# The Boundary Element Method in Acoustics: A Survey
## 6 Areas of Discussion
### Efficiency
#### 6.1.3 Iterative Methods and Preconditioning  
The scalability of the standard BEM in acoustics has been in question for several decades. The computational estimate relates two areas of particular concern, the \(O(n^{3})\) nature of LU factorisation and the \(O(n^{2})\) cost of computing the matrices. Hence, much of the current research on the BEM in acoustics is focused on reducing its computational burden, so that the methods can be casually applied to more significant problems and high frequencies. The conjugate gradient method was identified as a useful iterative solution method for the acoustic BEM [155; 375; 376]. The conjugate gradient method and related methods are generally termed Krylov subspace methods and these variants have been significantly tested on the linear systems arising from the boundary element method in acoustics [106; 377; 378; 379]. In general, if the underlying operator is compact, corresponding to a clustering of the corresponding matrix eigenvalues, then iterative methods, such as the Krylov subspace methods, work well [380]. However, one of the operators, \(N_{k}\), is not compact, the eigenvalues of its matrix equivalent are spread out, and hence the raw iterative methods are only applicable in particular cases, in which the hypersingular operator is not in play. Hence, in the spirit of generality that is sought in this work, the iterative methods are of limited value in solving the untreated equations. However, in the acoustic boundary element method, these iterative methods are usually applied following an intervention with the original linear system, termed preconditioning [381]. In the acoustic BEM, preconditioning has been advocated since its early days. The original concept can be derived from the integral equation formulations. By various substitutions with the boundary integral equations, operator identities can be derived. One of the most useful is the substitution of (74) and (75) into (70) (\(\mathbf{p}\in S\)) (or substituting (63) or (64) into (48) (for \(p\!\varepsilon S)\)) giving  
\[L_{k}N_{k}=\left(M_{k}-\frac{1}{2}I\right)\!\left(M_{k}+\frac{1}{2}I\right)= \left(M_{k}+\frac{1}{2}I\right)\!\left(M_{k}-\frac{1}{2}I\right)={M_{k}}^{2}- \frac{1}{4}I_{*}\]and hence the left-hand side of the Burton and Miller Equation (76) may be re-written as follows  
\[ \begin{split}\Big{\{}(M_{k}-\frac{1}{2}I+\mu L_{k}N_{k})\varphi \Big{\}}_{S}\Big{(}\mathbf{p};\mathbf{n}_{p}\Big{)}&=\Big{\{}(M_{k}- \frac{1}{2}I+\mu(M_{k}+\frac{1}{2}I)(M_{k}-\frac{1}{2}I)_{k})\varphi\Big{\}}_{S }\Big{(}\mathbf{p};\mathbf{n}_{p}\Big{)}\\ &=\Big{\{}(L_{k}+\mu L_{k}(M_{k}^{t}+\frac{1}{2}I)v\Big{\}}_{S} \Big{(}\mathbf{p};\mathbf{n}_{p}\Big{)},\end{split}  \tag{96}\]  
The equation is preconditioned, as the \(N_{k}\) operator has been eliminated and all the operators are compact. However, the motivation for this technique can be with the elimination of the hypersingular integration, with incidental preconditioning. Methods based on (96) along with \(L_{k}N_{k}=M_{k}^{t^{2}}-\frac{1}{2}\), that is obtained through the substitution of (53) and (54) into (61) or the substitution of (71) and (72) into (73) (\(\mathbf{p}\in S\)), are often termed the Calderon equations [382; 383]. Methods that make use of this substitution have been developed [155; 241; 157]. Alternative approaches in developing well-conditioned integral formulations for iterative solution have also been the subject of research [384; 385; 386; 387; 388]. Langou [389] develops methods for the iterative solution of linear systems with multiple right-hand sides, echoing the stated advantage of the LU factorisation method, discussed in Section 6.1.1. Matrix preconditioners are often based on constructing an approximate inverse and following a fixed point/defect correction/contraction method. Introducing a preconditioning matrix also introduces an \(O(n^{2})\) matrix multiplication at each step, but this can be reduced to \(O(n)\) if the approximate inverse matrix is sparse or banded [388; 378].