An $$\mathcal O (N \log N)$$O(NlogN) Fast Direct Solver for Partial Hierarchically Semi-Separable Matrices

TLDR: The key ingredients behind this fast solver are recursion, efficient low rank factorization using Chebyshev interpolation, and the Sherman–Morrison–Woodbury formula.

Abstract: This article describes a fast direct solver (i.e., not iterative) for partial hierarchically semi-separable systems. This solver requires a storage of $$\mathcal O (N \log N)$$ O ( N log N ) and has a computational complexity of $$\mathcal O (N \log N)$$ O ( N log N ) arithmetic operations. The numerical benchmarks presented illustrate the method in the context of interpolation using radial basis functions. The key ingredients behind this fast solver are recursion, efficient low rank factorization using Chebyshev interpolation, and the Sherman---Morrison---Woodbury formula. The algorithm and the analysis are worked out in detail. The performance of the algorithm is illustrated for a variety of radial basis functions and target accuracies.