Numerics of Riemann-Hilbert problems and operator determinants (B03)

黎曼-希尔伯特问题的数值和算子行列式（B03）

• 批准号: 223485477
• 金额: --
• 依托单位: Lehrstuhl Wissenschaftliches Rechnen (M3)
• 依托单位国家: 德国
• 项目类别: CRC/Transregios
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2019-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/223485477?language=en
• 关键词: Numerics; Riemann; Hilbert; problems; operator

The numerical treatment of Riemann-Hilbert problems, e.g., from the isomonodromy transform of discrete holomorphic functions, requires the study of the interplay of asymptotic (continuous) and pre-asymptotic (discrete) structures. Preconditioning based on contour deformations, lensing and g-functions leads to the computation of shortest pathes in families of weighted graphs subject to some homotopy constraints. Integrable kernels lead to equivalent formulations in terms of operator determinants. Numerical methods require smooth kernels that can be obtained from discrete symmetries. The basic concept of Hilbert transforms will be studied within the framework of discrete complex analysis.

Riemann-Hilbert问题的数值处理，例如，从离散全纯函数的isomonodromy变换，需要研究渐近（连续）和前渐近（离散）结构的相互作用。基于轮廓变形、透镜效应和g-函数的预处理导致在某些同伦约束下的加权图族中的最短路径的计算。可积核导致等价的公式在运营商的决定。数值方法需要可以从离散对称性获得的光滑内核。希尔伯特变换的基本概念将在离散复分析的框架内进行研究。