Parallel Performance of an Iterative Solver Based on the Golub-Kahan Bidiagonalization

Abstract

We present a scalability study of Golub-Kahan bidiagonalization for the parallel iterative solution of symmetric indefinite linear systems with a 2 × 2 block structure. The algorithms have been implemented within the parallel numerical library PETSc. Since a nested inner-outer iteration strategy may be necessary, we investigate different choices for the inner solvers, including parallel sparse direct and multigrid accelerated iterative methods. We show the strong and weak scalability of the Golub-Kahan bidiagonalization based iterative method when applied to a two-dimensional Poiseuille flow and to two-and three-dimensional Stokes test problems.

Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints

“…We observe that the speed-up reaches half of the ideal speed-up. Such a result is consistent to what we observed in [14].…”

“…Here, we focus on the aforementioned Craig's variant algorithm for the solution of saddle point systems, which is presented in Algorithm 1. As stopping criterion, we use a normalized lower bound estimate of the energy norm error e k := u − u k M described in [13,14]. The algorithm stops once this normalized lower bound undershoots a sufficiently small tolerance τ .…”

“…In a previous work [14], we have described our implementation of the generalized GKB solver into the parallel linear algebra library PETSc [15][16][17]. The GKB algorithm is low in memory consumption, as the iterates of the solution are computed on the fly and no basis needs to be stored.…”

“…However, the algorithm needs to solve linear systems of the size of the physical as well as the constraint block in each iteration, which defines the principal bottleneck of the method. In terms of scalability, we have investigated the parallel performance of the solver when applied to two test problems based on the Stokes equations provided in the PETSc examples in [14]. The solver showed high scalability (the best case scenario showed a speedup half as good as the theoretical speedup for a large number of processes), which was favored in this case by the availability of multigrid methods for solving the (1,1)-block corresponding to the discretization of the Laplace equation.…”

Parallel solution of saddle point systems with nested iterative solvers based on the Golub‐Kahan Bidiagonalization

“…As the reactor containment building test case from EDF was not yet available for the strong and weak scalability investigations, we have adapted Stokes sample problems provided in the PETSc distribution. In this work, we focus on extending and improving the parallel results presented in our conference contribution 8 with the following novel aspects: We ran the examples on a larger number of cores, that is, 1024, and we studied the scalability of a bigger matrix for the Poiseuille flow test case ( m ≈16.8·10 6 , n ≈ 8.4·10 6 ). We introduce a three‐dimensional Stokes example and we comment on its strong scalability. We discuss the weak scalability of the nested inner‐outer iterative variants of our solver on the three test cases in two‐ and three‐dimensions. By linking with the Intel Math Kernel Library (MKL) for executing dense linear algebra operations, we improve the previously obtained computation times, especially those for the employed parallel sparse direct solver MUMPS 9 (Multifrontal Massively Parallel sparse direct Solver). We obtained for example a speed‐up with a factor of more than 5 for the standalone MUMPS solver and of about 3 for GKB‐MUMPS for computations on two cores. We investigate the portability and present the performance of the algorithm on an Advanced Micro Devices (AMD) architecture. …”