Mixed precision Krylov solvers with the Jacobi preconditioner often show significant convergence degradation when the Jacobi preconditioner is computed in low precision such as FP16 and BF16. It is found that this convergence degradation is attributed to loss of diagonal dominance due to roundoff errors in data conversion. To resolve this issue, we propose a new data conversion method, which is designed to keep diagonal dominance of the original matrix data. The proposed method is tested by computing the Poisson equation using the conjugate gradient method, the general minimum residual method, and the biconjugate gradient stabilized method with the FP16/BF16 Jacobi preconditioner on NVIDIA V100 GPUs. Here, the new data conversion is implemented by switching the round-nearest, round-up, round-down, and round-towards-zero intrinsics in CUDA, and is called once before the main iteration. Therefore, the cost of the new data conversion is negligible. When the coefficients of matrix is continuously changed by scaling the linear system, the conventional data conversion based on the round-nearest intrinsic shows periodic changes of the convergence property depending on the difference of the roundoff errors between diagonal and off-diagonal coefficients. Here, the period and magnitude of the convergence degradation depend on the bit length of significand. On the other hand, the proposed data conversion method is shown to fully avoid the convergence degradation, and robust mixed precision computing is enabled for the Jacobi preconditioner without extra overheads.

We develop a convolutional neural network model to predict multi-resolution steady flow data. Based on the image-to-image translation model pix2pixHD, our model can predict high resolution flow fields from the set of patched signed distance functions. By patching the high resolution data, our model uses roughly the one third of memory used by pix2pixHD. The accuracy of our model is almost the same as the U-Net model using the unpatched high resolution data.

We develop a multigrid preconditioned conjugate gradient (MG-CG) solver for the pressure Poisson equation in a two-phase flow CFD code JUPITER. The JUPITER code is redesigned to realize efficient CFD simulations including complex boundaries and objects based on a block-structured Cartesian grid system. The code is written in CUDA, and is tuned to achieve high performance on GPU based supercomputers. The main kernels of the MG-CG solver achieve more than 90% of the roofline performance. The MG preconditioner is constructed based on the geometric MG method with a three-stage V-cycle, and a red-black SOR (RB-SOR) smoother and its variant with cache-reuse optimization (CR-SOR) are applied at each stage. The numerical experiments are conducted for two-phase flows in a fuel bundle of a nuclear reactor. Thanks to the block-structured data format, grids inside fuel pins are removed without performance degradation, and the total number of grids is reduced to 2.26 × 109, which is about 70% of the original Cartesian grid. The MG-CG solvers with the RB-SOR and CR-SOR smoothers reduce the number of iterations to less than 15% and 9% of the original preconditioned CG method, leading to 3.1- and 5.9-times speedups, respectively. In the strong scaling test, the MG-CG solver with the CR-SOR smoother is accelerated by 2.1 times between 64 and 256 GPUs. The obtained performance indicates that the MG-CG solver designed for the block-structured grid is highly efficient and enables large-scale simulations of two-phase flows on GPU based supercomputers.

The aerodynamics simulation code based on the lattice Boltzmann method (LBM) using forest-of-octrees-based block-structured adaptive mesh refinement (AMR) with temporary-fixed refinement was implemented, and its performance was evaluated on GPU-based supercomputers. Although the Space-Filling-Curve-based (SFC) domain partitioning algorithm for the octree-based AMR has been widely used on conventional CPU-based supercomputers, accelerated computation on GPU-based supercomputers revealed a bottleneck due to costly halo data communication. Our new tree cutting approach adopts a hybrid domain partitioning with the coarse structured block decomposition and the SFC partitioning in each block. This hybrid approach improved the locality and the topology of the partitioned sub-domains and reduced the amount of the halo communication to one-third of the original SFC approach. In the strong scaling test, the code achieved maximum ×1.82 speedup at the performance of 2207 MLUPS (mega-lattice update per second) on 128 GPUs (NVIDIA® Tesla® V100). In the weak scaling test, the code achieved 9620 MLUPS at 128 GPUs with 4.473 billion grid points, while keeping the parallel efficiency of 93.4% from 8 to 128 GPUs.

Stencil-based applications such as CFD have succeeded in obtaining high performance on GPU supercomputers. The problem sizes of these applications are limited by the GPU device memory capacity, which is typically smaller than the host memory. On GPU supercomputers, a locality improvement technique using temporal blocking method with memory swapping between host and device enables large computation beyond the device memory capacity. However, because the loop management of temporal blocking with data movement across these memories increase programming difficulty, the applying this methodology to the real stencil applications demands substantially higher programming cost. Our high-productivity stencil framework automatically applies temporal blocking to boundary exchange required for stencil computation and supports automatic memory swapping provided by a MPI/CUDA wrapper library. The framework-based application for the airflow in an urban city maintains 80% performance even with the twice larger than the GPU memory capacity and have demonstrated good weak scalability on the TSUBAME 2.5 supercomputer.

The weather prediction code demands large computational performance to achieve fast and high-resolution simulations. Skillful programming techniques are required for obtaining good parallel efficiency on GPU supercomputers. Our framework-based weather prediction code ASUCA has achieved good scalability with hiding complicated implementation and optimizations required for distributed GPUs, contributing to increasing the maintainability, ASUCA is a next-generation high resolution meso-scale atmospheric model being developed by the Japan Meteorological Agency. Our framework automatically translates user-written stencil functions that update grid points and generates both GPU and CPU codes. User-written codes are parallelized by MPI with intra-node GPU peer-to-peer direct access. These codes can easily utilize optimizations such as overlapping technique to hide communication overhead by computation. Our simulations on the GPU-rich supercomputer TSUBAME 2.5 at the Tokyo Institute of Technology have demonstrated good strong and weak scalability achieving 209.6 TFlops in single precision for our largest model using 4,108 NVIDIA K20X GPUs.