Séminaire POEMS sur les méthodes stochastiques et la quantification des incertitudes
14h: Anthony Nouy "Low-rank methods for high-dimensional approximation"
Tensor methods are among the most prominent tools for the numerical solution of high-dimensional problems where functions of multiple variables have to be approximated. Such high-dimensional approximation problems naturally arise in stochastic analysis and uncertainty quantification. In many practical situations, the approximation of high-dimensional functions is made computationally tractable by using rank-structured approximations. In this talk, we first recall basic concepts on the approximation in low-rank tensor formats. Then, we present constructive algorithms for the approximation in subspace-based tensor formats which can be interpreted as multidimensional extensions of reduced basis methods. These algorithms are illustrated on variational problems. Finally, we present algorithms using statistical learning methods for the approximation of a function in low-rank formats from simple evaluations of the function (black-box approach).
15h30: Ludovic Chamoin (with B. Marchand, C. Rey) "Data assimilation on nonlinear dynamical systems using Kalman filtering, model reduction, and the concept of modified Constitutive Relation Error"
The work deals with a model updating strategy based on the coupling between a probabilistic data assimilation method (unscented Kalman filtering) and a deterministic inverse problem approach (modified CRE). It constitutes an efficient tool when addressing observation, updating, and control on nonlinear dynamical systems with a limited number of (potentially corrupted) experimental data. In order to perform data assimilation in real-time, a model reduction method based on the Proper Generalized Decomposition (PGD) is also used. The strategy is applied to several mechanical problems, such as the control of damage evolution, and very recent developments (in particular adaptive techniques) are presented.