# Enhanced Upward Translations for Systems with Clusters
## 3 Enhanced Translations  
As mentioned, far fields of leaf sources, beyond the layer of the thickness \(0.25R_{e}\), are translated conventionally much more accurately than far fields of sources within the layer. Thus translations for the sources with distances to the leaf center less than \(0.75R_{e}\) may be translated conventionally. They may be excluded from the clustering process, presented in the previous section. The discussion focuses on sources comprising the distinguished clusters. For them improving the accuracy of S2M translations appears desirable. From now on, without loss of generality, to simplify notations, we assume that the leaf contains merely the sources comprising the clusters. The total number of these sources is \(N_{L}\), and the number of clusters is \(N_{cl}\). Their intensities, locations and normals of the representative sources, are defined by (1) for each of the clusters. Specifically, for the \(p\)-th cluster, they are \(\mathbf{q}_{clp}\), \(\mathbf{y}_{clp}\), and \(\mathbf{n}_{clp}\) (\(p=1,\ldots,N_{cl}\)). Recall how upward translations are performed conventionally by a KI-FMM (e.g., [18; 20]). We shall use the interpretation of a translation in terms of equivalent intensities [24]. For a given kernel \(\mathbf{K}(\mathbf{x},\mathbf{y})\), the exact far field \(\mathbf{f}(\mathbf{x})\), translated to a field point \(\mathbf{x}\) from leaf sources, is  
\[\mathbf{f}(\mathbf{x})=\sum_{j=1}^{N_{L}}\mathbf{K}\Big{(}\mathbf{x},\mathbf{y}_{j}\Big{)}\mathbf{q}_{j} \tag{2}\]  
In (2), \(N_{L}\) is the number of sources in the leaf, whose far fields are generated by the kernel \(\mathbf{K}(\mathbf{x},\mathbf{y})\), \(\mathbf{q}_{j}\) are their intensities, \(\mathbf{y}_{j}\) are points of their locations. When the kernel \(\mathbf{K}(\mathbf{x},\mathbf{y})\) depends on the normal at the source point \(\mathbf{y}=\mathbf{y}_{j}\), the normal \(\mathbf{n}_{j}\) assigned at \(\mathbf{y}_{j}\) is taken into account. A KI-FMM approximates the exact field (2) by replacing the field of the leaf sources by the field, generated by sources at an equivalent surface (ES):  
\[\mathbf{f}(\mathbf{x})\approx\sum_{s=1}^{N_{\epsilon}}\mathbf{K}(\mathbf{x},\mathbf{y}_{es})\mathbf{q} _{es} \tag{3}\]  
Herein, \(N_{\epsilon}\) is the total number of equivalent sources with intensities \(\mathbf{q}_{es}\) located at nodes \(\mathbf{y}_{es}\) on the ES. In cases when the kernel depends on the normal at the source point, the normal is orthogonal to the ES. The equivalent intensities \(\mathbf{q}_{es}\) are found by equating the right-hand sides of (3) to the exact field at \(N_{\epsilon}\) check points \(\mathbf{x}_{i}\) located at a check surface (ChS):  
\[\sum_{s=1}^{N_{\epsilon}}\mathbf{K}(\mathbf{x}_{i},\mathbf{y}_{es})\mathbf{q}_{es}=\sum_{j=1} ^{N_{L}}\mathbf{K}\Big{(}\mathbf{x}_{i},\mathbf{y}_{j}\Big{)}\mathbf{q}_{j}\,\ i=1,\ldots,N_{\epsilon} \tag{4}\]  
For certainty, the ES and ChS may be assumed spherical of radii \(R_{e}\) and \(R_{ch}\), respectively. The nodes are distributed uniformly; the normal at a node at ChS, like the normal at a point at ES, is orthogonal to the surface. Such choices minimize the translation error for a given number \(N_{\epsilon}\), as compared with other choices, say, with cubic ES [24]. The system (4) is solved by using its inverse matrix. The latter is calculated in advance and stays the same in all iterations. The found equivalent intensities \(\mathbf{q}_{es}\) are used in Equation (3), which approximates the far field (2) at any point outside the ChS. This completes the path of the conventional method of translation. The _modified_ translations employ the clusters and their representative sources to increase the accuracy of translated far fields. As mentioned, the clustering algorithm includes all \(N_{L}\) leaf sources, so that \(N_{L}=\sum_{p=1}^{N_{cl}}M_{clp}\).