A highly scalable dense linear system solver for multiple right-hand sides in data analytics

We describe PP-BCG, a parallel iterative solver for the solution of dense and symmetric positive-definite linear systems with multiple right-hand sides suitable for MPPs. Such linear systems appear in the context of stochastic estimation of the diagonal of the matrix inverse in Uncertainty Quantification and the trace of matrix products in statistical analysis. We propose a novel numerical scheme based on the block Conjugate Gradient algorithm combined with Galerkin projections and describe its implementation. We test the method on model covariance matrices from uncertainty quantification, where the solution of the linear systems is typically used to estimate the diagonal of the matrix inverse. Numerical experiments on an MPP illustrate the performance of the proposed scheme in terms of efficiency and convergence rate, as well as its effectiveness relative to block Conjugate Gradient and the Cholesky-based ScaLAPACK solver. In particular, on a 4 rack BG/Q with up to 65536 cores using dense matrices of order as high as 0.5×10 and 800 rhs, PP-BCG is 2-3 times faster. Keywords-Linear systems, multiple right-hand sides, block Conjugate Gradient, Galerkin projections, distributed memory environments, uncertainty quantification