Hierarchical Orthogonal Factorization: Sparse Least Squares Problems

In this work, we develop a fast hierarchical solver for solving large, sparse least squares problems. We build upon the algorithm, spaQR (sparsified QR Gnanasekaran and Darve in SIAM J Matrix Anal Appl 43(1):94–123, 2022), that was developed by authors to solve large linear systems. Our algorithm is built on top of Nested Dissection based multifrontal approach. use low-rank approximations frontal matrices sparsify vertex separators at every level elimination tree. Using two-step sparsification scheme, reduce number columns maintain ratio rows each front without introducing any additional fill-in. With improvised show runtime scales as $$\mathcal {O}(M \log N)$$ uses {O}(M)$$ memory store factorization. This achieved expense small controllable approximation error. The end result an approximate factorization matrix stored sequence orthogonal upper-triangular factors hence easy apply/solve with vector. Numerical experiments compare performance direct QR, inner-outer iterative method, CGLS method diagonal preconditioner Robust Incomplete Factorization (RIF) preconditioner.