Space-time parallel methods, also known more recently under the name PinT (Parallel in Time) methods, have a long history, but they have received a lot of attention over the past two decades. This is driven by the parallel hardware architectures that have now millions of cores, leading to saturation when parallelizing in space only. Parallelizing also the time direction is tempting, but this is very different from the space direction, because evolution problems satisfy a causality principle: the future is dependent on the past, and not the other way round.

I will show in my presentation that successful strategies for PinT methods depend strongly on the nature of the evolution problem. For hyperbolic problems, effective PinT methods are Domain Decomposition methods of Waveform Relaxation type, culminating in Unmapped Tent Pitching methods, ParaDiag methods, and also direct time parallel methods like ParaExp. Most of these methods can also be very effectively used for parabolic problems, but for such problems there are also highly successful multilevel methods, like Parareal and its variants. The currently best ones are however space-time multigrid methods. All these multilevel methods struggle however when applied to hyperbolic problems.

 Martin J. Gander obtained a Master in Computer Science from ETH Zürich in 1994 supervised by Ueli Maurer, a Master in Mathematics from Stanford University in 1995 supervised by George Papanicolaou, and a PhD in Scientific Computing and Computational Mathematics from Stanford University in 1997, supervised by Andrew M. Stuart. After postdoctoral studies at Ecole Polytechnique in Paris from 1998-1999 with Frederic Nataf and Laurence Halpern, he became professor with tenure at McGill University in Montreal from 1999-2004. Since then, he is full Professor of Mathematics at the University of Geneva, and was Vice-Dean of the Faculty of Science from 2009 to 2022. He is a SIAM Fellow since 2020, held the Jean-Morlet Chair of the CIRM in 2023 and an FSMP (Fondation Sciences Mathmématiques de Paris) professorship in 2023. He serves as an editor of several international journals, such as SIAM Journal on Scientific Computing (2009.12-2015.12), ETNA(2008.9-present) and SIAM Review (2016.9-present). His research interests are Numerical Analysis and Scientific Computing, Numerical Linear Algebra, Parallel Computing and Domain Decomposition, Time Parallelization, and Preconditioning.