论文标题
Ablowitz-Ladik方程的减少订购建模
Reduced-order modeling for Ablowitz-Ladik equation
论文作者
论文摘要
在本文中，为Ablowitz-Ladik方程（ALE）构建了降低的模型（ROMS），这是一种具有和没有阻尼的非线性Schrödinger方程（NLSE）的可集成的半差异化。两种啤酒都是非典型的保守和耗散性的哈密顿式系统，其泊松基质是根据州变量和二次汉密尔顿人的二次矩阵。全阶溶液是通过保守啤酒的能量保留中点规则和耗散啤酒的指数中点规则获得的。减小的溶液是通过保留降低的非典型汉密尔顿系统的偏度对称结构来纳入构建的，该结构通过将适当的正交分解（POD）与盖雷金投影施加。对于ROM的有效离线离线分解，泊松基质的二次非线性项是通过离散的经验插值方法（DEIM）近似的。通过使用张量技术进一步加速了降级溶液的计算。保守啤酒的汉密尔顿和动量保存，并保存耗散啤酒的耗散特性，保证了孤子溶液的长期稳定性。
In this paper, reduced-order models (ROMs) are constructed for the Ablowitz-Ladik equation (ALE), an integrable semi-discretization of the nonlinear Schrödinger equation (NLSE) with and without damping. Both ALEs are non-canonical conservative and dissipative Hamiltonian systems with the Poisson matrix depending quadratically on the state variables, and with quadratic Hamiltonian. The full-order solutions are obtained with the energy preserving midpoint rule for the conservative ALE and exponential midpoint rule for the dissipative ALE. The reduced-order solutions are constructed intrusively by preserving the skew-symmetric structure of the reduced non-canonical Hamiltonian system by applying proper orthogonal decomposition (POD) with the Galerkin projection. For an efficient offline-online decomposition of the ROMs, the quadratic nonlinear terms of the Poisson matrix are approximated by the discrete empirical interpolation method (DEIM). The computation of the reduced-order solutions is further accelerated by the use of tensor techniques. Preservation of the Hamiltonian and momentum for the conservative ALE, and preservation of dissipation properties of the dissipative ALE, guarantee the long-term stability of soliton solutions.
