# Applied Math Seminar (AMS)

#### 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.