Section: New Results

Objective 4 – Model Reduction

Model reduction aims at proposing efficient algorithmic procedures for the resolution (to some reasonable accuracy) of high-dimensional systems of parametric equations. This overall objective entails many different subtasks:

1) the identification of low-dimensional surrogates of the target “solution’’ manifold 2) The devise of efficient methodologies of resolution exploiting low-dimensional surrogates 3) The theoretical validation of the accuracy achievable by the proposed procedures

This year, we made several contributions to these subtasks. In most of our contributions, we deviated from the standard working hypothesis involving a linear subspace surrogate.

In a first group of publications, we concentrated our attention on the so-called “sparse’’ low-dimensional model. In this context, we have proposed several new algorithmic solutions to decrease the computational complexity associated to projection onto this low-dimensional model. These methodologies take place in the context of “screening’’ procedures for LASSO. We first introduced a new screening strategy, dubbed "joint screening test", which allows the rejection of a set of atoms by performing one single test, see [4]. Our approach enables to find good compromises between complexity of implementation and effectiveness of screening. Second, we proposed two new methods to decrease the computational cost inherent to the construction of the (so-called) "safe region". Our numerical experiments show that the proposed procedures lead to significant computational gains as compared to standard methodologies, see [11]. We finally showed in another work that the main concepts underlying screening procedures can be extended to different families of convex optimization problems, see [22].

Another avenue of research has been the study of the sparse surrogate in the context of “continuous’’ dictionaries, where the elementary signals forming the decomposition catalog are functions of some parameters taking its values in some continuously-valued domain. In this context, we contributed to the theoretical characterization of the performance of some well-known algorithmic procedure, namely “orthogonal matching pursuit’’ (OMP). More specifically, we proposed the first theoretical analysis of the behavior of OMP in the continuous setup, see [12], [17], [21]. We also provided a new connection between two popular low-rank approximations of continuous dictionaries, namely the “polar’’ and “SVD’’ approximations, see [9].

The tools exploited in the field of model-order reduction and sparsity have found some particular applicative field in geophysics and fluid mechanics. In [8], [7], we derived procedures based on sparse representations to localize the positions of particles in a moving fluid. In [16], [15], [14], [26], [27], we designed learning methodologies to learn the dynamical model underlying a set of observed data.