A parallel linear solver for multilevel Toeplitz systems with possibly several right-hand sides

Abstract

Abstract A Toeplitz matrix has constant diagonals; a multilevel Toeplitz matrix is defined recursively with respect to the levels by replacing the matrix elements with Toeplitz blocks. Multilevel Toeplitz linear systems appear in a wide range of applications in science and engineering. This paper discusses an MPI implementation for solving such a linear system by using the conjugate gradient algorithm. The implementation techniques can be generalized to other iterative Krylov methods besides conjugate gradient. These techniques include the use of an arbitrary dimensional process grid for handling the multilevel Toeplitz structure, a communication-hiding approach for performing matrix–vector multiplications, the incorporation of multilevel circulant preconditioning for accelerating convergence, an efficient orthogonalization manager for detecting linear dependence in block iterations, and an algorithmic rearrangement to eliminate all-reduce synchronizations. The combined use of these techniques leads to a scalable solver for large multilevel Toeplitz systems, possibly with several right-hand sides. We show experimental results on matrices of size up to the order of one billion with nearly perfect scaling by using up to 1024 MPI processes. We also demonstrate an application of the solver in parameter estimation for analyzing large-scale climate data.