论文标题
通过广义的高索引鞍动力搜索解决方案景观
Searching the solution landscape by generalized high-index saddle dynamics
论文作者
论文摘要
我们引入了一种广义的数值算法来构建解决方案景观，该途径是由所有固定点及其连接组成的途径图。基于梯度系统的基于高指数优化的缩小二聚体（HIOSD）方法，提出了广义的高点鞍动力学（GHISD）来计算动态系统的任何索引鞍座。索引 - $ K $鞍点的线性稳定性可以证明GHISD系统。向下搜索算法和向上搜索算法的组合应用于系统地构建解决方案景观，这不仅提供了一种强大而有效的方法来计算多个解决方案而不调整初始猜测，还可以揭示不同解决方案之间的关系。数值示例，包括三维示例和相位场模型，通过显示连接的途径图来证明解决方案景观的新概念。
We introduce a generalized numerical algorithm to construct the solution landscape, which is a pathway map consisting of all stationary points and their connections. Based on the high-index optimization-based shrinking dimer (HiOSD) method for gradient systems, a generalized high-index saddle dynamics (GHiSD) is proposed to compute any-index saddles of dynamical systems. Linear stability of the index-$k$ saddle point can be proved for the GHiSD system. A combination of the downward search algorithm and the upward search algorithm is applied to systematically construct the solution landscape, which not only provides a powerful and efficient way to compute multiple solutions without tuning initial guesses, but also reveals the relationships between different solutions. Numerical examples, including a three-dimensional example and the phase field model, demonstrate the novel concept of the solution landscape by showing the connected pathway maps.
