Data-driven Construction of Hierarchical Matrices with Nested Bases
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Hierarchical matrices provide a powerful representation for significantly reducing the computational complexity associated with dense kernel matrices. For general kernel functions, interpolation-based methods are widely used for the efficient construction of hierarchical matrices. In this paper, we present a fast hierarchical data reduction (HiDR) procedure with O(n) complexity for the memory-efficient construction of hierarchical matrices with nested bases where n is the number of data points. HiDR aims to reduce the given data in a hierarchical way so as to obtain O(1) representations for all nearfield and farfield interactions. Based on HiDR, a linear complexity ℋ^2 matrix construction algorithm is proposed. The use of data-driven methods enables better efficiency than other general-purpose methods and flexible computation without accessing the kernel function. Experiments demonstrate significantly improved memory efficiency of the proposed data-driven method compared to interpolation-based methods over a wide range of kernels. Though the method is not optimized for any special kernel, benchmark experiments for the Coulomb kernel show that the proposed general-purpose algorithm offers competitive performance for hierarchical matrix construction compared to several state-of-the-art algorithms for the Coulomb kernel.
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