Error bounds and analysis of proper orthogonal decomposition model reduction methods using snapshots from the solution and the time derivatives

TL;DR

This paper compares two proper orthogonal decomposition methods for reducing large ODE systems, deriving new error bounds and analyzing the benefits of including time derivative snapshots.

Contribution

It introduces new bounds for POD error analysis and demonstrates the advantages of using solution and derivative snapshots together.

Findings

Error bounds depend on singular values and snapshot spacings.

Including time derivative snapshots can improve reduction accuracy.

Numerical experiments validate the theoretical bounds and insights.

Abstract

Proper orthogonal decomposition methods for model reduction utilize information about the solution at certain time and parameter points to generate a reduced space basis. In this paper, we compare two proper orthogonal decomposition methods for reducing large systems of ODEs. The first method is based on collecting snapshots from the solutions only; the second method uses snapshots from both the solutions and their time derivatives. To compare the methods, we derive new bounds for the 2-norm of the approximation error induced by the each of the methods. The bounds are represented as a sum of two terms: the first depends on the size of the first neglected singular value while the second depends only on the spacings between the snapshots. We performed numerical experiments to compare the errors from the two model reduction methods applied to the semidiscretized FitzHugh-Nagumo system and investigated the relation between the behavior of the numerically observed error and the error bounds. We find that the error bounds, though not tight, provide insights and justification for using time derivative snapshots in POD model reduction for dynamical systems.