Biography

After receiving his PhD in 1995 from the Technical University of Denmark, Professor Hesthaven joined Brown University, USA where he became Professor of Applied Mathematics in 2005. In 2013 he joined EPFL as Chair of Computational Mathematics and Simulation Science and since 2017 as Dean of the School of Basic Sciences. His research interests focused on the development, analysis, and application of high-order accurate methods for the solution of complex time-dependent problems, often requiring high-performance computing. A particular focus of his research has been on the development of computational methods for problems of linear and non-linear wave problems and the development of reduced basis methods.

He has received several awards for both his research and his teaching, and has published 4 monographs and more than 125 research papers. He is on the editorial board of 8 journals and serves as Editor-in-Chief of SIAM J. Scientific Computing.

Parallel-in-Time Methods for Transport-Dominated Problems

In an attempt to overcome limits on strong scaling and take full advantage of large computing platforms,

there has been substantial research devoted to the development of parallel-in-time methods during the last decade. Among several techniques, the Parareal method has been the topic of substantial attention and
has been successfully applied to a range of different problems, including molecular dynamics and diffusion dominated problems.

However, the advances for transport dominated problems has been considerably slower with classic methods being plagued by stability problems or slow convergence.

We begin with a quick introduction to parallel-in-time methods and, in particular, the Parareal method and illustrate some of the successes and limitations of such methods. We then focus on transport dominated problems and discuss several past attempts that seek to enable the use of the Parareal method to accelerate the parallel solution of transport problems by overcoming what appears to be a stability problem. This leads to the initial conclusion that the quality of the coarse grid solver is particularly important for hyperbolic problems. As we shall demonstrate, such strategies can either be based on replacing the coarse solver or by carefully managing the work flow of the Parareal method in a general parallel environment to ensure a sufficiently accurate coarse solver while maintaining the potential for parallel speedup.

A second look at the Parareal algorithm reveals that the scheme primarily corrects for amplitude errors while phase errors, often dominating the solution of transport problems, remain essentially uncorrected. This supports the initial observation that an accurate coarse solver allows for rapid convergence as that introduces the required phase error corrections. The challenge remains that a phase accurate coarse solver is not likely to be achievable at low computational cost.

Based on this insight, we proposed a phase-corrected Parareal method and demonstrate that this converges substantially faster for purely hyperbolic problems and transport dominated problems without requiring the use of complex coarse solvers. It also offers a different and more general approach to formulate parallel-in-time methods based on assumptions on the error behavior.

We shall discuss the performance of these different approaches through a number of examples, including the demonstration of time-parallel efficiency exceeding 30\% for the fully parallel solution of nonlinear hyperbolic problems.