A Scalable Parallel LSQR Algorithm for Solving Large-scale Linear System for Tomographic Problems: A Case Study in Seismic Tomography

Abstract

Least Squares with QR-factorization (LSQR) method is a widely used Krylov subspace algorithm to solve sparse rectangu- lar linear systems for tomographic problems. Traditional parallel implementations of LSQR have the potential, depending on the non-zero structure of the matrix, to have significant communication cost. The communication cost can dramatically limit the scalability of the algorithm at large core counts. We describe a scalable parallel LSQR algorithm that utilizes the particular non-zero structure of matrices that occurs in tomographic problems. In particular, we specially treat the kernel component of the matrix, which is relatively dense with a random structure, and the damping component, which is very sparse and highly structured separately. The resulting algorithm has a scalable communication volume with a bounded number of communica- tion neighbors regardless of core count. We present scaling studies from real seismic tomography datasets that illustrate good scalability up to O(10, 000) cores on a Cray XT cluster.