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報告題目：Accelerating exponential integrators

主講人：Alexander Ostermann教授（奧地利因斯布魯克大學(xué)）

時間：2025年9月11日（周四）10:00 a.m.

地點：北院卓遠(yuǎn)樓305會議室

主辦單位：統(tǒng)計與數(shù)學(xué)學(xué)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.

主講人簡介：

Alexander Ostermann是奧地利因斯布魯克大學(xué)的數(shù)值分析和科學(xué)計算教授。他在奧地利因斯布魯克大學(xué)獲得了博士學(xué)位，并之后在瑞士日內(nèi)瓦大學(xué)擔(dān)任博士后研究員。研究重點是偏微分方程的數(shù)值解。近年來，他與Christian Lubich合作研究了隱式和線性隱式龍格-庫塔方法，與Marlis Hochbruck合作研究了指數(shù)積分器及其剛性階條件的發(fā)展，與Lukas Einkemmer合作研究了分裂方法中非平凡邊界條件的正確處理，與Katharina Schratz合作研究了具有極其粗糙初始數(shù)據(jù)的色散方程的數(shù)值積分器。

Alexander Ostermann曾擔(dān)任奧地利因斯布魯克大學(xué)數(shù)學(xué)、計算機(jī)科學(xué)和物理學(xué)院院長八年。他也是多個科學(xué)協(xié)會和委員會的成員。十多年來，他一直領(lǐng)導(dǎo)著奧地利因斯布魯克大學(xué)的科學(xué)計算研究領(lǐng)域。在這個職位上，他負(fù)責(zé)監(jiān)督大學(xué)的計算基礎(chǔ)設(shè)施，并擔(dān)任奧地利科學(xué)計算組織的奧地利最大的計算集群的董事會成員，近期還負(fù)責(zé)在因斯布魯克大學(xué)安裝量子計算機(jī)。