Accuracy Explicitly Controlled H2-Matrix Arithmetic in Linear Complexity and Fast Direct Solutions for Large-Scale Electromagnetic Analysis

The design of advanced engineering systems generally results in large-scale numerical problems, which require efficient computational electromagnetic (CEM) solutions. Among existing CEM methods, iterative methods have been a popular choice since conventional direct solutions are computationally expensive. The optimal complexity of an iterative solver is O(NN_itN_rhs) with N being matrix size, N_it the number of iterations and N_rhs the number of right hand sides. How to invert or factorize a dense matrix or a sparse matrix of size N in O(N) (optimal) complexity with explicitly controlled accuracy has been a challenging research problem. For solving a dense matrix of size N, the computational complexity of a conventional direct solution is O(N³); for solving a general sparse matrix arising from a 3-D EM analysis, the best computational complexity of a conventional direct solution is O(N²). Recently, an H²-matrix based mathematical framework has been developed to obtain fast dense matrix algebra. However, existing linear-complexity H²-based matrix-matrix multiplication and matrix inversion lack an explicit accuracy control. If the accuracy is to be controlled, the inverse as well as the matrix-matrix multiplication algorithm must be completely changed, as the original formatted framework does not offer a mechanism to control the accuracy without increasing complexity.

In this work, we develop a series of new accuracy controlled fast H² arithmetic, including matrix-matrix multiplication (MMP) without formatted multiplications, minimal-rank MMP, new accuracy controlled H² factorization and inversion, new accuracy controlled H² factorization and inversion with concurrent change of cluster bases, H²-based direct sparse solver and new HSS recursive inverse with directly controlled accuracy. For constant-rank H²-matrices, the proposed accuracy directly controlled H² arithmetic has a strict O(N) complexity in both time and memory. For rank that linearly grows with the electrical size, the complexity of the proposed H² arithmetic is O(NlogN) in factorization and inversion time, and O(N) in solution time and memory for solving volume IEs. Applications to large-scale interconnect extraction as well as large-scale scattering analysis, and comparisons with state-of-the-art solvers have demonstrated the clear advantages of the proposed new H² arithmetic and resulting fast direct solutions with explicitly controlled accuracy. In addition to electromagnetic analysis, the new H² arithmetic developed in this work can also be applied to other disciplines, where fast and large-scale numerical solutions are being pursued.