Emmanuel Trélat

Thierry Paul, Emmanuel Trélat

We develop a general operator-theoretic route that turns Kato-type quasilinear evolution systems on a Banach scale $(Z,X)$ into finite-dimensional interacting approximations. The construction proceeds in two steps. First, one introduces a regularized family $(A_\varepsilon,f_\varepsilon)$ indexed by a scale parameter $\varepsilon>0$, for which the drift $A_\varepsilon[t,z]z+f_\varepsilon[t,z]$ takes values in an output space $Y$ suitable for discretization. Second, one discretizes this regularized dynamics by a sampling-reconstruction pair $(P_N,R_N)$ and obtains an interacting ODE on a finite-dimensional state space $V_N\simeq\R^{dN}$. Our main abstract theorem provides a quantitative estimate of the discrepancy $y_\varepsilon^N-y$ between the lifted discrete solution and the exact one, separating the regularization error $χ(\varepsilon)$ from the discretization error $(1+L_\varepsilon)N^{-γ}$, where $L_\varepsilon$ measures the size of the regularized drift in the output norm. This makes explicit the trade-off between the regularization scale $\varepsilon$, the discretization scale $N$, and the possible deterioration of $L_\varepsilon$ as $\varepsilon\to 0$. As a running example, we focus on quasilinear PDEs on bounded Lipschitz domains with boundary conditions. We show that Burenkov's variable-step mollifiers provide a boundary-compatible kernelization: they regularize differential operators into explicit integral-interaction operators supported inside the domain and preserve boundary traces of sufficiently regular fields. In this setting one can choose an output space $Y$ for which $L_\varepsilon$ remains uniformly bounded, leading to algebraic convergence rates in $N$ for quasi-uniform discretizations.

Fatiha Alabau-Boussouira, Yannick Privat, Emmanuel Trélat

We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time discretization parameters, by adding appropriate numerical viscosity terms. Our main arguments use the optimal-weight convexity method and uniform observability inequalities with respect to the discretization parameters. We establish our results, first in the continuous setting, then for space semi-discrete models, and then for time semi-discrete models. The full discretization is inferred from the previous results. Our results cover, for instance, the Schrödinger equation with nonlinear damping, the nonlinear wave equation, the nonlinear plate equation, as well as certain classes of equations with nonlocal terms.

Cédric M. Campos, Sina Ober-Blöbaum, Emmanuel Trélat

In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute.