Batched Kronecker product for 2-D matrices and 3-D arrays on NVIDIA GPUs

TL;DR

This paper introduces a fast, memory-efficient GPU implementation for batched Kronecker products on small matrices and arrays, optimized for finite element applications and outperforming generic GEMM-based methods.

Contribution

A specialized GPU algorithm for batched Kronecker products that is faster and more memory-efficient than existing GEMM-based approaches, tailored for small matrix sizes.

Findings

Achieves up to 285 GFlop/s for single precision on matrices of size 16

Faster and uses less memory than generic GEMM-based methods

Effective for finite element polynomial degrees up to size 16

Abstract

We describe an interface and an implementation for performing Kronecker product actions on NVIDIA GPUs for multiple small 2-D matrices and 3-D arrays processed in parallel as a batch. This method is suited to cases where the Kronecker product component matrices are identical but the operands in a matrix-free application vary in the batch. Any batched GEMM (General Matrix Multiply) implementation, for example ours [1] or the one in cuBLAS, can also be used for performing batched Kronecker products on GPUs. However, the specialized implementation presented here is faster and uses less memory. Partly this is because a simple GEMM based approach would require extra copies to and from main memory. We focus on matrix sizes less than or equal to 16, since these are the typical polynomial degrees in Finite Elements, but the implementation can be easily extended for other sizes. We obtain 143 and 285 GFlop/s for single precision real when processing matrices of size 10 and 16, respectively on NVIDIA Tesla K20c using CUDA 5.0. The corresponding speeds for 3-D array Kronecker products are 126 and 268 GFlop/s, respectively. Double precision is easily supported using the C++ template mechanism.