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An efficient direct solver for volume integral equations with O ( N ) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal blocks of certain dense matrices have numerically low rank. Technically, the solver is inspired by previously developed direct solvers for integral equations based on “recursive skeletonization” and “Hierarchically Semi-Separable” (HSS) matrices, but it improves on the asymptotic complexity of existing solvers by incorporating an additional level of compression. The resulting solver has optimal O ( N ) complexity for all stages of the computation, as demonstrated by both theoretical analysis and numerical examples. The computational examples further display good practical performance in terms of both speed and memory usage. In particular, it is demonstrated that even problems involving 10 7 unknowns can be solved to precision 10 − 10 using a simple Matlab implementation of the algorithm executed on a single core. [ABSTRACT FROM AUTHOR]