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

Abstract: 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.