Bienvenue - Laboratoire Jacques-Louis Lions

22 janvier 2021 — 14h00
Exposé à distance retransmis par Zoom
Kathrin Smetana (Université de Twente)
(Randomized) multiscale methods for parabolic problems
Résumé
In this talk we discuss rigorously analyzed, localizable, and computationally efficient multiscale methods that yield reliable results without structural assumptions like scale separation for parabolic problems with coefficients that are arbitrarily rough in space and time.
To construct the local approximation spaces, we consider a compact transfer operator that maps to the space of all local solutions of the PDE and construct an approximation of the range of the operator using the left singular vectors of the operator. This yields approximation spaces that are optimal in the sense of Kolmogorov. To prove compactness of the transfer operator we combine a suitable parabolic Caccioppoli inequality with the compactness theorem of Aubin-Lions.
In this way, we will first construct local space-time approximation spaces, which are localized in space and cover the whole time interval, which we couple via a partition of unity. We derive rigorous local and global a priori error bounds. In detail, we bound the global approximation error in a graph norm by the local errors in the L^2(H^1)-norm, noting that the space the transfer operator maps to is equipped with this norm. Secondly, we will consider local approximation spaces that depend only on the spatial variable. Transferring methods from randomized numerical linear algebra to this setting allows us to construct these spaces very efficiently in parallel in time.
This is joint work with J. Schleuss (University of Münster) and L. ter Maat (University of Twente).