# Sparse Plane Wave Approximation of Acoustic Modes to Address Basis Mismatch
## 3 Proposed Combined Two-Stage Method
### Stage 2: Estimation of PWAs and Directions Using Parametric SBL
#### 3.2.3 Hidden Random Variable Inference
The discretized PW directions were obtained by solving the Thomson problem [33]. We examined two sets of pre-discretized PW directions of \(R=400\) and 1000, and the corresponding \(\text{x}_{\text{lim}}\)s are \(11^{\circ}\) and \(7^{\circ}\), respectively. Figure 3 shows the RMEs corresponding to the first 100 modes for different methods, with \(R=400\) and 1000. The methods evaluated were LS, \(\ell_{1}\)-norm relaxation (L1), L1-L1, and the proposed L1-parSBL. L1-L1 denotes the method of selecting dominant PW directions based on L1 and re-estimating the PWAs based on the low-dimensional dictionary, according to L1. We observed that L1-parSBL outperformed the other methods, and even L1-parSBL with \(R=400\) yielded better outcomes than the other methods with \(R=1000\). All methods except LS provided comparable RMEs for low-order modes. LS exhibited very poor performance and could not provide an acceptable reconstruction for modes above the 22 nd order (RME \(>0.3\)). Although a maximum of eight PW directions exist, smaller eigenwavelengths relative to the enclosure size cause the mode shapes to become more complex and more sensitive to the basis mismatch for higher-order modes. Thus, the RMEs of the other methods were increased. L1-parSBL could reduce the basis mismatch and achieved a better RME because it optimized the PW directions in Stage 2. In addition, the RMEs of different higher-order modes were considerably different, corresponding to different mode shapes. Axial and tangential modes tended to be better reconstructed than oblique modes. For oblique modes, more sample data points were needed or samples needed to be arranged properly to capture sufficient spatial information. Specifically, we considered the high-order oblique mode (1, 3, 1) as an example to compare the approximation performance of these methods. Figure 4a shows the desired mode shape. Figure 4b-d plots the relative errors of the approximation at field points (\(R=400\)), \(|\mathbf{\psi}_{\text{des}}-\mathbf{\psi}_{\text{rec}}|\oslash|\mathbf{\psi}_{\text{des}}|\), for LS, L1, and L1-parSBL, which achieved RME values of 0.73, 0.26, and 0.09, respectively. The result of L1-L1 was similar to that of L1 and is not shown here. It was observed that L1-parSBL produced a better reconstruction of the mode shape. In Figure 5a-d, note that the numbers of estimated PWs for LS and L1, \(\{\bar{\mathbf{w}}_{\text{LS}},\mathcal{S}\}\) and \(\{\bar{\mathbf{w}}_{\text{L1}},\mathcal{S}\}\), respectively, were both \(R=400\); in contrast, for L1-L1, there existed only \(R^{\prime}=23\) estimated PWs, \(\{\mathbf{w}_{\text{L1-L1}},\mathcal{S}^{\prime}\}\), whose directions were selected in Stage 1. Moreover, L1-parSBL yielded \(R^{\prime}=23\) estimated PWs, \(\{\mathbf{w}_{\text{L1-parSBL}},\mathcal{S}^{\prime\prime}\}\), whose directions

\[\ln\hat{Q}_{j}(\mathcal{H}_{j})=\mathbb{E}_{Q(\mathcal{H}\setminus\mathcal{H}_ {j})}[\ln P(\mathcal{H},\mathbf{\psi})], \tag{13}\]3 (Color online) Relative mean error (RME) of different methods in different modes (\(\mathbf{\cdot}\): \(R=400\); \(\mathbf{\cdot}\): \(R=1000\) ). were selected in Stage 1 and optimized in Stage 2. LS could not identify the actual PW directions because of which spatial aliasing artifacts were produced. The PWs could not be accurately estimated because of the basis mismatch, although L1 and L1-L1 could identify approximate directions. The proposed L1-parSBL efficiently optimized the PW directions to ensure that they were closer to the actual ones in Stage 2 and L1-parSBL estimated the PWAs more precisely than other methods. Similar results were obtained for the other modes as well. Thus, outcomes for only one mode are illustrated. The elapsed times of L1, L1-L1, and L1-parSBL for the PW approximation of the (1, 3, 1) mode are listed in \[\ln\hat{Q}_{j}(\mathcal{H}_{j})=\mathbb{E}_{Q(\mathcal{H}\setminus\mathcal{H}_ {j})}[\ln P(\mathcal{H},\mathbf{\psi})], \tag{13}\]4 The computer included two Intel Xeon central processing units with two 2.20 GHz processors.