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# 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*with_{it}N_{rhs})*N*being matrix size,*N*the number of iterations and_{it }*N*the number of right hand sides. How to invert or factorize a dense matrix or a sparse matrix of size_{rhs}*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(N*; for solving a general sparse matrix arising from a 3-D EM analysis, the best computational complexity of a conventional direct solution is^{3})*O(N*. Recently, an^{2})*H*-matrix based mathematical framework has been developed to obtain fast dense matrix algebra. However, existing linear-complexity^{2}*H*-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.^{2}In this work, we develop a series of new accuracy controlled fast

*H*arithmetic, including matrix-matrix multiplication (MMP) without formatted multiplications, minimal-rank MMP, new accuracy controlled^{2}*H*factorization and inversion, new accuracy controlled^{2}*H*factorization and inversion with concurrent change of cluster bases,^{2}*H*-based direct sparse solver and new^{2}*HSS*recursive inverse with directly controlled accuracy. For constant-rank*H*-matrices, the proposed accuracy directly controlled^{2}*H*arithmetic has a strict^{2}*O(N)*complexity in both time and memory. For rank that linearly grows with the electrical size, the complexity of the proposed*H*arithmetic is^{2}*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*H*arithmetic and resulting fast direct solutions with explicitly controlled accuracy. In addition to electromagnetic analysis, the new^{2}*H*arithmetic developed in this work can also be applied to other disciplines, where fast and large-scale numerical solutions are being pursued.^{2}