An efficient hardware-aware matrix-free implementation for finite-element discretized matrix-multivector products

Abstract

The finite-element (FE) discretization of a partial differential equation usually involves the construction of a FE discretized operator and computing its action on trial FE discretized fields for the solution of a linear system of equations or eigenvalue problems and is traditionally computed using global sparse-vector multiplication modules. However, recent hardware-aware algorithms for evaluating such matrix-vector multiplications suggest that on-the-fly matrix-vector products without building and storing the cell-level dense matrices reduce both arithmetic complexity and memory footprint and are referred to as matrix-free approaches. These approaches exploit the tensor-structured nature of the FE polynomial basis for evaluating the underlying integrals and, the current state-of-the-art matrix-free implementations deal with the action of FE discretized matrix on a single vector. These are neither optimal nor readily applicable for matrix-multivector products involving a large number of vectors. We discuss a computationally efficient and scalable matrix-free implementation procedure to compute the FE discretized matrix-multivector products on multi-node CPU and GPU architectures. The accuracy and performance of our implementation is assessed on the problem of computing FE overlap (mass) matrix-multivector multiplications as a representative benchmark example, and we observe superior performance of our implementation. For instance, computational gains up to 2.9x on GPU architectures and 6x on CPU-only architectures are observed for matrix-multivector products compared to the FE cell-level matrix-multiplication approach for the case of 100 vectors. We further benchmark our performance against the matrix-free module in deal.II and speedups up to 2x are observed for multivectors on CPU-only architectures and 1.5x on GPUs.Comment: 5 pages, 7 figure