This paper is devoted to the numerical solution of constrained energy minimization problems arising in computational physics and chemistry such as the Gross-Pitaevskii and Kohn-Sham models. In particular, we introduce the Riemannian Newton methods on the infinite-dimensional Stiefel and Grassmann manifolds. We study the geometry of these two manifolds, its impact on the Newton algorithms, and present expressions of the Riemannian Hessians in the infinite-dimensional setting, which are suitable for variational spatial discretizations. A series of numerical experiments illustrates the performance of the methods and demonstrates its supremacy compared to other well-established schemes such as the self-consistent field iteration and gradient descent schemes.

翻译：本文致力于数值求解计算物理与化学中出现的约束能量最小化问题，例如Gross-Pitaevskii模型和Kohn-Sham模型。特别地，我们在无限维Stiefel流形与Grassmann流形上引入黎曼牛顿方法。我们研究这两种流形的几何结构及其对牛顿算法的影响，并给出无限维情形下适用于变分空间离散化的黎曼Hessian表达式。一系列数值实验展示了该方法的性能，并证明其相较于自洽场迭代、梯度下降等成熟方案具有显著优势。