Research On The Application Of Compressed Sensing Theory In Computational Electromagnetics

In this dissertation, the application of compressed sensing(CS) theory in computational electromagnetics(CEM) is studied with the example of method of moments(MoM). This dissertation takes investigation on the construction and solution of the underdetermined MoM equation and aims to improve the efficiency of MoM by introducing the CS technology. The main work is as follows:Firstly, the basic principle of three main numerical methods of CEM and the research history and recent applications of CS theory are reviewed. The significance and detailed research method of introducing CS theory into CEM are discussed and analyzed. Then, the mathematical description of MoM is demonstrated, the most frequently-used basis and weight functions of MoM are listed, and the application of MoM in 2-D scattering problem is analyzed. The theoretical foundation and mathematical principle of CS theory are presented. Some key issues of CS theory are also given and discussed for providing the relevant theoretical background of the following research.Then, the particular case that both the basis and weight functions are linearly correlated is studied. Because of this linear correlation, the resulted MoM equation would has a sparse solution, and can be converted into underdetermined equation with the elementary row transformation. Thus, the CS technique can be introduced to achieve the adaptively mesh generation and local mesh refinement.Third, the more common cases that both the basis and weight functions are linearly independent are studied and a new construction method of the underdetermined equation is proposed. In the new method, an appropriate transform matrix is used to sparsify the solution based on some physical priori information of the problem. Then, the underdetermined equation is obtained by randomly dropping some weight functions. The effect of sparse transform matrix and dropping strategy on computational performances are investigated with numerical experiments.Forth, based on the discrete wavelet MoM, a deterministic construction method of the undetermined equation is proposed. With appropriate discrete wavelet transform, the impedance matrix, excitation vector, and solution vector can be localized. Thus, the resulted matrix equation has a sparse solution. Moreover, because of the significant correlation between the localized excitation and solution vectors, the distribution of the sparse elements of the solution vector can be predicted more accurately. The proposed method constructs the undetermined equation by retaining the weight functions corresponding to the support domain of the solution vector, so it avoids the effect of the uncertainty of the underdetermined equation on the stability of the numerical results.Finally, the fast method of filling the impedance matrix is studied based on CS theory. Because of the clear physical meaning of the impedance matrix, there is significant correlation among the elements. By constructing discrete sparse transform of the impedance matrix, a set of equations that are in connection with the column vectors of the impedance matrix and have sparse solutions can be obtained. Then, the randomly dropping method is utilized to get the corresponding underdetermined equations. Finally, the impedance matrix can be fast filled by solving these underdetermined equations with optimization method.