Y. Maday

Y. Maday, O. Mula

This paper introduces a generalization of the empirical interpolation method (EIM) and the reduced basis method (RBM) in order to allow their combination with data mining and data assimilation. The purpose is to be able to derive sound information from data and reconstruct information, possibly taking into account noise in the acquisition, that can serve as an input to models expressed by partial differential equations. The approach combines data acquisition (with noise) with domain decomposition techniques and reduced basis approximations.

Y. Maday, O. Mula, G. Turinici

Let $F$ be a compact set of a Banach space $\mathcal{X}$. This paper analyses the "Generalized Empirical Interpolation Method" (GEIM) which, given a function $f\in F$, builds an interpolant $\mathcal{J}_n[f]$ in an $n$-dimensional subspace $X_n \subset \mathcal{X}$ with the knowledge of $n$ outputs $(σ_i(f))_{i=1}^n$, where $σ_i\in \mathcal{X}'$ and $\mathcal{X}'$ is the dual space of $\mathcal{X}$. The space $X_n$ is built with a greedy algorithm that is adapted to $F$ in the sense that it is generated by elements of $F$ itself. The algorithm also selects the linear functionals $(σ_i)_{i=1}^n$ from a dictionary $Σ\subset \mathcal{X}'$. In this paper, we study the interpolation error $\max_{f\in F} \Vert f-\mathcal{J}_n[f]\Vert_{\mathcal{X}}$ by comparing it with the best possible performance on an $n$-dimensional space, i.e., the Kolmogorov $n$-width of $F$ in $\mathcal{X}$, $d_n(F,\mathcal{X})$. For polynomial or exponential decay rates of $d_n(F,\mathcal{X})$, we prove that the interpolation error has the same behavior modulo the norm of the interpolation operator. Sharper results are obtained in the case where $\mathcal X$ is a Hilbert space.

X. Dai, C. Le Bris, F. Legoll et al.

The parareal in time algorithm allows to efficiently use parallel computing for the simulation of time-dependent problems. It is based on a decomposition of the time interval into subintervals, and on a predictor-corrector strategy, where the propagations over each subinterval for the corrector stage are concurrently performed on the processors. In this article, we are concerned with the long time integration of Hamiltonian systems. Geometric, structure-preserving integrators are preferably employed for such systems because they show interesting numerical properties, in particular excellent preservation of the total energy of the system. Using a symmetrization procedure and/or a (possibly also symmetric) projection step, we introduce here several variants of the original plain parareal in time algorithm [Lions, Maday and Turinici 2001, Baffico, Bernard, Maday, Turinici and Zerah 2002, Bal and Maday 2002] that are better adapted to the Hamiltonian context. These variants are compatible with the geometric structure of the exact dynamics, and are easy to implement. Numerical tests on several model systems illustrate the remarkable properties of the proposed parareal integrators over long integration times. Some formal elements of understanding are also provided.

C. Le Bris, T. Lelievre, Y. Maday

We investigate mathematically a nonlinear approximation type approach recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to solve high dimensional partial differential equations. We show the link between the approach and the greedy algorithms of approximation theory studied e.g. in [R.A. DeVore and V.N. Temlyakov, Adv. Comput. Math., 1996]. On the prototypical case of the Poisson equation, we show that a variational version of the approach, based on minimization of energies, converges. On the other hand, we show various theoretical and numerical difficulties arising with the non variational version of the approach, consisting of simply solving the first order optimality equations of the problem. Several unsolved issues are indicated in order to motivate further research.

J. P. Argaud, B. Bouriquet, H. Gong et al.

The Empirical Interpolation Method (EIM) and its generalized version (GEIM) can be used to approximate a physical system by combining data measured from the system itself and a reduced model representing the underlying physics. In presence of noise, the good properties of the approach are blurred in the sense that the approximation error no longer converges but even diverges. We propose to address this issue by a least-squares projection with constrains involving a some a priori knowledge of the geometry of the manifold formed by all the possible physical states of the system. The efficiency of the approach, which we will call Constrained Stabilized GEIM (CS-GEIM), is illustrated by numerical experiments dealing with the reconstruction of the neutron flux in nuclear reactors. A theoretical justification of the procedure will be presented in future works.