Bayesian parameter identification in impedance boundary conditions for Helmholtz problems

TL;DR

The paper tackles Bayesian identification of acoustic impedance on boundary segments from noisy interior pressure measurements for the Helmholtz equation, treating the impedance $Z$ as a random variable and computing its moments via ratio estimators and Monte Carlo sampling. The forward problem is solved with a finite element discretization under impedance boundary conditions, and the Bayesian framework is proven well-posed with computable posterior moments that converge under discretization and sampling. Numerical experiments in 2D and 3D validate the theory: when data are generated from the impedance model, the impedance is recovered with high likelihood, whereas data from a nonlocal glass-wall boundary exhibit model-data misfit except near certain frequencies, which the analysis links to eigenmode structure. The work demonstrates a rigorous, probabilistic approach to impedance identification in room acoustics and provides insights into model consistency and the influence of resonances on inference.

Abstract

We consider the problem of identifying the acoustic impedance of a wall surface from noisy pressure measurements in a closed room using a Bayesian approach. The room acoustics is modeled by the interior Helmholtz equation with impedance boundary conditions. The aim is to compute moments of the acoustic impedance to estimate a suitable density function of the impedance coefficient. For the computation of moments we use ratio estimators and Monte-Carlo sampling. We consider two different experimental scenarios. In the first scenario, the noisy measurements correspond to a wall modeled by impedance boundary conditions. In this case, the Bayesian algorithm uses a model that is (up to the noise) consistent with the measurements and our algorithm is able to identify acoustic impedance with high accuracy. In the second scenario, the noisy measurements come from a coupled acoustic-structural problem, modeling a wall made of glass, whereas the Bayesian algorithm still uses a model with impedance boundary conditions. In this case, the parameter identification model is inconsistent with the measurements and therefore is not capable to represent them well. Nonetheless, for particular frequency bands the Bayesian algorithm identifies estimates with high likelihood. Outside these frequency bands the algorithm fails. We discuss the results of both examples and possible reasons for the failure of the latter case for particular frequency values.

Figures (16)

• Figure 1: Domain for model problem in 2D. Black dots indicate the grid of possible microphone positions.
• Figure 2: Discretization error for statistical parameters of $Z^{(1)}$ (left) and $Z^{(2)}$ (right) for $f=50\,$Hz.
• Figure 3: Sampling error for statistical parameters of $Z^{(1)}$ (left) and $Z^{(2)}$ (right) for $f=50\,$Hz.
• Figure 4: Prior density and density given by estimated expected value and standard deviation for $f=50$ Hz for $Z^{(1)}$ (left) and $Z^{(2)}$ (right).
• Figure 5: Prior density and density given by estimated expected value and standard deviation for $f=100$ Hz for $Z^{(1)}$ (left) and $Z^{(2)}$ (right).

Theorems & Definitions (27)

• Definition 2.1: Source and measurement domain, cf. engel2019application
• Proposition 2.2: cf. engel2019application
• proof
• Theorem 3.1: cf. engel2019application
• proof
• Proposition 3.2
• proof
• Remark 4.1
• Definition 4.2
• Proposition 4.3