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${\cal H}^2$-matrix constitutes a general mathematical framework forefficient computation of both partial-differential-equation andintegral-equation-based operators. Existing linear-complexity ${\cal H}^2$matrix-matrix product (MMP) algorithm lacks explicit accuracy control, whilecontrolling accuracy without compromising linear complexity is challenging. Inthis paper, we develop an accuracy controlled ${\cal H}^2$ matrix-matrixproduct algorithm by instantaneously changing the cluster bases during thematrix product computation based on prescribed accuracy. Meanwhile, we retainthe computational complexity of the overall algorithm to be linear. Differentfrom the existing ${\cal H}^2$ matrix-matrix product algorithm where formattedmultiplications are performed using the original cluster bases, in the proposedalgorithm, all additions and multiplications are either exact or computed basedon prescribed accuracy. Furthermore, the original ${\cal H}^2$-matrix structureis preserved in the matrix product. While achieving optimal complexity forconstant-rank matrices, the computational complexity of the proposed algorithmis also minimized for variable-rank ${\cal H}^2$-matrices. The proposed workserves as a fundamental arithmetic in the development of fast solvers forlarge-scale electromagnetic analysis. Applications to both large-scalecapacitance extraction and electromagnetic scattering problems involvingmillions of unknowns on a single core have demonstrated the accuracy andefficiency of the proposed algorithm.

Accuracy Controlled Structure-Preserving ${\cal H}^2$-Matrix-Matrix Product in Linear Complexity with Change of Cluster Bases