# Wire-Grid and Sparse MoM Antennas: Past Evolution, Present Implementation, and Future Possibilities
## Why MoM?
### Overview of Computional Methods
In this way, it is possible to find the inversion of the system matrix using the matrix splitting method. When the nested dissection ordering [54] is applied to the sparse part and the LU-decomposition is applied to the dense part, the total computational complexity is O(\(N^{1.5}\)) for two-dimensional problems and O(\(N^{2}\)) for three-dimensional ones. The memory requirements are O(\(N\)log\(N\)) for two-dimensional and O(\(N^{4/3}\)) for three-dimensional problems. One of the disadvantages of differential equation solvers is the mesh dispersion error that causes the wave to have a different phase velocity within the mesh compared to the exact solution [55; 56]. This error can be eliminated by using a higher mesh density but causes an increase in the computational cost. Since the error is cumulative, it is especially noticeable when simulating a large region or in the case of large scatterers. To eliminate this error, the density of the mesh must be increased as the size of the modeling region increases. For second-order precision designs, the mesh density in one dimension (number of points per wavelength) is defined as \((kd)^{0.5}\), where \(d\) is the diameter of the modeling region, and \(k\) is the wave number [57]. Consequently, the number of unknowns is \((kd)^{1.5}\) for one dimension. Correspondingly, for two dimensions the number of unknowns scales as \((kd)^{3}\), while for three dimensions it scales as \((kd)^{4.5}\). The solution lies in using an accurate higher-order differential equation solver [58; 59] or in combining a differential equation solver with an integral equation solver when there are large homogeneous regions. Among the existing numerical methods, MoM [60] is one of the most frequently used methods in modeling microwave and millimeter integrated circuits and is widely applied to the problems of electromagnetic compatibility [61]. The name \"method of moments\" is, according to some authors, unfortunate, because it has a slightly different meaning in modern applied mathematics [62]. However, when solving the problems of electromagnetism, this approach is historically still called MoM. When [PERSON] chose a name for the method he used, he borrowed it from the work [63] where the authors described a method for solving an integral equation and called it MoM. Earlier, a method with this name was proposed in [64] where polynomials (moments functions) were used as test functions for solving integral equations. However, it should be noted that within [PERSON]'s paper, he used the term \"method of moments\" to refer to the method of weighted residuals (MWRs), despite the fact that he limited himself only to linear electromagnetic problems [65]. In the version of MoM proposed by [PERSON], piecewise constant (impulse) functions are used as basic functions and Dirac functions as test functions. This is also known as the collocation method. [PERSON] in [66] made a generalization that allows us to consider the collocation, Galerkin, and least-squares methods as special cases of MoM. According to some authors, FEM and even FDM methods can also be considered as special cases of MoM [67]. In applied mathematics, however, this approach is commonly referred to in another way. Thus, in 1956, the work [68] proposed the term \"weighted residuals\", which generalizes a whole family of these methods. A general discussion of this approach can be found in [69; 70]. The development of the MWRs is manifested in the works of [66; 71; 72; 73; 74; 75].