# Harbir Antil - Research

## Reduced Order Modeling of Nonlinear and Parametric Systems

## Goal

Projection based methods lead to reduced order models (ROMs) with dramatically reduced
numbers of equations and unknowns.
However, for nonlinear or parametrically varying problems the cost of evaluating
these ROMs still depends on the size of the full order model
and therefore is still expensive. The Discrete Empirical Interpolation Method (DEIM)
further approximates the nonlinearity in the projection based ROM.
The resulting DEIM ROM nonlinearity depends only on a few components of
the original nonlinearity. If each component of the original nonlinearity depends only
on a few components of the argument, the resulting DEIM ROM can be
evaluated efficiently at a cost that is independent of the size of the original problem.
For systems obtained from finite difference approximations, the $i$th component of the
original nonlinearity often depends only on the $i$th component of the argument. This is different
for systems obtained using finite element methods, where the dependence is determined by
the mesh and by the polynomial degree of the finite element subspaces. This paper describes two
approaches of applying DEIM in the finite element context, one applied to the assembled and
the other to the unassembled form of the nonlinearity. We carefully examine how the DEIM
is applied in each case, and the substantial efficiency gains obtained by the DEIM.
In addition, we demonstrate how to apply DEIM to obtain ROMs for a class
of parameterized system that arises, e.g., in shape optimization. The evaluations of the
DEIM ROMs are substantially faster than those of the standard projection based ROMs.
Additional gains are obtained with the DEIM ROMs when one has
to compute derivatives of the model with respect to the parameter.
## Collaborators

## Achievements

- We provide two novel approaches of applying DEIM in the finite element context.
- We demonstrate how to apply DEIM to obtain ROMs for a class of parameterized systems that arises, e.g., in shape optimization.

## Related Publications

**H. Antil, M. Heinkenschloss, and D. C. Sorensen.**

* Application of Discrete Empirical Interpolation Method to Reduced Order
Modeling of Nonlinear and Parametric Systems. *

A. Quarteroni and G. Rozzas (eds.), Reduced Order Methods for Modeling and Computational Reduction, Model. Simul.& Appl. Vol. 9, 2014, pp. 101-136, Springer Italia, Milan.