Performance of the Block Jacobi-Davidson Method for the Solution of Large Eigenvalue Problems on Modern Clusters

Abstract

We investigate a block Jacobi-Davidson method for computing a few exterior eigenpairs of a large sparse matrix. The block method typically requires more matrix-vector and vector-vector operations than the standard algorithm. However, this is more than compensated by the performance gains through better data reusage on modern CPUs, which we demonstrate by detailed performance engineering and numerical experiments. The key ingredients to achieving high performance consist in both kernel optimizations and a careful design of the algorithm that allows using blocked operations in most parts of the computation. We show the performance gains of the block algorithm with our hybrid parallel implementation for a variety of matrices on up to 5\,120 CPU cores. A new development we discuss in this context is a highly accurate and efficient block orthogonalization scheme that exploits modern hardware features and mixed precision arithmetic.