非对称加权厄米函数标准内积的格拉姆矩阵
On the Gram matrix of standard inner products of asymmetrically-weighted Hermite functions
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中文总结 AI 辅助
本文推导了非对称加权厄米函数无限格拉姆矩阵的元素、乔列斯基分解及逆矩阵的显式公式，分析了有限格拉姆矩阵及其舒尔补的渐近性质，并将其应用于弗拉索夫-泊松系统伽辽金谱方法的分析与实现。
中文摘要 AI 辅助
设A为与非对称加权（AW）厄米函数的标准L2内积相关的无限格拉姆矩阵，我们推导其元素的显式表达式及其乔列斯基分解，进一步表明该分解在缩放的Bargmann-Fock基上具有自然解释，还得到了A的逆矩阵的显式公式。随后我们考虑对应的有限格拉姆矩阵，分析其渐近性质及其舒尔补的渐近性质。该分析由等离子体物理的数值方法推动，特别是应用于弗拉索夫-泊松（VP）系统的伽辽金谱方法。作为应用，我们展示如何将推导的格拉姆矩阵公式和渐近结果用于VP系统伽辽金谱方法的分析与实现。
英文摘要
Let A denote the infinite Gram matrix associated with the standard L2 inner product of asymmetrically-weighted (AW) Hermite functions. We derive an explicit representation of its entries and its Cholesky factorization. We further show that this factorization admits a natural interpretation on a scaled Bargmann-Fock basis. An explicit formula for the inverse of A is also obtained. We then consider the corresponding finite Gram matrix and analyze its asymptotic property, as well as that of its Schur complement. The analysis is motivated by numerical methods for plasma physics, in particular Galerkin spectral methods applied to the Vlasov-Poisson (VP) system. As an application, we demonstrate how the derived Gram matrix formulas and asymptotic results can be exploited in the analysis and implementation of a Galerkin spectral method for the VP system.
发表机构
- Inria（法国国家信息与自动化研究所）
- University of Pau and Pays de l’Adour（波城与阿杜尔地区大学）
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