DEIM vs. leverage scores for time-parallel construction of problem-adapted basis functions

TL;DR

This paper compares DEIM and leverage scores for selecting time points in a parallel algorithm that constructs problem-adapted basis functions for time-dependent problems, aiming to improve efficiency and accuracy.

Contribution

It introduces a deterministic DEIM-based method for time point selection within a parallel basis construction algorithm, replacing the previous probabilistic leverage score approach.

Findings

DEIM-based selection performs comparably or better than leverage scores.

Numerical experiments demonstrate the effectiveness of the DEIM approach.

The method enhances the parallel construction of problem-adapted basis functions.

Abstract

To tackle heterogeneous time-dependent problems, an algorithm that constructs problem-adapted basis functions in an embarrassingly parallel and local manner in time has recently been proposed in [Schleuss, Smetana, ter Maat, SIAM J. Sci. Comput., 2022+]. Several simulations of the problem are performed for only few time steps in parallel by starting at different, randomly drawn start time points. For this purpose, data-dependent probability distributions that are based on the (time-dependent) data functions of the problem, such as leverage scores, are employed. In this paper, we suggest as a key new contribution to perform a deterministic time point selection based on the (discrete) empirical interpolation method (DEIM) within the proposed algorithm. In numerical experiments we investigate the performance of a DEIM based time point selection and compare it to the leverage score sampling approach.