论文标题
Riemannian Multigrid Line搜索低级问题
Riemannian multigrid line search for low-rank problems
论文作者
论文摘要
由于涉及PDE的问题的离散化而引起的大规模优化问题有时会允许通过低级矩阵可以很好地近似的解决方案。在本文中，我们将通过直接在低级矩阵上解决优化问题来利用这种低级别的近似属性。特别是，我们引入了一种新的多级算法，其中优化变量被限制在固定级矩阵的riemannian歧管上。与大多数其他多级算法相反，在每个级别上选择等级以控制由于低级别截断而导致的扰动，我们可以在整个迭代过程中固定等级（以及计算复杂性）。此外，基于沃尔夫条件的线路搜索的经典实现使计算一个解决方案，其中数值精度仅限于机器epsilon的平方根。在这里，我们提出了对Hager和Zhang线搜索的Riemannian歧管的扩展，该搜索使用了近似的Wolfe条件，该条件可以在机器Epsilon的顺序上计算解决方案。数值实验证明了所提出的框架的计算效率。
Large-scale optimization problems arising from the discretization of problems involving PDEs sometimes admit solutions that can be well approximated by low-rank matrices. In this paper, we will exploit this low-rank approximation property by solving the optimization problem directly over the set of low-rank matrices. In particular, we introduce a new multilevel algorithm, where the optimization variable is constrained to the Riemannian manifold of fixed-rank matrices. In contrast to most other multilevel algorithms where the rank is chosen adaptively on each level in order to control the perturbation due to the low-rank truncation, we can keep the ranks (and thus the computational complexity) fixed throughout the iterations. Furthermore, classical implementations of line searches based on Wolfe conditions enable computing a solution where the numerical accuracy is limited to about the square root of the machine epsilon. Here, we propose an extension to Riemannian manifolds of the line search of Hager and Zhang, which uses approximate Wolfe conditions that enable computing a solution on the order of the machine epsilon. Numerical experiments demonstrate the computational efficiency of the proposed framework.
