论文标题
探索能量景观的临界点：从低维示例到相位晶体PDE
Exploring Critical Points of Energy Landscapes: From Low-Dimensional Examples to Phase Field Crystal PDEs
论文作者
论文摘要
在目前的工作中，我们探讨了一些根发现方法在一系列典型示例中的应用。我们考虑的方法包括：（a）所谓的连续时间Nesterov（CTN）流量方法； （b）其称为平方操作方法（SOM）的变体； （c）上述两种方法的联合作用，采用所谓的通气方法。还提出了更传统的方法，例如牛顿的方法（及其具有放通的变体）。我们的玩具示例始于一个天真的一个自由度（DOF）系统，以提供土地的范围。随后，我们转向一个二-DOF系统，该系统是由软物质结晶的无限二二维，相位晶体（PFC）模型的降低所激发的。一旦阐明了2-DOF系统的景观，我们将转向完整的PDE模型，并说明了低维示例的见解如何导致PDE级别的新颖解决方案与软物质结晶的完整框架相关的PDE级别。
In the present work we explore the application of a few root-finding methods to a series of prototypical examples. The methods we consider include: (a) the so-called continuous-time Nesterov (CTN) flow method; (b) a variant thereof referred to as the squared-operator method (SOM); and (c) the the joint action of each of the above two methods with the so-called deflation method. More traditional methods such as Newton's method (and its variant with deflation) are also brought to bear. Our toy examples start with a naive one degree-of-freedom (dof) system to provide the lay of the land. Subsequently, we turn to a 2-dof system that is motivated by the reduction of an infinite-dimensional, phase field crystal (PFC) model of soft matter crystallisation. Once the landscape of the 2-dof system has been elucidated, we turn to the full PDE model and illustrate how the insights of the low-dimensional examples lead to novel solutions at the PDE level that are of relevance and interest to the full framework of soft matter crystallization.
