AI 中文总结
针对离散能量最小化问题的多水平优化方法在流形约束下受限，本文引入基于粗模型的黎曼多水平优化扩展，含度量兼容向量转移算子，制定算法并证明收敛性，通过应用展示效果，相比单级黎曼优化显著减少计算时间。
AI 中文摘要
多水平优化方法对离散能量最小化问题非常有效，但其欧几里得公式不适用于流形约束。我们基于一个粗模型引入了黎曼多水平优化的扩展，该粗模型与精细目标一阶相干，并在温和的回缩凸性假设下产生下降方向。框架包括用于在不同水平间传递一阶信息的度量兼容向量转移算子。我们制定了两级和多级算法，并使用黎曼Zoutendijk型论证证明了全局收敛性。在Kohn-Sham密度泛函理论、Gross-Pitaevskii基态计算和二元连续切割中的应用展示了该方法。实验表明，与单级黎曼优化相比，计算时间显著减少。
英文摘要
Multilevel optimization methods are highly effective for discretized energy minimization problems, but their Euclidean formulation does not directly apply to manifold constraints. We introduce a Riemannian extension of multilevel optimization based on a coarse model that is first-order coherent with the fine-level objective and yields descent directions under mild retraction-convexity assumptions. The framework includes metric-compatible vector transfer operators for passing first-order information between levels, covering both intrinsic and extrinsic constructions. We formulate two-level and multilevel algorithms and prove global convergence using a Riemannian Zoutendijk-type argument. Applications to Kohn--Sham density functional theory, Gross--Pitaevskii ground-state computation, and binary continuous cuts demonstrate the method on Stiefel, ellipsoid and Bernoulli manifolds. The experiments show significant reductions in computational time compared with single-level Riemannian optimization.
