# The Boundary Element Method in Acoustics: A Survey
## 5 Extending the Boundary Element Method in Acoustics
### Inverse Problems  
An inverse problem in acoustics, involves measuring properties of the acoustic field and processing that information in order to determine something about its origin. Over recent decades, the main text that guides inverse acoustic (and electromagnetic) (scattering) problems is that of Colton and Kress [16], now in its third edition. Colton and Kress provide a mathematical analysis of inverse problems, and, in acoustics, their focus is on determining the shape of the boundary of a scatterer from the (disturbed) sound pressure at points in the far-field. An example of an application of this is identifying bodies on the sea floor [327]. Inverse problems are ill-posed. This means a range of solutions can give rise to the same measurements, in contrast to the forward problems, studied so far, which are usually well-posed, with a unique solution. In practice, the linear system that is returned by the boundary element method (or any standard method) is significantly ill-conditioned. If a conventional method is used to solve the linear system then the solution will not be acceptable. In the author's previous work on the inverse diffusion problem (classically, the backward heat conduction problem) [328], also included in Visser [237]) it was discussed that there is effectively insufficient information in the data to determine the solution of an inverse problem. The notion of an 'acceptable' solution, the bias of the observer, provides the final constraint that enables the ill-posed problem to be substituted by a nearby well-posed problem. This returns an acceptable solution, even though it is a less accurate solution of the discrete equations. In the literature, this technique is termed _regularisation_. Acoustic holography--determining the sound field near the source from acoustic properties at a distance from the source--is one of the main application areas of the inverse boundary element method. In general, an array of pressure or velocity transducers provides the data and the inversion returns the acoustic properties on the surface. Acoustic holography can determine the sources of noise from a radiating structure, which can help guide a re-design. For example, starting with the discrete equivalent of Equation (71),  
\[\underline{\varphi}_{E}=L_{ES,k}\,\widehat{\sigma_{S}}\, \tag{93}\]  
the field data \(\underline{\varphi}_{E}\) (effectively the sound pressure, see Section 2.1) is related to the layer potential \(\sigma_{S}\). If we could find a discrete approximation to \(\sigma_{S}\), then the approximation to the surface potential (sound pressure) and velocity could be found by the matrix-vector multiplication of the discrete equivalent of Equations (53) and (54) and the surface intensity could then be found by the Equation (5). However, as discussed, the solution of the linear system will not yield acceptable results. Even if the number of data points massively exceeds the number of elements, the system is stll effectively underdetermined. Perhaps the most popular method of resolving the ill-posedness is to use _Tikonov regularisation_. This involves minimising the residual in system, along with a penalty function that constrains the variability of the solution in some sense. For example, Equation (93) is replaced by  
\[min\,\,\widehat{\underline{\sigma}_{S}}\left\{\left\|\underline{\varphi}_{E}- L_{ES,k}\,\widehat{\underline{\sigma}_{S}}\right\|+\eta^{2}\left\|\widehat{ \underline{\sigma}_{S}}\right\|^{2}\right\}, \tag{94}\]  
where \(\eta\) is a parameter that can be 'tuned' to achieve and acceptable solution and the norm is the 2-norm. A related regularisation method is termed truncated singular value decomposition (TSVD). For example, the singular value decomposition (SVD) of the \(L_{ES,k}\) matrix factorises it as follows:  
\[L_{ES,k}=U\Sigma V^{H}, \tag{95}\]  
where \(U\) is a \(n_{E}\times n_{S}\) matrix, \(V\) is \(n_{S}\times n_{S}\) and \(\Sigma\) is a diagonal matrix containing \(n_{S}\) singular values \(s_{i}\), non-negative values in non-decreasing order.