非线性偏微分方程多解计算的大范围收敛性算法及其应用研究

This project is aimed to investigate the algorithms with global convergence for nonlinear partial differential equations with multiple solutions. First based on the subspace S spanned by the previously found solutions, several skilful augmented singular transforms (AST) will be introduced and the so-called augmented partial Newton methods (APNM) are implemented to solve the corresponding augmented singular equations. This approach changes the structure of the singular line of the Newton method for the original problems. It not only guarantees the global convergence of the algorithms, but also makes sure that the solutions obtained are new. It is a brand new idea and can be understood as: when the Newton flow fails to pass through its singular line on the “ground”, our method will “fly over it in the sky”. It is worthwhile to point out that this approach is adaptable for the computation of multiple solutions of nonlinear differential equations with either variational or non-variational structures. On the other hand, various line-search rules will be modified and combined with the local min-max methods (LMM) to establish a new algorithm with global convergence to find the multiple solutions of nonlinear partial equations with variational structure. The system of theoretical analysis will be constructed for the approaches above. Further, these methods will be used to simulate the ground states or excited states of some practical problems, e.g.， the Gross-Pitaevskii equation, the Swift-Hohenberg equation and the transition pathway of energy surfaces in computational chemistry,et. al..

本项目研究非线性偏微分方程多解问题的大范围收敛性算法。首先基于已找到的解所张成的子空间S，通过巧妙地引入几类增广奇异变换，再利用增广部分牛顿法求解相应的增广奇异方程。该方法改变了求解原问题的牛顿法的奇异线的结构，不仅能保证算法的大范围收敛性，而且使得所计算出来的解一定为新解。这是一种全新的思路，直观上可以理解为：当牛顿流在“地面上”不能穿越奇异线时，我们的方法将从“空中”飞越奇异线。值得指出的是该方法同时适用于有变分结构和没有变分结构的非线性偏微分方程多解的计算。此外，本项目将改进线搜索准则，并使之与极小极大方法结合，发展一类新的计算具有变分结构的非线性偏微分方程多解问题的大范围收敛性算法。本项目不仅将建立上述算法的理论分析体系，并将其应用于某些实际问题的基态或激发态的模拟，如Gross-Pitaevskii方程，Swift-Hohenberg方程以及计算化学中的能量面过渡态等。