A Posteriori Error Estimation for the p-curl Problem

We derive a posteriori error estimates for a semi-discrete finite element approximation of a nonlinear eddy current problem arising from applied superconductivity, known as the $p$-curl problem. In particular, we show the reliability for non-conforming N\'{e}d\'{e}lec elements based on a residual type argument and a Helmholtz-Weyl decomposition of $W^p_0(\text{curl};\Omega)$. As a consequence, we are also able to derive an a posteriori error estimate for a quantity of interest called the AC loss. The nonlinearity for this form of Maxwell's equation is an analogue of the one found in the $p$-Laplacian. It is handled without linearizing around the approximate solution. The non-conformity is dealt by adapting error decomposition techniques of Carstensen, Hu and Orlando. Geometric non-conformities also appear because the continuous problem is defined over a bounded $C^{1,1}$ domain while the discrete problem is formulated over a weaker polyhedral domain. The semi-discrete formulation studied in this paper is often encountered in commercial codes and is shown to be well-posed. The paper concludes with numerical results confirming the reliability of the a posteriori error estimate.Comment: 32 page

Optimal Design of Validation Experiments for the Prediction of Quantities of Interest

Numerical predictions of quantities of interest measured within physical systems rely on the use of mathematical models that should be validated, or at best, not invalidated. Model validation usually involves the comparison of experimental data (outputs from the system of interest) and model predictions, both obtained at a specific validation scenario. The design of this validation experiment should be directly relevant to the objective of the model, that of predicting a quantity of interest at a prediction scenario. In this paper, we address two specific issues arising when designing validation experiments. The first issue consists in determining an appropriate validation scenario in cases where the prediction scenario cannot be carried out in a controlled environment. The second issue concerns the selection of observations when the quantity of interest cannot be readily observed. The proposed methodology involves the computation of influence matrices that characterize the response surface of given model functionals. Minimization of the distance between influence matrices allow one for selecting a validation experiment most representative of the prediction scenario. We illustrate our approach on two numerical examples. The first example considers the validation of a simple model based on an ordinary differential equation governing an object in free fall to put in evidence the importance of the choice of the validation experiment. The second numerical experiment focuses on the transport of a pollutant and demonstrates the impact that the choice of the quantity of interest has on the validation experiment to be performed.Comment: 31 pages, 10 figure

Multi-level Neural Networks for Accurate Solutions of Boundary-Value Problems

The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy, classical discretization methods such as the finite element methods, remains a significant challenge. Deep learning methods usually struggle to reliably decrease the error in their approximate solution. A new methodology to better control the error for deep learning methods is presented here. The main idea consists in computing an initial approximation to the problem using a simple neural network and in estimating, in an iterative manner, a correction by solving the problem for the residual error with a new network of increasing complexity. This sequential reduction of the residual of the partial differential equation allows one to decrease the solution error, which, in some cases, can be reduced to machine precision. The underlying explanation is that the method is able to capture at each level smaller scales of the solution using a new network. Numerical examples in 1D and 2D are presented to demonstrate the effectiveness of the proposed approach. This approach applies not only to physics informed neural networks but to other neural network solvers based on weak or strong formulations of the residual.Comment: 34 pages, 20 figure

Prospection aérienne dans les Yvelines

Concernant la Préhistoire, de nouvelles minières de silex ont été photographiées à la limite des communes de Flins-sur-Seine et d’Aubergenville. La prospection a permis d’observer les puits de mines où l’activité d’extraction à cet endroit est connue depuis longtemps au vu des amas de déchets de taille qui jonchent le sol. Ce silex de Flins est reconnu comme étant de bonne qualité et il fut abondamment utilisé ; on le retrouve, en prospection, dans toute la région. Parmi les sites fossoyés, l..

Croisy-sur-Seine – Prieuré Saint-Léonard Saint-Martin

Des travaux de restauration menés en 1998, à l’intérieur et à l’extérieur de cet ancien prieuré donné, en 1121, par Pierre de Nemours, évêque de Paris, au prieuré cistercien de Saint-Léonard de Noblat en Limousin, ont permis d’observer dans des tranchées des restes osseux probablement issus de sépultures déjà déplacées. À l’intérieur de l’édifice, des creusements, déjà effectués lors de notre passage, ont livré des ossements fortement perturbés, probablement lorsque l’église fut fermée au cul..

Les Alluets-le-Roi – Église Saint-Nicolas

L’église paroissiale Saint-Nicolas est attestée vers 1250 (Pouillé de Chartres). L’édifice est du début du xiiie s., avec des remaniements vers les xve-xvie s. Les sculptures de la nef (chapiteaux du premier art gothique) sont particulièrement remarquables. La municipalité, soucieuse de faire des économies, avait fait vider le transept durant les week-ends par des habitants motivés. Ainsi, une couche d’environ 40 cm avait déjà été évacuée lors de notre première visite. En accord avec le Servi..

Guyancourt – Église Saint-Victor

À la suite d’un dépôt de dossier de restauration et de pose d’un chauffage dans le sol, en 1991, une demande de surveillance des travaux avait été formulée par notre Service et approuvée par le ministère de la Culture. En 1998, le Service archéologique départemental des Yvelines a pu effectuer une surveillance intermittente des travaux durant les mois de janvier et février. Si la famille de Guyancourt apparaît en 1262, l’édifice ne figure pas dans le Pouillé du xiiie s. Il est bâti en pierre ..