Cache-Aware Matrix Polynomials

Efficient solvers for partial differential equations are among the most important areas of algorithmic research in high-performance computing. In this paper we present a new optimization for solving linear autonomous partial differential equations. Our approach is based on polynomial approximations for exponential time integration, which involves the computation of matrix polynomial terms (M(p)v) in every time step. This operation is very memory intensive and requires targeted optimizations. In our approach, we exploit the cache-hierarchy of modern computer architectures using a temporal cache blocking approach over the matrix polynomial terms. We develop two single-core implementations realizing cache blocking over several sparse matrix-vector multiplications of the polynomial approximation and compare it to a reference method that performs the computation in the traditional iterative way. We evaluate our approach on three different hardware platforms and for a wide range of different matrices and demonstrate that our approach achieves time savings of up to 50% for a large number of matrices. This is especially the case on platforms with large caches, significantly increasing the performance to solve linear autonomous differential equations.