# Enhanced Upward Translations for Systems with Clusters
## 1 Introduction
Instead of using multipole expansions of kernels, the translations are performed by means of integration over specially chosen _equivalent_ surfaces. The integrand is the product of the kernel by the equivalent density. The latter is found by equating the field, corresponding to the integral to the field known on a specially chosen _check_ surface. Details of the method are provided in the textbook [20]. Suggested initially for static and asymptotically smooth kernels, later on, the method has been extended to highly oscillating kernels [28]. The extension employs scaling the problem by wavelength. Since the method does not depend on a particular kernel, the authors called it the kernel-independent fast multipole method (KI-FMM). This terminology was accepted in later papers using and developing the method (e.g., [21; 23; 24; 25; 28; 29; 30]). Henceforth we shall also use the author's interpretation of the kernel independence although it is notably less general than that provided by the ACA algorithm. For our theme, two features of the KI-FMM are significant. Firstly, it explicitly employs geometrical data on the mutual location of boundary elements. This makes it possible to evaluate the matrix of distances, which is basic in problems concerning clusters [31]. The problems of rock and fracture mechanics are of the same kind. Secondly, in our paper [24] it is established that there is no need to integrate the product of equivalent _density_ by kernel over the equivalent surface. Instead, we may use equivalent _intensities_ of pointed sources located on the equivalent surfaces. The intensities are promptly found by using checkpoints on the check surface. The interpretation of KI-FMM in terms of equivalent intensities, having clear physical meaning, further extends the options to account for clusters. We shall use both features of the KI-FMM in the next section to give a strict definition of the cluster and to suggest a simple algorithm which rapidly generates the clusters. The need to use the concept of clusters [31] becomes evident from the numerical results reported in our paper [24]. They show the drastic (in some orders) loss of the accuracy of S2M translations for sources near the equivalent surfaces of leaves. However, there have been no suggestions, or even discussion, of how to improve the accuracy. Our objective is to numerically distinguish clusters within a leaf and to accurately translate their fields to the check surface of the parent cell. The purpose is reached by using two innovations. The first of them consists of appropriately ordering the leaf sources and grouping them in clusters by using data on their location and intensities. The second consists of introducing and employing modified translations, which translate far fields of each of the clusters more accurately than conventional S2M translations. Throughout the following discussion, we shall use the concepts and terminology of FMM, explained in detail in the textbooks [19; 20]. In particular, the far field is assumed as the combined field, generated by the sources in a tree cell outside the closest neighbors of the cell (e.g., [19; 21; 28; 29; 30]). For reasons explained, the exposition is provided for the FMM in its kernel-independent form. The comment on the extension to the analytical FMM is provided in the last paragraph of the paper. The paper is organized as follows. Section 2 contains guiding considerations for grouping the leaf sources into clusters in accordance with the objective of increasing the accuracy of S2M translations. They yield an algorithm for appropriately distinguishing the clusters. Each of them is characterized by its representative point source. In Section 3, the representative source of a cluster is used to modify S2M translation by means of the \"subtract and add\" approach. This improves the accuracy of approximate far fields, generated by equivalent sources located at an equivalent surface of the leaf.