University of Geneva

We present discrete methods for computing low-rank approximations of time-dependent tensors that are the solution of a differential equation. The approximation format can be Tucker, tensor trains, MPS or hierarchical tensors. We will consider two types of discrete integrators: projection methods based on quasi-optimal metric projection, and splitting methods based on inexact solutions of substeps. For both approaches we show numerically and theoretically that their behaviour is superior compared to standard methods applied to the so-called gauged equations. In particular, the error bounds are robust in the presence of small singular values of the tensor’s matricisations. Based on joint work with Emil Kieri, Christian Lubich, and Hanna Walach.

In the first part of this talk, we will give a very gentle introduction to the Ising model. Then , we will explain a very simple proof of a combinatorial formula for the 2D Ising model partition function using the language of Kac-Ward matrices. This approach can be used for general weighted graphs embedded in surfaces, and extends to the study of several other observables. This is a joint work with Dima Chelkak and Adrien Kassel.
 

Domain decomposition methods have been developed in various contexts, and with very different goals in mind. I will start my presentation with the historical inventions of the Schwarz method, the Schur methods and Waveform Relaxation. I will show for a simple model problem how all these domain decomposition methods function, give precise results for the model problem, and also explain the most general convergence results available currently for these methods. I will conclude with the parareal algorithm as a new variant for parallelization of evolution problems in the time direction.