# Past Computational Mathematics and Applications Seminar

We propose a strategy which allows computing eigenvalue enclosures for the Maxwell operator by means of the finite element method. The origins of this strategy can be traced back to over 20 years ago. One of its main features lies in the fact that it can be implemented on any type of regular mesh (structured or otherwise) and any type of elements (nodal or otherwise). In the first part of the talk we formulate a general framework which is free from spectral pollution and allows estimation of eigenfunctions.

We then prove the convergence of the method, which implies precise convergence rates for nodal finite elements. Various numerical experiments on benchmark geometries, with and without symmetries, are reported.

A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. Considering a probabilistic model of sparse signals, we show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations.

This is joint work with Rodolphe Jenatton & Francis Bach.

The context of data assimilation in oceanography will be described as well as the computational challenges associated with it. A class of numerical linear algebra methods is described whose purpose is to exploit the problem structure in order to reduce the computational burden and provide provable convergence results for what remains a (very large) nonlinear problem. This class belongs to the Krylov-space family of methods and the special structure used is the imbalance between the dimensions of the state space and the observation space. It is also shown how inexact matrix-vector products can be exploited. Finally, preconditioning issues and resulting adaptations of the trust-region methodology for nonlinear minimization will also be outlined.

By Serge Gratton, Selime Gurol, Philippe Toint, Jean Tshimanga and Anthony Weaver.