François-Henry Rouet

Zixi Xu, Léopold Cambier, François-Henry Rouet et al. · stanford

The efficient compression of kernel matrices, for instance the off-diagonal blocks of discretized integral equations, is a crucial step in many algorithms. In this paper, we study the application of Skeletonized Interpolation to construct such factorizations. In particular, we study four different strategies for selecting the initial candidate pivots of the algorithm: Chebyshev grids, points on a sphere, maximally-dispersed and random vertices. Among them, the first two introduce new interpolation points (exo-vertices) while the last two are subsets of the given clusters (endo- vertices). We perform experiments using three real-world problems coming from the multiphysics code LS-DYNA. The pivot selection strategies are compared in term of quality (final rank) and efficiency (size of the initial grid). These benchmarks demonstrate that overall, maximally-dispersed vertices provide an accurate and efficient sets of pivots for most applications. It allows to reach near-optimal ranks while starting with relatively small sets of vertices, compared to other strategies.

Christopher Gorman, Gustavo Chávez, Pieter Ghysels et al.

We present new algorithms for the randomized construction of hierarchically semi-separable matrices, addressing several practical issues. The HSS construction algorithms use a partially matrix-free, adaptive randomized projection scheme to determine the maximum off-diagonal block rank. We develop both relative and absolute stopping criteria to determine the minimum dimension of the random projection matrix that is sufficient for the desired accuracy. Two strategies are discussed to adaptively enlarge the random sample matrix: repeated doubling of the number of random vectors, and iteratively incrementing the number of random vectors by a fixed number. The relative and absolute stopping criteria are based on probabilistic bounds for the Frobenius norm of the random projection of the Hankel blocks of the input matrix. We discuss parallel implementation and computation and communication cost of both variants. Parallel numerical results for a range of applications, including boundary element method matrices and quantum chemistry Toeplitz matrices, show the effectiveness, scalability and numerical robustness of the proposed algorithms.