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Error Estimation and Stopping Criteria for Krylov-Based Model Order Reduction in Acoustics

Siyang Hu, Nick Wulbusch, Alexey Chernov, Tamara Bechtold

TL;DR

The paper tackles selecting the optimal reduced order for Krylov-based MOR in acoustics by mathematically validating a heuristic error estimator derived from the difference between two consecutive Krylov ROMs. It shows that this estimator approximates the true relative error near the expansion point, with the relation $E_r(s)=\hat{E}_r(s)+O((s-s_0)^{r+1})$, and demonstrates this on 2D and 3D Helmholtz problems. Numerically, the estimator tracks true error well in a simple 2D setting but underestimates the error in a more challenging 3D scenario, highlighting conditioning effects and the influence of expansion-point placement. The work supports using the consecutive-ROM difference as a stopping criterion, while recommending smoothing and optimized expansion-point distribution for robust performance in practice.

Abstract

Depending on the frequency range of interest, finite element-based modeling of acoustic problems leads to dynamical systems with very high dimensional state spaces. As these models can mostly be described with second order linear dynamical system with sparse matrices, mathematical model order reduction provides an interesting possibility to speed up the simulation process. In this work, we tackle the question of finding an optimal order for the reduced system, given a desired accuracy. To do so, we revisit a heuristic error estimator based on the difference of two reduced models from two consecutive Krylov iterations. We perform a mathematical analysis of the estimator and show that the difference of two consecutive reduced models does provide a sufficiently accurate estimation for the true model reduction error. This claim is supported by numerical experiments on two acoustic models. We briefly discuss its feasibility as a stopping criterion for Krylov-based model order reduction.

Error Estimation and Stopping Criteria for Krylov-Based Model Order Reduction in Acoustics

TL;DR

The paper tackles selecting the optimal reduced order for Krylov-based MOR in acoustics by mathematically validating a heuristic error estimator derived from the difference between two consecutive Krylov ROMs. It shows that this estimator approximates the true relative error near the expansion point, with the relation , and demonstrates this on 2D and 3D Helmholtz problems. Numerically, the estimator tracks true error well in a simple 2D setting but underestimates the error in a more challenging 3D scenario, highlighting conditioning effects and the influence of expansion-point placement. The work supports using the consecutive-ROM difference as a stopping criterion, while recommending smoothing and optimized expansion-point distribution for robust performance in practice.

Abstract

Depending on the frequency range of interest, finite element-based modeling of acoustic problems leads to dynamical systems with very high dimensional state spaces. As these models can mostly be described with second order linear dynamical system with sparse matrices, mathematical model order reduction provides an interesting possibility to speed up the simulation process. In this work, we tackle the question of finding an optimal order for the reduced system, given a desired accuracy. To do so, we revisit a heuristic error estimator based on the difference of two reduced models from two consecutive Krylov iterations. We perform a mathematical analysis of the estimator and show that the difference of two consecutive reduced models does provide a sufficiently accurate estimation for the true model reduction error. This claim is supported by numerical experiments on two acoustic models. We briefly discuss its feasibility as a stopping criterion for Krylov-based model order reduction.

Paper Structure

This paper contains 8 sections, 2 theorems, 24 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical Modeling of Acoustic Systems
  3. Krylov-based Model Order Reduction
  4. Estimation of Approximation Error
  5. Numerical Experiments
  6. 2D Helmholtz problem
  7. 3D Helmholtz Problem: Speaker Plate
  8. Conclusion & Outlook

Key Result

Theorem 4.1

Let $s_0=ik_0$ and $s$ be sufficiently close to $s_0$ so that $\bm{G}(s)$ is well-defined according to eq:g(s). Let $\bm{G}_r$ be the transfer function of the reduced order model of order $r$ according to eq:gr(s) obtained after $r$ iterations of a Krylov subspace-based MOR method and suppose that $

Figures (5)

  • Figure 1: Domain for model problem. Points illustrate measurements points at which the error is computed in the numerical experiments.
  • Figure 2: True error vs error estimator (sup-norm) for the 2D Helmholtz problem and ROMs of dimensions every 100 iterations up to 800.
  • Figure 3: Domain for the 3D Helmholtz problem involving a speaker and a plate. The pointed line illustrates measurement points (all nodes on the line) at which the error is computed in the numerical experiments.
  • Figure 4: Comparison of FOM versus ROM plotted over different wave numbers $k$ and their relative difference.
  • Figure 5: True error vs error estimator (2-norm) for the 3D Helmholtz problem and ROMs of dimensions up to 800.

Theorems & Definitions (4)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof