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Lukas Renelt: Thursday, 24th July 2025 – 10h30 to 12h00 Abstract: The numerical solution of partial differential equations (PDEs) is one of the main research fields in computational mathematics where a vast variety of numerical methods have been developed. Applications include the simulation of reactive transport, groundwater flow, electromagnetism or even the computation of quantum states. These equations often additionally depend on physical parameters such as coefficients or boundary data which can significantly influence the solutions behaviour. This poses an additional challenge if solutions for many different parameter values are required such as in parameter studies, optimization tasks, inverse problems or uncertainty quantification. While methods such as the finite element method, finite volumes or discontinuous Galerkin approaches work well for given fixed parameters, they are prohibitively costly if solutions for thousands of different parameters are needed. To address this challenge, model order reduction methods have been developed which aim at approximating the highly complex set of all possible solutions jointly by a small (linear) subspace. In this talk, we will give a general introduction to the methodology with particular focus on the Reduced Basis (RB) approach highlighting both the abstract analysis and also showing concrete realizations. In the second part of the talk, we will present recent results when applying the method to parametrized Friedrichs’ systems – a large abstract class of linear PDE problems including for example convection-diffusion-reaction, linear transport, linear elasticity or the time-harmonic Maxwell equations. From a theoretical point of view, these problems are particularly interesting as their Friedrichs’ formulation involves parameter-dependent function spaces – a setting which has not been explored by the model order reduction community thus far. We present a novel theoretical framework and highlight the connections to the established theory with implications beyond Friedrichs’ systems. Additionally, a normal-equation-based discretization is introduced and used…

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Désolé, cet article est seulement disponible en Anglais Américain.

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Désolé, cet article est seulement disponible en Anglais Américain.

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Baptiste Plaquevent-Jourdain: Monday, 7th July 2025 – 10h30 to 12h00 Abstract: The initial goal of this thesis is the resolution of complementarity problems. These problems are reformulated here by the minimum C-function, which is piecewise thus nondifferentiable and leads to nonsmooth systems. The globalization of local methods for such equations (semi smooth Newton for instance) generally faces the difficulty that the computed directions are not necessarily descent direction for the associated merit function, used in lineasearch methods (whereas for smooth equations, the opposite of the gradient always works). In the case of the minimum C-function, a recent method replaces the pseudo-linearization direction by a direction found in a suitable convex polyhedron, guaranteed to be nonempty by some stringent regularity condition. The initial objective was to remove the regularity condition, as for the smooth systems, by using a Levenberg-Marquardt approach. The piecewise smooth aspect of the merit function induced by the minimum implies to choose a certain piece, this choice being discussed towards the end of the presentation. When trying to better understand this method and the generalization of the derivative, the B(ouligand)-differential, for the minimum function, it appeared that, for the simple case of linear problems (or affine), the inherent structure of the B-differential is the one of an arrangement or hyperplanes. This problem, very classic in combinatorial geometry, that we discovered here, is actually surprisingly rich and deep. We propose improvements on a state-of-the-art algorithm identifying the chambers. In particular, « primal-dual » variants, linking the chambers of an arrangement with the circuits of the associated matroid, seem promising. This long detour is actually relevant for the nonsmooth method and the choice of the « piece ».

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Romain Mottier: Thursday, 17th July 2025 – 10h30 to 12h00 Abstract: The objective of this Thesis is to develop approximation methods for the simulation of elasto-acoustic wave propagation. The spatial discretization of these equations relies on Hybrid High-Order (HHO) methods, which offer several advantageous properties, such as the ability to handle general meshes, computational efficiency, and high accuracy. An energy-norm error estimate is derived in the space semi-discrete setting. The time discretization is based on Runge–Kutta schemes. Particular attention is given to the stability of explicit schemes and to the efficiency of both explicit and implicit approaches. Numerical results are presented, including a geophysical application involving a complex geometry and strong heterogeneities. A second important aspect explored in this Thesis concerns the use of so-called unfitted meshes for elliptic problems with curved interfaces. These meshes do not conform to the physical interfaces, leading to mesh cells that may be intersected by them. This approach simplifies the mesh design. A method based on polynomial extension is introduced and analyzed in order to handle ill-cut cells. This approach is less intrusive than the more classical one based on the agglomeration of ill-cut cells.

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Emile Parolin: Thursday, 03rd July 2025 at 10:30 Abstract: Efficiently solving partial differential equation problems with highly heterogeneous coefficients is challenging due to the need for fine meshes, leading to large-scale and ill-conditioned linear systems. Domain decomposition methods address this by constructing efficient preconditioners that solve independent local problems in parallel. A key aspect to achieve scalability and robustness in these methods is the incorporation of a suitable coarse space. This presentation begins with an overview of one-level preconditioners, then introduces a novel algebraic construction of adaptive coarse spaces for two-level methods. The analysis is broadly applicable, encompassing non-hermitian and indefinite problems, as well as symmetric preconditioners like the additive Schwarz method and non-symmetric ones such as the restricted additive Schwarz method. It can also account for both exact and inexact subdomain solves. The coarse space is constructed by solving local generalized eigenproblems within each subdomain and applying a carefully chosen operator to the selected eigenvectors to obtain a local discrete solution. This is based on a joint work with Frédéric Nataf and Pierre-Henri Tournier.

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