论文标题
对于解决方案景观的约束高指数马鞍动力学和平等约束
Constrained high-index saddle dynamics for the solution landscape with equality constraints
论文作者
论文摘要
我们提出了一种受约束的高索引鞍动力学（CHISD）方法，以搜索能量功能的索引 - $ k $鞍点，受到平等约束。借助Riemannian歧管工具，CHISD衍生在Minimax框架中，其线性稳定性在索引 - $ K $鞍点上得到了证明。为了确保歧管属性，使用缩回和向量传输来数值实现CHISD。然后，我们通过将CHISD与向下和向上的搜索算法相结合来构建解决方案景观，从而在存在相等性约束的情况下构建解决方案景观。我们将汤姆森问题和玻色网凝结作为数值示例，以证明所提出的方法的效率。
We propose a constrained high-index saddle dynamics (CHiSD) method to search for index-$k$ saddle points of an energy functional subject to equality constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax framework, and its linear stability at an index-$k$ saddle point is proved. To ensure the manifold property, the CHiSD is numerically implemented using retractions and vector transport. Then we present a numerical approach by combining CHiSD with downward and upward search algorithms to construct the solution landscape in the presence of equality constraints. We apply the Thomson problem and the Bose-Einstein condensation as numerical examples to demonstrate the efficiency of the proposed method.
