  We are interested in solving linear systems arising from three applications:
(1) kernel methods in machine learning, (2) discretization of boundary integral
equations from mathematical physics, and (3) Schur complements formed in the
factorization of many large sparse matrices. The coefficient matrices are often
data-sparse in the sense that their off-diagonal blocks have low numerical
ranks; specifically, we focus on "hierarchically off-diagonal low-rank (HODLR)"
matrices. We introduce algorithms for factorizing HODLR matrices and for
applying the factorizations on a GPU. The algorithms leverage the efficiency of
batched dense linear algebra, and they scale nearly linearly with the matrix
size when the numerical ranks are fixed. The accuracy of the HODLR-matrix
approximation is a tunable parameter, so we can construct high-accuracy fast
direct solvers or low-accuracy robust preconditioners. Numerical results show
that we can solve problems with several millions of unknowns in a couple of
seconds on a single GPU.
