Ludovic Chamoin

Adrien Le Coënt, Laurent Fribourg, Nicolas Markey et al.

We present a correct-by-design method of state-dependent control synthesis for linear discrete-time switching systems. Given an objective region R of the state space, the method builds a capture set S and a control which steers any element of S into R. The method works by iterated backward reachability from R. More precisely, S is given as a parametric extension of R, and the maximum value of the parameter is solved by linear programming. The method can also be used to synthesize a stability control which maintains indefinitely within R all the states starting at R. We explain how the synthesis method can be performed in a distributed manner. The method has been implemented and successfully applied to the synthesis of a distributed control of a concrete floor heating system with 11 rooms and 2^11 = 2048 switching modes.

Jean Ruel, Frédéric Legoll, Arthur Lebée et al.

Effective models for slender structures derived from well-known plate (or shell) theories are justified within the limit of a small thickness, and may therefore prove limited for intermediate slenderness. On the other hand, direct 3D simulation of such structures is sub-optimal because it does not take advantage of the presence of small dimensions in some directions and is sometimes too costly and ill-conditioned. In this context, the Proper Generalized Decomposition (PGD) method, a model order reduction method based on a modal representation of the solution with separation of variables, makes it possible to obtain a 3D solution with 2D resolution complexity. In this work, an analysis of the links between the PGD reduced order model and the solution provided by plate theory is carried out using asymptotic expansion. It is shown that, in the limit of large slenderness, the first mode of the PGD exhibits Kirchhoff-Love type kinematics, but only corresponds to the asymptotic solution in very special cases of loading and boundary conditions. To capture the asymptotic solution, a new PGD strategy is introduced consisting of computing the first two modes simultaneously. We also demonstrate that the PGD is subject to shear locking, and we show how to deal with it. Numerical experiments are provided, demonstrating the interest of this approach and confirming the theoretical analysis.

Jean Ruel, Ludovic Chamoin, Frédéric Legoll et al.

Model order reduction techniques have become an attractive approach for obtaining fast approximations of multidimensional problems. Besides computational efficiency, ensuring the reliability of the resulting approximations is of primary importance. This work focuses on the certification of PGD-based reduced-order models based on the separation of spatial variables, which are particularly well suited to plate and shell geometries. Considering diffusion problems defined in plate-like domains, we introduce a guaranteed global error estimate associated with the PGD approximation. To this end, the error bounds are derived from the Constitutive Relation Error (CRE) method. The main difficulty of this approach lies in the construction of equilibrated fluxes, for which a dedicated procedure is proposed. Based on the resulting estimator, an adaptive strategy is developed to control both the discretization error and the number of PGD modes. This certification procedure is further extended to the error control in quantities of interest. We provide several numerical examples illustrating the reliability and efficiency of our procedure.

Daniel Martin Xavier, Ludovic Chamoin, Jawher Jerray et al.

The early stopping strategy consists in stopping the training process of a neural network (NN) on a set $S$ of input data before training error is minimal. The advantage is that the NN then retains good generalization properties, i.e. it gives good predictions on data outside $S$, and a good estimate of the statistical error (``population loss'') is obtained. We give here an analytical estimation of the optimal stopping time involving basically the initial training error vector and the eigenvalues of the ``neural tangent kernel''. This yields an upper bound on the population loss which is well-suited to the underparameterized context (where the number of parameters is moderate compared with the number of data). Our method is illustrated on the example of an NN simulating the MPC control of a Van der Pol oscillator.

Ludovic Chamoin, Frederic Legoll

We introduce quantitative and robust tools to control the numerical accuracy in simulations performed using the Multiscale Finite Element Method (MsFEM). First, we propose a guaranteed and fully computable a posteriori error estimate for the global error measured in the energy norm. It is based on dual analysis and the Constitutive Relation Error (CRE) concept, with recovery of equilibrated fluxes from the approximate MsFEM solution. Second, the estimate is split into several indicators, associated to the various MsFEM error sources, in order to drive an adaptive procedure. The overall strategy thus enables to automatically identify an appropriate trade-off between accuracy and computational cost in the MsFEM numerical simulations. Furthermore, the strategy is compatible with the offline/online paradigm of MsFEM. The performances of our approach are demonstrated on several numerical experiments.