优化薛定谔泛函的索伯列夫梯度：在量子力学与非线性光学中的应用
Optimizing Schroedinger functionals using Sobolev gradients: Applications to Quantum Mechanics and Nonlinear Optics
AI总结：
本文提出使用索伯列夫梯度作为预处理器，结合快速傅里叶变换，高效最小化薛定谔泛函，显著提升量子力学与非线性光学中求解最小能量和稳态解的收敛速度。
摘要：
本文研究了索伯列夫梯度技术在最小化多个薛定谔泛函问题中的应用，这些泛函与量子力学和非线性光学中及时且困难的非线性问题相关。我们证明这些梯度在最小化方法中充当传统下降方向选择的预处理器，并展示了一种使用离散傅里叶基和快速傅里叶变换来计算它们的计算成本低廉的方法。我们证明，索伯列夫预处理在寻找最小能量解以及稳态解方面，相较于传统技术提供了巨大的收敛性改进，并建议将该方法推广到使用任意线性算子。
英文摘要：
In this paper we study the application of the Sobolev gradients technique to the problem of minimizing several Schrödinger functionals related to timely and difficult nonlinear problems in Quantum Mechanics and Nonlinear Optics. We show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete Fourier basis and a Fast Fourier Transform. We show that the Sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
