SpMV and BiCG-Stab optimization for a class of hepta-diagonal-sparse matrices on GPU

Abstract

The abundant data parallelism available in many-core GPUs has been a key interest to improve accuracy in scientific and engineering simulation. In many cases, most of the simulation time is spent in linear solver involving sparse matrix–vector multiply. In forward petroleum oil and gas reservoir simulation, the application of a stencil relationship to structured grid leads to a family of generalized hepta-diagonal solver matrices with some regularity and structural uniqueness. We present a customized storage scheme that takes advantage of generalized hepta-diagonal sparsity pattern and stencil regularity by optimizing both storage and matrix–vector computation. We also present an in-kernel optimization for implementing sparse matrix–vector multiply (SpMV) and biconjugate gradient stabilized (BiCG-Stab) solver. In-kernel is intended to avoid the multiple kernels invocation associated with the use of the numerical library operators. To keep in-kernel, a lock-free inter-block synchronization is used in which completing thread blocks are assigned some independent computations to avoid repeatedly polling the global memory. Other optimizations enable combining reductions and collective write operations to memory. The in-kernel optimization is particularly useful for the iterative structure of BiCG-Stab for preserving vector data locality and to avoid saving vector data back to memory and reloading on each kernel exit and re-entry. Evaluation uses generalized hepta-diagonal matrices that derives from a range of forward reservoir simulation’s structured grids. Results show the profitability of proposed generalized hepta-diagonal custom storage scheme over standard library storage like compressed sparse row, hybrid sparse, and diagonal formats. Using proposed optimizations, SpMV and BiCG-Stab have been noticeably accelerated compared to other implementations using multiple kernel exit–re-entry when the solver is implemented by invoking numerical library operators.