报告题目：Skew-symmetric differentiation matrices and spectral methods on the real line
报告时间：2018年10月15日（周一）下午15: 20--16: 20
报告地点：尚贤楼706会议室
报告人：Prof. Arieh Iserles (剑桥大学)
主持人：蒋勇教授
专家简介：Arieh Iserles教授是国际著名数值分析学家、英国剑桥大学分析中心主任、应用数学和理论物理系终身教授。在微分方程的李群方法、高振荡积分的高效计算等方面做出了突出贡献。1999年获得挪威Lars Onsager奖，2012年获得英国伦敦数学学会授予的David Crighton奖，2012年应邀在第6届欧洲数学大会上作邀请报告。Arieh Iserles 教授现任或曾任国际著名期刊《Acta Numerica》、《Foundations of Computational Mathematics》、《IMA Journal of Numerical Analysis》的主编，以及国际著名计算数学期刊《Numerische Mathematic》、《Advances in Computational Mathematics》、《Calcolo》等的编辑。
报告内容摘要：A most welcome feature of orthogonal bases employed in spectral methods is that their differentiation matrix is skew symmetric, since this makes energy conservation automatic in conservative time-evolving problems. A familiar example is given by Hermite functions, which are dense in $L(-/infty,/infty)$ and give raise to a skew-symmetric, tridiagonal differentiation matrix.In this talk, describing joint work with Marcus Webb (KU Leuven), we present full characterisation of all orthogonal systems acting on $L2(-/infty,/infty)$, dense either there or in a Paley—Wiener space, and that have a differentiation matrix which is skew-symmetric, tridiagonal and irreducible. We also present a constructive algorithm for their generation — essentially, given any symmetric Borel measure on $(-/infty,/infty)$ or on $(-a,a)$ for some $a>0$, there exists a unique (up to rescaling) basis of this kind and it can be generated constructively. We conclude with a number of examples, related to Konoplev, Carlitz and Freud measures.Finally, we address the more general question of skew-Hermitian differentiation matrices. This brings us to very recent work on a variant of Malmquist—Takenaka basis, which appears to tick every desirable box: an orthonormal system dense in $L2(-/infty,/infty)$, with tridiagonal skew-Hermitian differentiation matrix and whose generalised Fourier coefficients can be computed with a single FFT.
数学与统计学院
2018年10月12日
