GPU Acceleration of Communication Avoiding Chebyshev Basis Conjugate Gradient Solver for Multiphase CFD Simulations

Flynn’s Reconciliation

“…For example, one of the pillars of scientific simulations in the peta-, respectively, exascale eras are communication-avoiding algorithms that scale well over thousands of nodes. One prominent case is the block-Jacobi preconditioner [2] that has recently been augmented with mixed precision [4]. Computing and using this preconditioner requires kernels (2)(3)(4).…”

“…One prominent case is the block-Jacobi preconditioner [2] that has recently been augmented with mixed precision [4]. Computing and using this preconditioner requires kernels (2)(3)(4). As one more expensive alternative to Jacobi preconditioning, incomplete block-matrix factorizations [50] require kernel (1) to perform Schur downdates.…”

“…Line 8 leads to [0, 0, 1, 1]. To avoid write conflicts, we split the loop progressively into index sets [[0], [1,2], [3]]. As long as there are no dependencies on earlier iterations of this loop, the split index set can be optimized by reordering iterations, leading to [[0, 2], [1,3]].…”

“…These kernels are executed on large batches of relatively small input problems, e.g., a batched dense matrix multiplication in BERT-like networks. Besides ML, batched kernels are used in, e.g., sparse linear algebra [2] or integer optimization [6]. Still, when using operations that are not part of vendor's libraries inside a computational graph on accelerators, users have to implement these high-performance kernels themselves, either manually, using architecture-specific compilers [43] or code synthesizers [52] for different architectures.…”