Dataflow acceleration of Krylov subspace sparse banded problems

Most of the efforts in the FPGA community related to sparse linear algebra focus on increasing the degree of internal parallelism in matrix-vector multiply kernels. We propose a parametrisable dataflow architecture presenting an alternative and complementary approach to support acceleration of banded sparse linear algebra problems which benefit from building a Krylov subspace. We use banded structure of a matrix A to overlap the computations Ax, A²x, ..., A^kx by building a pipeline of matrix-vector multiplication processing elements (PEs) each performing Aⁱx. Due to on-chip data locality, FLOPS rate sustainable by such pipeline scales linearly with k. Our approach enables trade-off between the number k of overlapped matrix power actions and the level of parallelism in a PE. We illustrate our approach for Google PageRank computation by power iteration for large banded single precision sparse matrices. Our design scales up to 32 sequential PEs with floating point accumulation and 80 PEs with fixed point accumulation on Stratix V D8 FPGA. With 80 single-pipe fixed point PEs clocked at 160Mhz, our design sustains 12.7 GFLOPS.