trial step to generate next iterate, we prove that the iterates generated by the new algorithm are not bounded away from stationary points. And also assuming that the second-order sufficient condition holds at some nondegenerate stationary point, we prove the Q-linear convergence of the objective function values.Preliminary numerical experience for problems with bound constraints from the CUTEr collection is also reported.
