arXiv subjects

Erin C. Carson

Publications and source records attributed to Erin C. Carson.

2 recordsLinked to original sources

arXiv2019

An Adaptive $s$-step Conjugate Gradient Algorithm with Dynamic Basis Updating

The adaptive $s$-step CG algorithm is a solver for sparse, symmetric positive definite linear systems designed to reduce the synchronization cost per iteration while still achieving a user-specified accuracy requirement. In this work, we improve the adaptive $s$-step conjugate gradient algorithm by use of iteratively updated estimates of the largest and smallest Ritz values, which give approximations of the largest and smallest eigenvalues of $A$, using a technique due to Meurant and Tich{\' y} [G. Meurant and P. Tich{\' y}, Numer. Algs. (2018), pp.~1--32]. The Ritz value estimates are used to dynamically update parameters for constructing Newton or Chebyshev polynomials so that the conditioning of the $s$-step bases can be continuously improved throughout the iterations. These estimates are also used to automatically set a variable related to the ratio of the sizes of the error and residual, which was previously treated as an input parameter. We show through numerical experiments that in many cases the new algorithm improves upon the previous adaptive $s$-step approach both in terms of numerical behavior and reduction in number of synchronizations.

math.NA↗

arXiv2019

Predict-and-recompute conjugate gradient variants

The standard implementation of the conjugate gradient algorithm suffers from communication bottlenecks on parallel architectures, due primarily to the two global reductions required every iteration. In this paper, we study conjugate gradient variants which decrease the runtime per iteration by overlapping global synchronizations, and in the case of pipelined variants, matrix-vector products. Through the use of a predict-and-recompute scheme, whereby recursively-updated quantities are first used as a predictor for their true values and then recomputed exactly at a later point in the iteration, these variants are observed to have convergence behavior nearly as good as the standard conjugate gradient implementation on a variety of test problems. We provide a rounding error analysis which provides insight into this observation. It is also verified experimentally that the variants studied do indeed reduce the runtime per iteration in practice and that they scale similarly to previously-studied communication-hiding variants. Finally, because these variants achieve good convergence without the use of any additional input parameters, they have the potential to be used in place of the standard conjugate gradient implementation in a range of applications.

math.NA↗