This talk is devoted to additive Schwarz methods for convex optimization. First, we propose an abstract framework for additive Schwarz methods for convex optimization. The framework's flexibility allows it to handle composite optimization problems and inexact local solvers. Moreover, it establishes a sharp convergence theory that agrees with the classical theory when addressing linear problems.

We consider non-conforming discretizations of the stationary Stokes equation in two and three dimensions by Crouzeix-Raviart type elements. The original definition in the seminal paper by M. Crouzeix and P.-A. Raviart in 1973 is implicit and also contains substantial freedom for a concrete choice.

In this talk, we discuss problems and numerical methods arising in the calculus of variations and energy minimization. Among numerous applications, energy minimization is a core element of Machine Learning algorithms. Within the field of nonlinear PDEs, the calculus of variations has received a lot of attention from the analysis point of view.  Although quite interesting and challenging,  the numerical analysis of these problems is much less developed.

After a brief introduction to the field of Computational Cardiology and cardiac reentry, we introduce and study some scalable domain decomposition preconditioners for cardiac reaction-diffusion models, discretized with splitting semi-implicit techniques in time and isoparametric finite elements in space.

Coffee Time: 15:30-16:00. It has been a long-standing debate that “Is deep learning alchemy or science? ”, since the success of deep learning mainly relies on various engineering design and tricks, and lack of theoretical foundation. Unfortunately, the underlying mechanism of deep learning is still mysterious, severely limiting its further development from both theoretical and application aspects.

In this talk, I will explain the problem, its solution, and some subsequent work generalizing, extending and improving the ProxSkip method in various ways. We study distributed optimization methods based on the local training (LT) paradigm - achieving improved communication efficiency by performing richer local gradient-based training on the clients before parameter averaging - which is of key importance in federated learning. Looking back at the progress of the field in the last decade, we identify 5 generations of LT methods: 1) heuristic, 2) homogeneous, 3) sublinear, 4) linear, and 5) accelerated. The 5th generation, initiated by the ProxSkip method of Mishchenko et al (2022) and its analysis, is characterized by the first theoretical confirmation that LT is a communication acceleration mechanism.

In this talk, we shall explain how transportation networks emerge as a self-regulating process, with a particular focus on applications in biology (leaf venation in plants, neuronal networks in animals). We start by introducing a purely diffusive model with tensor-valued diffusivity, derived as a gradient flow of a broad class of entropy dissipations. The introduction of a prescribed electric potential leads to the Fokker-Planck equation. We show that with quadratic entropy density modeling Joule heating, the model is convex with respect to the diffusivity tensor.