Starting from a high-level problem description in terms of partial differential equations using abstract tensor notation, the Chemora framework discretizes, optimizes, and generates complete high performance codes for a wide range of compute architectures. Chemora extends the capabilities of Cactus, facilitating the usage of large-scale CPU GPU systems in an efficient manner for complex applications, without low-level code tuning. Chemora achieves parallelism through MPI and multi-threading, combining OpenMP and CUDA. Optimizations include high-level code transformations, efficient loop traversal strategies, dynamically selected data and instruction cache usage strategies, and JIT compilation of GPU code tailored to the problem characteristics. The discretization is based on higher-order finite differences on multi-block domains. Chemora's capabilities are demonstrated by simulations of black hole collisions. This problem provides an acid test of the framework, as the Einstein equations contain hundreds of variables and thousands of terms.
Notable work on an example stencil application was selected as a finalist of the Gordon Bell Prize in SC 2011 as the first peta-scale result @cite_22 . demonstrated very high performance of 1.017 PFlop s in single precision using 4,000 GPUs along with 16,000 CPU cores on TSUBAME 2.0. Nevertheless, a set of new and more advanced optimization techniques introduced in the framework as well as its capabilities to generate highly efficient multi-GPU stencil computing codes from a high-level problem description make this framework even more attractive for users of large-scale hybrid systems.
@cite_22
2104853465
Many numerical codes now under development to solve Einstein's equations of general relativity in @math -dimensional spacetimes employ the standard ADM form of the field equations. This form involves evolution equations for the raw spatial metric and extrinsic curvature tensors. Following Shibata and Nakamura, we modify these equations by factoring out the conformal factor and introducing three connection functions.'' The evolution equations can then be reduced to wave equations for the conformal metric components, which are coupled to evolution equations for the connection functions. We evolve small amplitude gravitational waves and make a direct comparison of the numerical performance of the modified equations with the standard ADM equations. We find that the modified form exhibits much improved stability.