Optimal grouping of basis functions

The method of moments (MoM) is widely used for the solution of EM scattering problems, but one of its greatest limitations is the resulting large and dense impedance matrix. This imposes constraints both in terms of storage and solution-complexity of the impedance matrix. Most of the methods to relieve these constraints are concerned with rendering the matrix sparse, or otherwise exploit its structure. An approach relevant to this paper is that of using special basis functions such that the resulting impedance matrix will have only small number of dominant terms. Thresholding the matrix will lead to a sparse matrix, with yet almost no degradation in the result. An example is the use of wavelets as basis functions. In the category of selecting special basis functions also falls the impedance matrix localization (IML) method of Canning (1990, 1993). In the IML one performs a basis-transformation in order to transform the impedance matrix into one with only a few dominant terms, and then performs a threshold operation to arrive at a sparse matrix. A different point of view was suggested in the impedance matrix compression (IMC) method. In the IMC one seeks to use basis functions such that the resulting solution vector is sparse. This, in turn, enables the use of a much smaller (in dimensions) impedance matrix, which is much easier to solve than the original impedance matrix. The author looks into the question of which basis functions should be used for the IMC, in order to have the most possible sparse solution vector.