10–13 Jun 2024
Inria Center at the University of Lille
Europe/Paris timezone

Learning based reduction methods in the context of PDE constrained optimization

11 Jun 2024, 14:30
45m
Amphitheater, Building B (Inria Center at the University of Lille)

Amphitheater, Building B

Inria Center at the University of Lille

Parc scientifique de la Haute-Borne, 40 avenue Halley, 59650 Villeneuve d'Ascq – France

Speaker

Mario Ohlberger (Universität Münster)

Description

Model order reduction for parameterized partial differential equations is a
very active research area that has seen tremendous development in recent years
from both theoretical and application perspectives. A particular promising ap-
proach is the reduced basis method that relies on the approximation of the solu-
tion manifold of a parameterized system by tailored low dimensional approxima-
tion spaces that are spanned from suitably selected particular solutions, called
snapshots. With speedups that can reach several orders of magnitude, reduced
basis methods enable high fidelity real-time simulations for certain problem
classes and dramatically reduce the computational costs in many-query appli-
cations. While the ”online efficiency” of these model reduction methods is very
convincing for problems with a rapid decay of the Kolmogorov n-width, there
are still major drawbacks and limitations. Most importantly, the construction
of the reduced system in a so called ”offline phase” is extremely CPU-time and
memory consuming for large scale systems. For practical applications, it is thus
necessary to derive model reduction techniques that do not rely on a classical
offline/online splitting but allow for more flexibility in the usage of computa-
tional resources. In this talk we focus on learning based reduction methods in
the context of PDE constrained optimization and inverse problems and evaluate
their overall efficiency. We discuss learning strategies, such as adaptive enrich-
ment as well as a combination of reduced order models with machine learning
approaches in the contest of time dependent problems. Concepts of rigorous cer-
tification and convergence will be presented, as well as numerical experiments
that demonstrate the efficiency of the proposed approaches.

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