Summary The Gram–Schmidt process uses orthogonal projection to construct the A = QR factorization of a matrix. When Q has linearly independent columns, the operator P = I − Q(QTQ)−1QT defines an orthogonal projection onto Q⊥. In finite precision, Q loses orthogonality as the factorization progresses. A family of approximate projections is derived with the form P = I − QTQT, with correction matrix T. When T = (QTQ)−1, and T is triangular, it is postulated that the best achievable orthogonality is 𝒪(ε)κ(A). We present new variants of modified (MGS) and classical Gram–Schmidt algorithms that require one global reduction step. An interesting form of the projector leads to a compact WY representation for MGS. In particular, the inverse compact WY MGS algorithm is equivalent to a lower triangular solve. Our main contribution is to introduce a backward normalization lag into the compact WY representation, resulting in a 𝒪(ε)κ([r0,AVm]) stable Generalized Minimal Residual Method (GMRES) algorithm that requires only one global reduce per iteration. Further improvements in performance are achieved by accelerating GMRES on GPUs.

We investigate two single-reduce orthogonalization schemes for both s-step and pipelined GMRES. The first is based on classical Gram Schmidt with reorthogonalization (CGS2), and the second on modified Gram Schmidt (MGS). Standard iterated CGS2 requires three global reductions. In standard MGS, the number of global reductions is proportional to the number of vectors against which we are orthogonalizing. In both cases, we can reduce this to a single global reduction, including reorthogonalization for accuracy.Our implementation is based on Trilinos software components, and therefore, is portable to different machine architectures with a single code base. We first demonstrate solver performance on the Intel Haswell nodes of the NERSC Cori Supercomputer. For these experiments, we integrated our solvers into Nalu-wind, a computational fluid dynamics application. At each time step, Nalu uses GMRES with a smoothed aggregation algebraic multigrid (SA-AMG) preconditioner to solve a pressure Poisson linear system. In this experiment, sstep GMRES reduced Nalu's total GMRES solve time by a factor of 1.4×.We then benchmarked the single-reduce orthogonalization schemes on the ORNL Summit supercomputer. In these experiments, our low-synchronization CGS2 and MGS improved the s-step GMRES performance by a factor of 2.4× and 10.1× on 384 NVIDIA V100 GPUs, respectively, while on the IBM Power9 CPUs, they improved the stability of the pipelined GMRES without increasing the iteration time.

Please cite this article in press as: A. Amritkar et al., Recycling Krylov subspaces for CFD applications and a new hybrid recycling solver, J. Comput. Phys. (2015), http://dx. AbstractWe focus on robust and efficient iterative solvers for the pressure Poisson equation in incompressible Navier-Stokes problems. Preconditioned Krylov subspace methods are popular for these problems, with BiCGStab and GMRES(m) most frequently used for nonsymmetric systems. BiCGStab is popular because it has cheap iterations, but it may fail for stiff problems, especially early on as the initial guess is far from the solution. Restarted GMRES is better, more robust, in this phase, but restarting may lead to very slow convergence. Therefore, we evaluate the rGCROT method for these systems. This method recycles a selected subspace of the search space (called recycle space) after a restart. This generally improves the convergence drastically compared with GMRES(m). Recycling subspaces is also advantageous for subsequent linear systems, if the matrix changes slowly or is constant. However, rGCROT iterations are still expensive in memory and computation time compared with those of BiCGStab. Hence, we propose a new, hybrid approach that combines the cheap iterations of BiCGStab with the robustness of rGCROT. For the first few time steps the algorithm uses rGCROT and builds an effective recycle space, and then it recycles that space in the rBiCGStab solver.We evaluate rGCROT on a turbulent channel flow problem, and we evaluate both rGCROT and the new, hybrid combination of rGCROT and rBiCGStab on a porous medium flow problem. We see substantial performance gains on both problems.

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Adjoint methods have become the state of the art in recent years for both functional error estimation and adaptation. But, since most engineering applications rely upon multiple functionals to assess a physical process or system, an adjoint solution must be obtained for each functional of interest which can increase the overall computational cost significantly. In this paper, new techniques are presented for functional error estimation and adaptation which provide the same error estimates and adaptation indicators as a conventional adjoint method, but do so much more efficiently, especially when multiple functionals must be examined. For functional error estimation, the adjoint solve is replaced by the solution of an error transport equation for the local solution error and an inner product with the functional linearization. This method is shown to produce the same functional error estimate as an adjoint solve. For functional-based adaptation, the bilinear form of the adjoint correction to the functional is exploited in a manner so as to obtain the adjoint variables in an approximate sense but still accurate enough to form useful adaptation indicators. These new methods for functional error estimation and adaptation are tested using the quasi-1D nozzle problem.