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This article introduces HODLR2D, a new hierarchical low-rank representationfor a class of dense matrices arising out of $N$ body problems in twodimensions. Using this new hierarchical framework, we propose a new fastmatrix-vector product that scales almost linearly. We apply this fastmatrix-vector product to accelerate the iterative solution of large denselinear systems arising out of radial basis function interpolation anddiscretized integral equation. The space and computational complexity ofHODLR2D matrix-vector products scales as $\mathcal{O}(pN \log(N))$, where $p$is the maximum rank of the compressed matrix subblocks. We also prove that $p\in \mathcal{O}(\log(N)\log(\log(N)))$, which ensures that the storage andcomputational complexity of HODLR2D matrix-vector products remain tractable forlarge $N$. Additionally, we also present the parallel scalability of HODLR2D aspart of this article.

HODLR2D: A new class of Hierarchical matrices