非线性电元功能计算梯度流和非线性动力法 (Gradient Flows and Nonlinear Power Methods for the Computation of Nonlinear Eigenfunctions)

This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how convergence of (discretized) approximations can be verified. We review several flows from literature, which were proposed to compute nonlinear eigenfunctions, and show that they all relate to normalized gradient flows. Furthermore, we show that the implicit Euler discretization of gradient flows gives rise to a nonlinear power method of the proximal operator and prove their convergence to nonlinear eigenfunctions. Finally, we prove that $\Gamma$-convergence of functionals implies convergence of their ground states, which is important for discrete approximations.

翻译：本章描述Banach 空间的梯度流和非线性电力方法如何用于解决非线性电子元值依赖性电子元值问题,以及如何核实(分解的)近似的趋同性。我们审查了文献中的一些流程,这些流程旨在计算非线性电子元件,并表明它们都与正常的梯度流有关。此外,我们表明,梯度流的隐性电极分解产生了准线性操作员的非线性电源方法,并证明了它们与非线性电子元件的趋同性。最后,我们证明,用$\Gamma$-converggging of 函数的趋同性意味着它们的地面状态的趋同性,这对于离散性近光体十分重要。