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Block Low-Rank Matrices with Shared Bases : Potential and Limitations of the BLR2 Format
We investigate a special class of data sparse rank-structured matrices that combine a flat block low-rank (BLR) partitioning with the use of shared (called nested in the hierarchical case) bases. This format is to $\mathcal{H}^2$ matrices what BLR is to $\mathcal{H}$ matrices: we therefore call it the BLR$^2$ matrix format. We present algorithms for the construction and LU factorization of BLR$^2$ matrices, and perform their cost analysis---both asymptotically and for a fixed problem size. With weak admissibility, BLR$^2$ matrices reduce to block separable matrices (the flat version of HBS/HSS). Our analysis and numerical experiments reveal some limitations of BLR$^2$ matrices with weak admissibility, which we propose to overcome with two approaches: strong admissibility, and the use of multiple shared bases per row and column.
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