Iterative representing set selection for nested cross approximation
This work provides a more automated and efficient approach for matrix approximation in computational science, eliminating the need for pre-selected proxy surfaces.
The paper presents a new fast algebraic method for obtaining an H²-approximation of a matrix from its entries, achieving linear complexity in matrix size and polynomial in ranks, with numerical experiments confirming effectiveness and robustness.
A new fast algebraic method for obtaining an $\mathcal{H}^2$-approximation of a matrix from its entries is presented. The main idea behind the method is based on the nested representation and the maximum-volume principle to select submatrices in low-rank matrices. A special iterative approach for the computation of so-called representing sets is established. The main advantage of the method is that it uses only the hierarchical partitioning of the matrix and does not require special "proxy surfaces" to be selected in advance. The numerical experiments for the electrostatic problem and for the boundary integral operator confirm the effectiveness and robustness of the approach. The complexity is linear in the matrix size and polynomial in the ranks. The algorithm is implemented as an open-source Python package that is available online.