  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.
