# Randomized Discrete Empirical Interpolation Method for Nonlinear Model Reduction

## TL;DR

This paper introduces randomized algorithms for the discrete empirical interpolation method (DEIM) to improve efficiency in nonlinear model reduction, providing theoretical accuracy guarantees and demonstrating practical benefits through numerical experiments.

## Contribution

It develops randomized techniques for DEIM basis computation and component selection, enhancing efficiency and accuracy in nonlinear model reduction.

## Key findings

- Randomized algorithms significantly reduce computational cost.
- Theoretical bounds confirm the accuracy of randomized DEIM.
- Numerical experiments demonstrate improved efficiency and comparable accuracy.

## Abstract

Discrete empirical interpolation method (DEIM) is a popular technique for nonlinear model reduction and it has two main ingredients: an interpolating basis that is computed from a collection of snapshots of the solution and a set of indices which determine the nonlinear components to be simulated. The computation of these two ingredients dominates the overall cost of the DEIM algorithm. To specifically address these two issues, we present randomized versions of the DEIM algorithm. There are three main contributions of this paper. First, we use randomized range finding algorithms to efficiently find an approximate DEIM basis. Second, we develop randomized subset selection tools, based on leverage scores, to efficiently select the nonlinear components. Third, we develop several theoretical results that quantify the accuracy of the randomization on the DEIM approximation. We also present numerical experiments that demonstrate the benefits of the proposed algorithms.