报告题目：Accelerating exponential integrators

主讲人：Alexander Ostermann教授（奥地利因斯布鲁克大学）

时间：2025年9月11日（周四）10:00 a.m.

地点：北院卓远楼305会议室

主办单位：统计与数学学院

摘要：Exponential integrators are a well-established class of time integration schemes for the numerical solution of large systems of evolution equations. Unlike other time integration schemes, they solve the linear part of the problem exactly and discretize the nonlinearity with an explicit scheme. When the nonlinearity is small, this results in highly accurate schemes with excellent stability properties.Exponential integrators require computing the action of certain matrix functions (such as exponential and trigonometric functions) on vectors. Fast computations often require a particular form of the discretization matrix, which may conflict with the local linearization typically used to control the Lipschitz constant of the nonlinearity. For small problems, matrix functions are often computed explicitly, but for large problems, iterative methods such as Krylov subspace methods or Lagrange interpolation at Leja points are used. When these operations are computed efficiently, exponential integrators perform well. In important situations, acceleration techniques can be used to improve performance on modern HPC systems. This talk introduces two recent approaches: μ-mode integrators for evolution equations in Kronecker form and accelerated methods using simplified linearization.The μ-mode integrator is related to splitting methods and is based on one-dimensional precomputed exponentials. This technique can also be used to efficiently compute the spectral transform when a fast transform is not available. The accelerated integrator uses matrix functions from a related (but simpler) problem that can be computed cheaply. Numerical experiments in two and three dimensions demonstrate the effectiveness of these two new approaches.

主讲人简介：

Alexander Ostermann是奥地利因斯布鲁克大学的数值分析和科学计算教授。他在奥地利因斯布鲁克大学获得了博士学位，并之后在瑞士日内瓦大学担任博士后研究员。研究重点是偏微分方程的数值解。近年来，他与Christian Lubich合作研究了隐式和线性隐式龙格-库塔方法，与Marlis Hochbruck合作研究了指数积分器及其刚性阶条件的发展，与Lukas Einkemmer合作研究了分裂方法中非平凡边界条件的正确处理，与Katharina Schratz合作研究了具有极其粗糙初始数据的色散方程的数值积分器。

Alexander Ostermann曾担任奥地利因斯布鲁克大学数学、计算机科学和物理学院院长八年。他也是多个科学协会和委员会的成员。十多年来，他一直领导着奥地利因斯布鲁克大学的科学计算研究领域。在这个职位上，他负责监督大学的计算基础设施，并担任奥地利科学计算组织的奥地利最大的计算集群的董事会成员，近期还负责在因斯布鲁克大学安装量子计算机。