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On the Convergence of Sobolev Gradient Flow for the Gross–Pitaevskii Eigenvalue Problem
SIAM Journal on Numerical Analysis ( IF 2.9 ) Pub Date : 2024-03-04 , DOI: 10.1137/23m1552553 Ziang Chen 1 , Jianfeng Lu 2 , Yulong Lu 3 , Xiangxiong Zhang 4
SIAM Journal on Numerical Analysis ( IF 2.9 ) Pub Date : 2024-03-04 , DOI: 10.1137/23m1552553 Ziang Chen 1 , Jianfeng Lu 2 , Yulong Lu 3 , Xiangxiong Zhang 4
Affiliation
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 667-691, April 2024.
Abstract. We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross–Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross–Pitaevskii energy functional with respect to the [math]-metric and two other equivalent metrics on [math], including the iterate-independent [math]-metric and the iterate-dependent [math]-metric. We first prove the energy dissipation property and the global convergence to a critical point of the Gross–Pitaevskii energy for the discrete-time [math] and [math]-gradient flow. We also prove local exponential convergence of all three schemes to the ground state.
中文翻译：
Gross-Pitaevskii 特征值问题的 Sobolev 梯度流的收敛性
SIAM 数值分析杂志，第 62 卷，第 2 期，第 667-691 页，2024 年 4 月。
摘要。我们研究了三个投影 Sobolev 梯度流到 Gross-Pitaevskii 特征值问题的基态的收敛性。它们被构造为 Gross-Pitaevskii 能量泛函相对于 [math] 度量和 [math] 上的其他两个等效度量的梯度流，包括迭代无关的 [math] 度量和迭代相关的 [math] 度量。 ]-公制。我们首先证明了离散时间[数学]和[数学]梯度流的能量耗散性质和全局收敛到Gross-Pitaevskii能量的临界点。我们还证明了所有三种方案到基态的局部指数收敛。
更新日期：2024-03-05
Abstract. We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross–Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross–Pitaevskii energy functional with respect to the [math]-metric and two other equivalent metrics on [math], including the iterate-independent [math]-metric and the iterate-dependent [math]-metric. We first prove the energy dissipation property and the global convergence to a critical point of the Gross–Pitaevskii energy for the discrete-time [math] and [math]-gradient flow. We also prove local exponential convergence of all three schemes to the ground state.
中文翻译：
Gross-Pitaevskii 特征值问题的 Sobolev 梯度流的收敛性
SIAM 数值分析杂志，第 62 卷，第 2 期，第 667-691 页，2024 年 4 月。
摘要。我们研究了三个投影 Sobolev 梯度流到 Gross-Pitaevskii 特征值问题的基态的收敛性。它们被构造为 Gross-Pitaevskii 能量泛函相对于 [math] 度量和 [math] 上的其他两个等效度量的梯度流，包括迭代无关的 [math] 度量和迭代相关的 [math] 度量。 ]-公制。我们首先证明了离散时间[数学]和[数学]梯度流的能量耗散性质和全局收敛到Gross-Pitaevskii能量的临界点。我们还证明了所有三种方案到基态的局部指数收敛。
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