# Wire-Grid and Sparse MoM Antennas: Past Evolution, Present Implementation, and Future Possibilities
## Why MoM?
### MoM Acceleration
The main results on the theory of dense matrix approximation were obtained by the research group of [PERSON]. During the work under the grant RNF 14-11-00806 (2014-2016), the authors, in particular, investigated the application of algebraic methods of approximation of large dense matrices to increase the computational efficiency of methods for solving integral equations of mathematical physics. Moreover, in the work [94], the researchers developed and implemented algorithms for constructing mosaic-skeleton approximations, the iterative method of GMRES in relation to the solution of SLAEs with an unchanged matrix in mosaic-skeleton format and several right-hand sides. The developed algorithms are applied to solving problems of scattering of electromagnetic waves on ideally conducting surfaces of complex shape. [PERSON]'s works are devoted to the solution of very large sparse SLAEs obtained by approximating the boundary value problem by means of a finite-volume exponential scheme, with the help of new methods under development such as the method of bisconjugate residuals, the dual method of bisconjugate residuals, and the stabilized method of bisconjugate residuals. Thus, the work [95] considers a set of algorithmic and technological problems related to the development, research, and application of high-performance parallel methods for solving large SLAEs with sparse matrices. A number of the presented original results are related to the development of iterative processes in the Krylov subspaces, as well as principles of their preconditioning and scalable parallelization based on additive domain decomposition algorithms. The concept of the Krylov library is described as integrated open software for a wide range of linear algebra problems. There are ideas of developing iterative methods for solving sparse SLAEs [96; 97] and employing the approximation of the original dense matrix, not only to reduce the required memory size, but also to speed up the iterative process by using the approximated matrix as a preconditioner. The main results of improving on [PERSON]'s ideas were obtained by [PERSON] and [PERSON] [98]. In work [99], the method of moments in the frequency domain was applied to model electrically large problems. To speed up the computations, a parallel version of LU decomposition was used, and the unknowns were grouped into blocks that were approximated by matrices of small rank using adaptive cross-approximation. The proposed method was tested on problems with the number of unknowns reaching four million. In the work [100], two block iterative methods for solving sparse SLAEs obtained by finite difference and element methods were proposed. Their computational costs for solving SLAEs with a constant matrix and several right-hand sides were considered. The effectiveness of the proposed methods was demonstrated by the example of electromagnetic analysis of two test three-dimensional structures. It as shown that when using an effective preconditioner, it is preferable to use block methods to sequentially solve all SLAEs with different vectors of free terms. In the work [101], new methods for solving a number of SLAEs with a shift (with matrices differing from the original matrix by a scalar multiplied by a unit matrix) and many right-hand sides were proposed.