Section: New Results

Objective 4 – Model reduction

In [19], [7], we propose new methodologies to decrease the computational cost of safe screening tests for LASSO. We first introduce a new screening strategy, dubbed "joint screening test", which allows the rejection of a set of atoms by performing one single test. Our approach enables to find good compromises between complexity of implementation and effectiveness of screening. Second, we propose 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.

Model-order reduction methods tackle the following general approximation problem: find an "easily-computable" but accurate approximation of some target solution $h$. In order to achieve this goal, standard methodologies combine two main ingredients: i) a set of problem-specific constraints; ii) some "simple" prior model on the set of target solutions. The most common prior model encountered in the literature assume that the target solution $h$ is "close" to some low-dimensional subspace. Recently, several contributions have shown that refined prior models (based on a set of embedded approximation subspaces) may lead to enhanced approximation performance. Unfortunately, to date, no theoretical results have been derived to support the good empirical performance observed in these contributions. The goal of [18] is to fill this gap. More specifically, we provide a mathematical characterization of the approximation performance achievable by some particular "multi-space" decoder and emphasize that, in some specific setups, this "multi-space" decoder has provably better recovery guarantees than its standard counterpart based on a single approximation subspace.

In [20], we deal with the estimation of rare event probabilities using importance sampling (IS), where an optimal proposal distribution is computed with the cross-entropy (CE) method. Although, IS optimized with the CE method leads to an efficient reduction of the estimator variance, this approach remains unaffordable for problems where the repeated evaluation of the score function represents a too intensive computational effort. This is often the case for score functions related to the solution of parametric partial differential equations (PPDE) with random inputs. This work proposes to alleviate computation by adapting a score function approximation along the CE optimization process. The score function approximation is obtained by selecting the surrogate of lowest dimensionality, whose accuracy guarantees to pass the current CE optimization stage. The adaptation of the surrogate relies on certified upper bounds on the error norm. An asymptotic analysis provides some theoretical guarantees on the efficiency and convergence of the proposed algorithm. Numerical results demonstrate the gain brought by the adaptive method in the context of pollution alerts and a system modelled by a PPDE.

In [2], we deal with model order reduction of PPDE. We consider the specific setup where the solutions of the PPDE are only observed through a partial observation operator and address the task of finding a good approximation subspace of the solution manifold. We provide and study several tools to tackle this problem. We first identify the best worst-case performance achievable in this setup and propose simple procedures to approximate this optimal solution. We then provide, in a simplified setup, a theoretical analysis relating the achievable reduction performance to the choice of the observation operator and the prior knowledge available on the solution manifold.

In [3], we deal with model order reduction of parametrical dynamical systems. We consider the specific setup where the distribution of the system's trajectories is unknown but the following two sources of information are available: (i) some "rough" prior knowledge on the system's realisations; (ii) a set of "incomplete" observations of the system's trajectories. We propose a Bayesian methodological framework to build reduced-order models (ROMs) by exploiting these two sources of information. We emphasise that complementing the prior knowledge with the collected data provably enhances the knowledge of the distribution of the system's trajectories. We then propose an implementation of the proposed methodology based on Monte-Carlo methods. In this context, we show that standard ROM learning techniques, such e.g. Proper Orthogonal Decomposition or Dynamic Mode Decomposition, can be revisited and recast within the probabilistic framework considered in this paper. We illustrate the performance of the proposed approach by numerical results obtained for a standard geophysical model.