散焦Gross--Pitaevskii基态问题的一种高效分裂方法的全局收敛性
Global convergence of an efficient splitting method for the defocusing Gross--Pitaevskii ground state problem
- Purdue University（普渡大学）
- University of Maryland（马里兰大学）
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中文总结 AI 辅助
针对散焦Gross--Pitaevskii基态问题，提出基于Davis--Yin分裂的高效迭代格式，证明其全局收敛性，并在GPU上实现高效求解。
中文摘要 AI 辅助
针对计算散焦Gross--Pitaevskii能量的基态，我们提出并分析了两种基于Davis--Yin三算子分裂的高效格式，该分裂方法显式处理势项和相互作用项，并通过预解式处理动能。第一种格式每次迭代需对$I-\gamma\Delta$求逆，第二种格式需对$I-\gamma\Delta+\gamma V_1$求逆，其中$V_1$表示势的可分离部分，在结构化网格上这两个算子均可通过简单的快速GPU求解器求逆。对于单调离散拉普拉斯算子，包括二阶有限差分格式和满足适当角度条件的单纯形网格上的集中线性有限元方法，我们证明了对于任意正的归一化初始向量，以及任意小于显式阈值的常数步长，该方法全局收敛到唯一的正离散基态。相比之下，先前被证明全局收敛到基态的方法都需要求逆一个依赖于迭代变量的更困难的椭圆算子。在三维测试中，单个GPU上最多包含$999^3$个未知量，一个简单的变步长规则使所提出的分裂格式在实践中高效，其墙钟时间与同样仅求逆移位拉普拉斯算子的黎曼共轭梯度方法相当，并且对初始猜测的选择更加鲁棒。
英文摘要
For computing the ground state of the defocusing Gross--Pitaevskii energy, we propose and analyze two efficient schemes based on the Davis--Yin three-operator splitting, which treats the potential and interaction terms explicitly, and the kinetic energy by a resolvent. One iteration costs one inversion of $I-γΔ$ for the first scheme, and of $I-γΔ+γV_1$ for the second, with $V_1$ denoting the separable part of the potential, and on structured meshes both operators can be inverted by simple fast GPU solvers. For monotone discrete Laplacians, including the second-order finite difference scheme and the lumped linear finite element method on simplicial meshes with suitable angle conditions, we prove global convergence to the unique positive discrete ground state, for every positive normalized initial vector, for any constant step size below an explicit threshold. In contrast, the methods previously proven to converge globally to the ground state all invert a more difficult elliptic operator that depends on the iteration variable. In three-dimensional tests with up to $999^3$ unknowns on one GPU, a simple variable step size rule makes the proposed splitting schemes efficient in practice, comparable in wall-clock time to Riemannian conjugate gradient methods that also invert only a shifted Laplacian operator, and much more robust with respect to the choice of the initial guess.
