计算曲面上微分算子的谱性质

中文摘要 AI 辅助

在许多应用中，从医学图像分析到纳米尺度量子结构，计算曲面上的微分算子谱至关重要。然而，对无限维算子及底层曲面的离散化可能导致不准确和误导性的结果。本文介绍了SurfSpec算法套件，用于计算曲面上的算子特征值和谱测度，并带有误差控制和收敛性保证。我们结合基于复轮廓积分和残差的方法来计算离散谱和高频特征模态，包括非线性特征值问题。然后，我们计算谱测度与有理核的卷积以处理连续谱。在这两种情况下，关键工具是作用于函数的算子预解式，该预解式使用求解曲面偏微分方程的高阶方法进行计算。我们在各种微分算子和曲面上展示了我们的算法，包括一个三角龙模型。

英文摘要

Computing spectra of differential operators on surfaces is crucial in many applications, ranging from medical image analysis to nanoscale quantum structures. However, discretization of the infinite-dimensional operator and the underlying surface can lead to inaccurate and misleading results. This paper introduces the SurfSpec suite of algorithms to compute eigenvalues and spectral measures of operators on surfaces with error control and convergence guarantees. We combine methods based on complex contour integration and residuals to compute discrete spectra and high-frequency eigenmodes, including of nonlinear eigenvalue problems. We then compute convolutions of spectral measures with rational kernels to deal with continuous spectra. In both cases, the key tool is the operator resolvent applied to functions, which is computed using high-order methods that solve surface PDEs. We illustrate our algorithms on a variety of differential operators and surfaces, including a triceratops.

发表机构

• University of Cambridge（剑桥大学）
• Flatiron Institute（平顿研究所）

机构由 AI 辅助整理，请以论文原文为准。