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.

翻译：根据所关注的频率范围，基于有限元的声学问题建模会生成具有极高维状态空间的动态系统。由于这些模型大多可用具有稀疏矩阵的二阶线性动态系统描述，数学模型降阶技术为加速仿真过程提供了一种有效途径。本文致力于解决在给定期望精度条件下，如何确定降阶系统最优阶数的问题。为此，我们重新审视了一种基于相邻两次Krylov迭代所得降阶模型差异的启发式误差估计器。通过对该估计器进行数学分析，我们证明相邻两次降阶模型的差值确实能为真实模型降阶误差提供足够精确的估计。这一结论通过两个声学模型的数值实验得到验证。我们进一步探讨了将其作为Krylov基模型降阶停止准则的可行性。