Randomized Strong Recursive Skeletonization: Simultaneous Compression and LU Factorization of Hierarchical Matrices using Matrix-Vector Products

Abstract

The hierarchical matrix framework partitions matrices into subblocks that are either small or of low numerical rank, enabling linear storage complexity and efficient matrix-vector multiplication. This work focuses on the $H^2$-matrix format constructed under the strong admissibility condition, which has two key properties: (1) a compressed representation that approximates far-field interactions with low-rank blocks while near-field interactions are stored densely, and (2) a nested basis structure that reuses basis matrices across hierarchy levels. Although these matrices support fast Cholesky and LU factorizations, implementing them - especially for 3D PDE discretizations - remains challenging due to the nested recursions and recompressions involved. This paper introduces an algorithm that simultaneously compresses and factorizes a general ${H}^{2}$ matrix, using only the action of the matrix and its adjoint on vectors. The number of required matrix-vector products is independent of the matrix size, and depends only on the problem geometry and a rank parameter. The resulting LU factorization is invertible and can serve as an approximate direct solver, with accuracy influenced by the spectral properties of the matrix. To achieve competitive sample complexity, the method employs dense Gaussian test matrices without explicitly encoding structured sparsity. Samples are drawn only once at the start of the algorithm; as the factorization proceeds, structure is dynamically introduced into the test matrices through efficient linear algebraic operations. Numerical experiments demonstrate robustness to indefiniteness and ill-conditioning, as well as the efficiency of the method for integral and differential equations in 2D and 3D.