Earth system models are developed with a tight coupling to target hardware, often containing specialized code predicated on processor characteristics. This coupling stems from using imperative languages that hard-code computation schedules and layout. We present a detailed account of optimizing the Finite Volume Cubed-Sphere Dynamical Core (FV3), improving productivity and performance. By using a declarative Python-embedded stencil domain-specific language and data-centric optimization, we abstract hardware-specific details and define a semi-automated workflow for analyzing and optimizing weather and climate applications. The workflow utilizes both local and full-program optimization, as well as user-guided fine-tuning. To prune the infeasible global optimization space, we automatically utilize repeating code motifs via a novel transfer tuning approach. On the Piz Daint supercomputer, we scale to 2,400 GPUs, achieving speedups of up to 3.92× over the tuned production implementation at a fraction of the original code.

Most compilers have a single core intermediate representation (IR) (e.g., LLVM) sometimes complemented with vaguely defined IR-like data structures. This IR is commonly low-level and close to machine instructions. As a result, optimizations relying on domain-specific information are either not possible or require complex analysis to recover the missing information. In contrast, multi-level rewriting instantiates a hierarchy of dialects (IRs), lowers programs level-by-level, and performs code transformations at the most suitable level. We demonstrate the effectiveness of this approach for the weather and climate domain. In particular, we develop a prototype compiler and design stencil- and GPU-specific dialects based on a set of newly introduced design principles. We find that two domain-specific optimizations (500 lines of code) realized on top of LLVM’s extensible MLIR compiler infrastructure suffice to outperform state-of-the-art solutions. In essence, multi-level rewriting promises to herald the age of specialized compilers composed from domain- and target-specific dialects implemented on top of a shared infrastructure.

We present a new parallel algorithm for solving triangular systems with multiple right hand sides (TRSM). TRSM is used extensively in numerical linear algebra computations, both to solve triangular linear systems of equations as well as to compute factorizations with triangular matrices, such as Cholesky, LU, and QR. Our algorithm achieves better theoretical scalability than known alternatives, while maintaining numerical stability, via selective use of triangular matrix inversion. We leverage the fact that triangular inversion and matrix multiplication are more parallelizable than the standard TRSM algorithm. By only inverting triangular blocks along the diagonal of the initial matrix, we generalize the usual way of TRSM computation and the full matrix inversion approach. This flexibility leads to an efficient algorithm for any ratio of the number of right hand sides to the triangular matrix dimension. We provide a detailed communication cost analysis for our algorithm as well as for the recursive triangular matrix inversion. This cost analysis makes it possible to determine optimal block sizes and processor grids a priori. Relative to the best known algorithms for TRSM, our approach can require asymptotically fewer messages, while performing optimal amounts of computation and communication in terms of words sent.