In this paper, we discuss strategies for computing subsets of eigenvectors of matrices corresponding to subdomains of finite element meshes achieving compromise between two contradicting goals. The subset of eigenvectors is required in the construction of coarse spaces used in algebraic multigrid methods (AMG) as well as in certain domain decomposition (DD) methods. The quality of the coarse spaces depends on the number of eigenvectors, which improves the approximation properties of the coarse space and impacts the overall performance and convergence of the associated AMG or DD algorithms. However, a large number of eigenvectors affects negatively the sparsity of the corresponding coarse matrices, which can become fairly dense. The sparsity of the coarse matrices can be controlled to a certain extent by the size of the subdomains (union of finite elements) referred to as agglomerates. If the size of the agglomerates is too large, then the cost of the eigensolvers increases and eventually can become unacceptable for the purpose of constructing the AMG or DD solvers. This paper investigates strategies to optimize the solution of the partial eigenproblems of interest. In particular, we examine direct and iterative eigensolvers for computing those subsets. Our experiments with a well-known model of an oil-reservoir simulation benchmark indicate that iterative eigensolvers can lead to significant improvements in the overall performance of an AMG solver that exploits such spectral construction of coarse spaces.

Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few fast algorithms have been efficiently implemented or used in practical applications. However, there exist many practical alternatives to Strassen's algorithm with varying performance and numerical properties. Fast algorithms are known to be numerically stable, but because their error bounds are slightly weaker than the classical algorithm, they are not used even in cases where they provide a performance benefit. We argue in this paper that the numerical sacrifice of fast algorithms, particularly for the typical use cases of practical algorithms, is not prohibitive, and we explore ways to improve the accuracy both theoretically and empirically. The numerical accuracy of fast matrix multiplication depends on properties of the algorithm and of the input matrices, and we consider both contributions independently. We generalize and tighten previous error analyses of fast algorithms and compare their properties. We discuss algorithmic techniques for improving the error guarantees from two perspectives: manipulating the algorithms, and reducing input anomalies by various forms of diagonal scaling. Finally, we benchmark performance and demonstrate our improved numerical accuracy.

We study the performance of a two-level algebraic-multigrid algorithm, with a focus on the impact of the coarse-grid solver on performance. We consider two algorithms for solving the coarse-space systems: the preconditioned conjugate gradient method and a new robust HSS-embedded low-rank sparse-factorization algorithm. Our test data comes from the SPE Comparative Solution Project for oil-reservoir simulations. We contrast the performance of our code on one 12-core socket of a Cray XC30 machine with performance on a 60-core Intel Xeon Phi coprocessor. To obtain top performance, we optimized the code to take full advantage of fine-grained parallelism and made it thread-friendly for high thread count. We also developed a bounds-and-bottlenecks performance model of the solver which we used to guide us through the optimization effort, and also carried out performance tuning in the solver’s large parameter space. As a result, significant speedups were obtained on both machines.