  This article introduces HODLR2D, a new hierarchical low-rank representation
for a class of dense matrices arising out of $N$ body problems in two
dimensions. Using this new hierarchical framework, we propose a new fast
matrix-vector product that scales almost linearly. We apply this fast
matrix-vector product to accelerate the iterative solution of large dense
linear systems arising out of radial basis function interpolation and
discretized integral equation. The space and computational complexity of
HODLR2D 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 and
computational complexity of HODLR2D matrix-vector products remain tractable for
large $N$. Additionally, we also present the parallel scalability of HODLR2D as
part of this article.
