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

Summary The Golub‐Kahan bidiagonalization is widely used in the singular value decomposition of rectangular matrices and has been generalized to an iterative solver for symmetric indefinite linear systems with a two‐by‐two block structure. In this work, we present a scalability study of this generalized solver as implemented in a recent release of the parallel numerical library PETSc (Portable, Extensible Toolkit for Scientific Computation). We present an improved solver performance for the two‐dimensional (2D) Stokes equations as compared to previous work. Furthermore, we investigate the performance of different parallel inner solvers in the outer Golub‐Kahan iteration for a three‐dimensional Stokes problem. The study includes parallel sparse direct solvers and multigrid methods. When increasing the number of cores for a fixed total problem size, the solver exhibits good speedups of up to 50% at the 1024 core count. For the tests in which the total problem size grows while the workload in each core stays constant, the parallel performance of the solver scales almost linearly with the increase in the core counts. In particular, the computation time increases only by about 15% when the number of cores increases from 80 to 1024 for a 2D test case.