The parallel-in-time integration of wave-type equations is well known to be a difficult task. When applying classical waveform-relaxation (WR) and parareal type methods, one generally experiences rapid error growth before reaching convergence in a finite number of iterations. This negative behavior prevents, in general, the successful application of these domain decomposition methods. In this talk, the focus is on WR-type methods. Classical WR convergence analyses use classical Laplace/Fourier techniques. However, these approaches provide analyses for unbounded time intervals, and do not allow one to describe precisely the WR converge behavior on finite time intervals. In this talk, we present a novel analysis based on the methods of characteristics, which allows us, on the one hand, to obtain a detailed characterization of the error growth along with the iterations and, on the other hand, to introduce a new parallel-in-time computational strategy. Numerical experiments support our new theoretical and numerical findings. This is a joint work with Martin J. Gander and Ilario Mazzieri.[-]

In this talk, we present a three-level variant of the parareal algorithm that uses three propagators at the fine, intermediate and coarsest levels. The fine and intermediate levels can both be run in parallel, only the coarsest level propagation is completely sequential. We interpret our algorithm as a variant of three-level MGRIT, and we present a convergence analysis that uses parareal-type assumptions, i.e., those that involve Lipschitz constants on the propagators. We present numerical experiments to illustrate how sharp the estimates are for various time dependent problems.[-]

When thinking about parallel in time schemes, one often tends to view time as a variable to discretize within a numerical scheme (that usually involves a time marching strategy). In this talk, I propose to review alternative strategies where time can be seen as a parameter so that computing the PDE solution at a given time would consist in evaluating closed formulas or in solving tasks of very low computational cost that do not involve any time marching. This type of approach is by nature entirely parallelizable. It can be achieved by either leveraging analytic formulas (whose existence strongly depends on the nature of the PDE), or by learning techniques such as model order reduction. For the later strategy, convection dominated problems are challenging (just like in classical PinT schemes such as parareal) and I will present recent contributions to address this type of problems.[-]

This talk consists of two parts, one elementary and one related to the solution of complicated systems of evolutionary partial differential equations. In the first part we show how preconditioning for all-at-once descriptions of linear time-dependent differential equations can defeat the everpresent danger of high-index nilpotency associated with the principle of causality. In particular we will describe some theory for periodic preconditioning of initial value problems that establishes it as a viable Parallelin-time (PinT) approach. The second part builds on much excellent work on PinT methods for scalar parabolic PDEs such as the diffusion equation to propose PinT methods for more complicated evolutionary PDE systems. We will explain the idea with reference to the time-dependent incompressible Stokes and Navier-Stokes equations and indicate it's more broad applicability. This part of the talk is joint with Federico Danieli and Ben Southworth.[-]

I will give an elementary introduction of basic deep learning models and training algorithms from a mathematical viewpoint. In particular, I will relate some basic deep learning models with finite element and multigrid methods. I will also touch on some advanced topics to demonstrate the potential of new mathematical insight and analysis for improving the efficiency of deep learning technologies and, in particular, for their application to numerical solution of partial differential equations.[-]

This presentation will be kept at a basic level, both continuous and algebraic versions of the methods will be given in their most common variants and the main ingredients of domain decomposition methods will be presented. The content will follow the lines of the chapters 1 and 3 from the domain decomposition book. A short introduction to Freefem software will be given which will allow the students to use quickly the codes illustrating the methods.
Outcomes: At the end of this first lecture, students will have a basic understanding of the methods but also of their implementation.[-]

Domain decomposition methods are meant to be used as parallel solvers and scalability (behaviour independent of the number of subdomains/processors) and robustness with respect to the physical parameters are very important issues. An introduction to coarse spaces and two-level methods for symmetric positive definite (SPD) problems will be given together with the presentation of a few variants of domain decomposition preconditioners (AS, RAS, ORAS, SORAS). The content will follow chapters 4 and 5 from the book, although more recent research results will also be included.
Outcomes: Students will be able to understand the use and the impact of the two-level methods both for scalability and robustness (even if at this stage the codes are sequential).[-]