Domain decomposition research intensified in the early nineties, and there is still substantial research activity in this field. There has been however an important shift in domain decomposition, and I will explain three new interesting research directions that are pursued very actively at the moment, and give newest results:
1. Iterative solvers for time harmonic wave propagation: time harmonic wave propagation problems are very hard to solve by iterative methods. All classical iterative methods, like Krylov methods, multigrid, and also domain decomposition methods, fail for the key model problem, the Helmholtz equation. There are new, highly promising domain decomposition methods for such problems, which I will present, and I will also state precisely under which conditions they can work well, and when they still fail.
2. Coarse space components: domain decomposition analysis has lacked behind multigrid in the precise understanding of the interaction between the domain decomposition smoother and coarse space solver, and all classical domain decomposition solvers need Krylov acceleration to be effective, while multigrid does not. I will present a new spectral analysis of the Schwarz iteration operator, which allows us to achive as an accurate understanding of two level Schwarz methods as the seminal Fourier analysis of multigrid methods.
3. Time parallelization: new computing architectures have too many computing cores to parallelize only in space for evolution problems. I will present time and space-time domain decomposition methods and explain which can be effective for parabolic and hyperbolic problems.[-]

In an era where Artificial Intelligence (AI) is permeating virtuallly every single field of science and engineering, it is becoming critical to members of the numerical linear algebra community to understand and embrace AI , and to contribute to its advancement, and more broadly to the advancement of machine learning. What is fascinating and rather encouraging is that Numerical Linear Algebra (NLA) is at the core of machine learning and AI. In this talk we will give an overview of Deep Learning with an emphasis on Large Language Models (LLMs) and Transformers [3, 4]. The very first step of LLMs is to convert the problem into one that can he exploited by numerical methods, or to be more accurate, by optimization techniques. All AI methods rely almost entirely on essentially 4 ingredients: data, optimization methods, statistical intuition, and linear algebra. Thus, the first task is to map words or sentences into tokens which are then imbedded into Euclidean spaces. From there on, the models refer to vectors and matrices. We will show a few examples of important developments in ML, that were heavily based on linear algebra ideas. Among these, we will briefly discuss LoRa [1] a technique in which low-rank approximation was used to reduce computational cost in some models, leading to gains of a few orders of magnitude. Another contribution that used purely algebraic arguments and that had a major impact on LLMs is the article [2]. Here the main discovery is that the nonlinear ""self-attention"" in LLMs can be approximated linearly, resulting in huge savings in computations, as the computational complexity was decreased from $O\left(n^2\right)$ to $O(n)$.The talk will be mostly a survey of known recent methods in AI with the primary goal of unraveling the mathematics of Transformers. A secondary goal is to initiate a discussion on the issue of how NLA specialitst can participate in AI research.[-]

The aim of this talk is to present a new variation formulation of the time-dependent many-body electronic Schrödinger equation with Coulombic singularities. More precisely, its solution can actually be expressed as the solution of a global space-time quadratic minimization problem that proves to be useful for several tasks:
1) first, it is amenable to Galerkin time-space discretization schemes, using an appropriate least-square formulation
2) it enables to yield a new variational principle for the construction dynamical low-rank approximations, that is different from the classical Dirac-Frenkel variational principle
3) it enables to obtain fully certified a posteriori error estimators between the exact solution and approximate solutions.
The present analysis can be applied to the electronic many-body time-dependent Schrödinger equation with an arbitrary number of electrons and interaction potentials with Coulomb singularities.[-]

In these two lectures, I will introduce the main algorithms used in today's noisy and tomorrow's fault-tolerant quantum computers. After a quick introduction to gate-based quantum computation, I will review basic primitives like the quantum Fourier transform and their use in algorithms such as quantum phase estimation, with applications to the factoring problem (Shor's algorithm) and energy estimation in quantum physics. Then, I will turn to the challenges of decoherence in quantum computers, to the variational algorithms that have been designed to mitigate its effects (including the variational quantum eigensolver, VQE), and to their limitations and some counter-measures like error mitigation. In the hands-on session, we will implement a phase estimation algorithm as well as a VQE algorithm applied to a quantum chemistry problem.[-]

One of the oldest and currently most promising application areas for quantum devices is quantum simulation. Popularised by Feynman in the early 1980s, it is important for the efficient simulation – compared to its classical counterpart – of one special partial differential equation (PDE): Schrodinger's equation. This is possible because quantum devices themselves naturally obey Schrodinger's equation. Just like with large-scale quantum systems, classical methods for other high-dimensional and large-scale PDEs often suffer from the curse-of-dimensionality, which a quantum treatment might in certain cases be able to mitigate. Aside from Schrodinger's equation, can quantum simulators also efficiently simulate other PDEs? To enable the simulation of PDEs on quantum devices that obey Schrodinger's equations, it is crucial to first develop good methods for mapping other PDEs onto Schrodinger's equations.After a brief introduction to quantum simulation, I will address the above question by introducing a simple and natural method for mapping other linear PDEs onto Schrodinger's equations. It turns out that by transforming a linear partial differential equation (PDE) into a higher-dimensional space, it can be transformed into a system of Schrodinger's equations, which is the natural dynamics of quantum devices. This new method – called /Schrodingerisation/ – thus allows one to simulate, in a simple way, any general linear partial differential equation and system of linear ordinary differential equations via quantum simulation.This simple methodology is also very versatile. It can be used directly either on discrete-variable quantum systems (qubits) or on analog/continuous quantum degrees of freedom (qumodes). The continuous representation in the latter case can be more natural for PDEs since, unlike most computational methods, one does not need to discretise the PDE first. In this way, we can directly map D-dimensional linear PDEs onto a (D + 1)-qumode quantum system where analog Hamiltonian simulation on (D + 1) qumodes can be used. It is the quantum version of analog computing and is more amenable to near-term realisation.These lectures will show how this Schrodingerisation method can be applied to linear PDEs, systems of linear ODEs and also linear PDEs with random coefficients, where the latter is important in the area of uncertainty quantification. Furthermore, these methods can be extended to solve problems in linear algebra by transforming iterative methods in linear algebra into the evolution of linear ODEs. It can also be applicable to certain nonlinear PDEs. We will also discuss many open questions and new research directions.[-]

Reduced MHD models in Tokamak geometry are convenient simplifications of full MHD and are fundamental for the numerical simulation of MHD stability in Tokamaks. This presentation will address the mathematical well-posedness and the justification of the such models.
The first result is a systematic design of hierachies of well-posed reduced MHD models. Here well-posed means that the system is endowed with a physically sound energy identity and that existence of a weak solution can be proved. Some of these models will be detailed.
The second result is perhaps more important for applications. It provides understanding on the fact the the growth rate of linear instabilities of the initial (non reduced) model is lower bounded by the growth rate of linear instabilities of the reduced model.
This work has been done with Rémy Sart.[-]

Tensor methods have emerged as an indispensable tool for the numerical solution of high-dimensional problems in computational science, and in particular problems arising in stochastic and parametric analyses. In many practical situations, the approximation of functions of multiple parameters (or random variables) is made computationally tractable by using low-rank tensor formats. Here, we present some results on rank-structured approximations and we discuss the connection between best approximation problems in tree-based low-rank formats and the problem of finding optimal low-dimensional subspaces for the projection of a tensor. Then, we present constructive algorithms that adopt a subspace point of view for the computation of sub-optimal low-rank approximations with respect to a given norm. These algorithms are based on the construction of sequences of suboptimal but nested subspaces.

Keywords: high dimensional problems - tensor numerical methods - projection-based model order reduction - low-rank tensor formats - greedy algorithms - proper generalized decomposition - uncertainty quantification - parametric equations[-]

We combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid. Computing the GMsFEM solution involves calculating the residual and the Jacobian on the fine grid. As such, we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid. The resulting reduced-order approach enables a significant reduction in the flow problem size while accurately capturing the behavior of fully-resolved solutions. We consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully-implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high-contrast porous media.

Keywords: generalized multiscale finite element method - nonlinear PDEs - heterogeneous porous media - discrete empirical interpolation - proper orthogonal decomposition[-]