Applied Math Seminar

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Nonlinear model reduction for transport-dominated problems

Speaker: Benjamin Peherstorfer, Courant

Location: TBA

Date: Friday, February 26, 2021, 2:30 p.m.

Synopsis:

Solution manifolds induced by transport-dominated problems
such as hyperbolic conservation laws typically exhibit nonlinear
structures. This means that traditional model reduction methods based
on linear approximations in subspaces are inefficient when applied to
these problems. This presentation discusses model reduction methods
for constructing nonlinear reduced models that seek approximations on
manifolds, rather than in subspaces, and so lead to efficient
dimensionality reduction even for transport-dominated problems. First,
we will discuss an online adaptive approach that exploits locality in
space and time to efficiently adapt piecewise linear approximations of
the solution manifolds. Second, we present an approach that derives
reduced approximations that are nonlinear by explicitly composing
global transport dynamics with locally linear approximations of the
solution manifolds. The compositions can be interpreted as
one-hidden-layer neural networks. Numerical results demonstrate that
the proposed approaches achieve speedups even for problems where
traditional, linear reduced models are more expensive to solve than
the high-dimensional, full model.