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Integral equations are commonly encountered when solving complex physicalproblems. Their discretization leads to a dense kernel matrix that is block orhierarchically low-rank. This paper proposes a new way to build a low-rankfactorization of those low-rank blocks at a nearly optimal cost of$\mathcal{O}(nr)$ for a $n \times n$ block submatrix of rank r. This is done byfirst sampling the kernel function at new interpolation points, then selectinga subset of those using a CUR decomposition and finally using this reduced setof points as pivots for a RRLU-type factorization. We also explain how thisimplicitly builds an optimal interpolation basis for the Kernel underconsideration. We show the asymptotic convergence of the algorithm, explain hisstability and demonstrate on numerical examples that it performs very well inpractice, allowing to obtain rank nearly equal to the optimal rank at afraction of the cost of the naive algorithm.

Fast Low-Rank Kernel Matrix Factorization through Skeletonized Interpolation