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Accuracy Explicitly Controlled H2-Matrix Arithmetic in Linear Complexity and Fast Direct Solutions for Large-Scale Electromagnetic Analysis
thesisposted on 17.10.2019 by Miaomiao Ma
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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(NNitNrhs) with N being matrix size, Nit the number of iterations and Nrhs 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.