The H-LDLT Fast Direct Solution Based On Electric Field Integral Equation

In computational electromagnetics, with the increase in electrical size and structural complexityof the target object, the implement of the method of moments (MOM) to solve electric field integralequation requires lots of memory and computational time. Hierarchical matrix, which is based on adata-sparse representation of dense matrix, can efficiently solve linear equations of electrodynamicproblems. This thesis proposes an efficient H-matrix LDLT factorization direct solver based onelectric field integral equation(EFIE), which can decrease the memory consumption andcomputational complexity respectively to O (N logN)and O (N log2N). In addition, thecomputational time and memory consumption of the algorithm are almost only half of H-LUfactorization’s.Firstly, this thesis introduces H-matrix and H-matrix arithmetic, including the concept ofH-matrix and the operations defined in H-matrix arithmetic. The detailed numerical procedure ofthe LU factorization is given, and the complexity of the various H-matrix algorithms are analyzed.As to various methods to construct H-matrix, two fast low-rank compression methods are applied toconstruct H-matrix, including: matrix decomposition algorithm-singular value decompositionalgorithm(MDA-SVD), adaptive cross approximation-singular value decompositionalgorithm(ACA-SVD) and then the two methods performance comparison is given. Consideringthat H-matrix formed by two fast low-rank compression methods is symmetrical, the thesisproposes H-LDLT direct solver. The numerical results confirm that the memory consumption andcomputation time of the algorithm is most only half of H-LU factorization’s. Finally, a methodcalled linear iterative improvement of solution accuracy is proposed for the improvement of theapproximate solution, which is computed by low accuracy H-matrix LDLT factorization. Thenumerical results confirm that compared to normal H-matrix LDLT factorization, it can improve thesolution accuracy with less memory consumption and computation time.