论文标题
用本地Lipschitz条件缩小二聚体鞍动力学的数学和数值分析
Mathematical and numerical analysis to shrinking-dimer saddle dynamics with local Lipschitz conditions
论文作者
论文摘要
我们向缩小的鞍座动力学提出了数学和数值研究，以查找解决方案景观中的任何索引鞍点。由于Hessian在马鞍动力学中的二聚体近似，局部Lipschitz的假设和鞍座动力学的强大非线性仍然是精致分析的挑战，例如溶液的界限和二聚体误差的界限。我们解决这些问题以在适当的放松参数下绑定解决方案，基于我们证明数值离散化的错误估计值通过匹配二聚体长度和时间步长大小来缩小与收缩二聚体鞍座动力学。此外，采用理查森外推以获得高阶近似。 The inherent reason of requiring the matching of the dimer length and the time step size lies in that the former serves a different mesh size from the later, and thus the proposed numerical method is close to a fully-discrete numerical scheme of some spacetime PDE model with the Hessian in the saddle dynamics and its dimer approximation serving as a "spatial operator" and its discretization, respectively, which in turn indicates the PDE nature of the马鞍动力学。
We present a mathematical and numerical investigation to the shrinkingdimer saddle dynamics for finding any-index saddle points in the solution landscape. Due to the dimer approximation of Hessian in saddle dynamics, the local Lipschitz assumptions and the strong nonlinearity for the saddle dynamics, it remains challenges for delicate analysis, such as the the boundedness of the solutions and the dimer error. We address these issues to bound the solutions under proper relaxation parameters, based on which we prove the error estimates for numerical discretization to the shrinking-dimer saddle dynamics by matching the dimer length and the time step size. Furthermore, the Richardson extrapolation is employed to obtain a high-order approximation. The inherent reason of requiring the matching of the dimer length and the time step size lies in that the former serves a different mesh size from the later, and thus the proposed numerical method is close to a fully-discrete numerical scheme of some spacetime PDE model with the Hessian in the saddle dynamics and its dimer approximation serving as a "spatial operator" and its discretization, respectively, which in turn indicates the PDE nature of the saddle dynamics.
