École normale supérieure de Cachan

About: École normale supérieure de Cachan is a education organization based out in Cachan, Île-de-France, France. It is known for research contribution in the topics: Decidability & Finite element method. The organization has 2717 authors who have published 5585 publications receiving 175925 citations.

Optimized high-order splitting methods for some classes of parabolic equations

Abstract: We are concerned with the numerical solution obtained by splitting methodsof certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients. It has been demonstrated that this se cond-order barrier can be overcome by using splitting methods with complex-valued coefficients (with positive real parts). In this way, method s of orders 3 to 14 by using the Suzuki‐Yoshida triple (and quadruple) jump composition procedure have been explicitly built. Here we reconsider this technique an d show that it is inherently bounded to order 14 and clearly sub-optimal with respect to error constants. As an alternative, we solve directly the algebraic equations arising from the order conditions and construct methods of orders 6 and 8 that are the most accurate ones available at present time, even when low accuracies are desired. We also show that, in the general case, 14 is not an order barrier for splitting method s with complex coefficients with positive real part by building explicitly a method of order 16 as a composition of methods of order 8.

Smoothing Effects for Classical Solutions of the Full Landau Equation

Abstract: In this work, we consider the smoothness of the solutions to the full Landau equation. In particular, we prove that any classical solutions (such as the ones obtained by Guo in a “close to equilibrium” setting) become immediately smooth with respect to all variables. This shows that the Landau equation is a nonlinear and nonlocal analog of an hypoelliptic equation.

ATL with Strategy Contexts: Expressiveness and Model Checking

Abstract: We study the alternating-time temporal logics ATL and ATL* extended with strategy contexts: these make agents commit to their strategies during the evaluation of formulas, contrary to plain ATL and ATL* where strategy quantifiers reset previously selected strategies. We illustrate the important expressive power of strategy contexts by proving that they make the extended logics, namely ATLsc and ATLsc*, equally expressive: any formula in ATLsc* can be translated into an equivalent, linear-size ATLsc formula. Despite the high expressiveness of these logics, we~prove that their model-checking problems remain decidable by~designing a tree-automata-based algorithm for model-checking ATLsc* on the full class of $n$-player concurrent game structures.