Statistical reduced order modelling for the parametric Helmholtz equation

TL;DR

The paper addresses uncertainty-quantified, frequency-domain PDE simulations by extending statFEM with a reduced-order modeling (ROM) framework. It introduces statROM, which uses Krylov moment matching to derive fast ROMs and an adjoint-based ROM error estimator to explicitly include ROM biases in the statistical data model, enabling accurate Bayesian conditioning on sensor data. Key contributions include the derivation of ROM-aware posterior updates via Sherman–Morrison–Woodbury, quasi-Monte Carlo-based ROM priors for high-dimensional inputs, and two numerical experiments (1D and 2D Helmholtz problems) showing improved posterior accuracy and faster convergence compared to standard statFEM on ROM priors. The approach enables efficient, uncertainty-aware inference for frequency-dependent wave problems using ROMs, with practical gains in speed and accuracy for multi-frequency sweeps.

Abstract

Predictive modeling involving simulation and sensor data at the same time, is a growing challenge in computational science. Even with large-scale finite element models, a mismatch to the sensor data often remains, which can be attributed to different sources of uncertainty. For such a scenario, the statistical finite element method (statFEM) can be used to condition a simulated field on given sensor data. This yields a posterior solution which resembles the data much better and additionally provides consistent estimates of uncertainty, including model misspecification. For frequency or parameter dependent problems, occurring, e.g. in acoustics or electromagnetism, solving the full order model at the frequency grid and conditioning it on data quickly results in a prohibitive computational cost. In this case, the introduction of a surrogate in form of a reduced order model yields much smaller systems of equations. In this paper, we propose a reduced order statFEM framework relying on Krylov-based moment matching. We introduce a data model which explicitly includes the bias induced by the reduced approximation, which is estimated by an inexpensive error indicator. The results of the new statistical reduced order method are compared to the standard statFEM procedure applied to a ROM prior, i.e. without explicitly accounting for the reduced order bias. The proposed method yields better accuracy and faster convergence throughout a given frequency range for different numerical examples.

Figures (13)

• Figure 1: Workflow diagram of the proposed statROM method. In an offline phase, the reduced order projection basis is constructed for the primal and adjoint problem and each sample of the input parameters. In an online phase the FE matrices are reduced and the system is solved in the reduced space at reduced cost compared to the full order model. The adjoint error indicator contributes to the quality of the subsequent data assimilation. Fast parameter sweeps over the frequency $\omega$ can be carried out in the online phase based on the sampled ROMs.
• Figure 2: Full order and ROM priors for the 1D problem. As the ROM is constructed with only $m=5$ moments matched around the expansion frequency, the approximation error is large. Observation data are collected on a finely discretized reference FE solution on $11$ equally spaced sensor locations. At each sensor, multiple noisy measurements (red dots) are recorded.
• Figure 3: Comparison of the quality of different ROM priors for the 1D problem. With $m=15$ moments matched, the approximation error is negligible compared to $m=5$ moments. The difference between the full order reference mean and the respective ROM prior mean is plotted in gray lines. Notice that for $m=15$ there is no visible deviation from the full order mean, even though a far smaller system of equations was solved. In the $2\sigma$ bands, for $m=15$ only a small error is visible compared to the $m=5$ case.
• Figure 4: Left: The approximation error of the ROM mean quickly converges to a constant quantity at $m=15$ matched moments. Therefore, a ROM with $m\geq16$ can be considered to provide an exact representation of the full order model. For the computation of the error norm, the ROM solution $\tilde{\bm{u}}(\omega)$ and the full order solution $\bm{u}(\omega)$ are evaluated at $\omega/2\pi = 460\mathrm{Hz}$ while the expansion frequency is $\omega/2\pi = 100\mathrm{Hz}$. Right: For higher numbers of matched moments $m$ around $\bar{\omega}/2\pi=100$Hz, the ROM yields smaller errors throughout the frequency range. With $m=15$, the error is similarly low throughout the frequency range.
• Figure 5: Left: Convergence of the ROM error estimator at the training points with increasing number of matched moments. A relative error of $0.1\%$ is reached at about $15$ matched moments. Right: For the example from before with $\bar{\omega}/2\pi=100$Hz, $m=5$, the ROM error is approximated with an adjoint error indicator which is evaluated at $14$ training points. A GP is trained to give an approximation of the ROM error in the whole field. Notice that while the rough shape of the error is well approximated, there is still a significant difference to the exact ROM error. Increasing the number of test points to $22$ and choosing $m=6$ instead of $m=5$ matched moments leads to a better approximation of the ROM error. The amplitude of the error naturally decreases because of the larger ROM basis.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3