Augmented Block-Arnoldi Recycling CFD Solvers

One of the limitations of recycled GCRO methods is the large amount of computation required to orthogonalize the basis vectors of the newly generated Krylov subspace for the approximate solution when combined with those of the recycle subspace. Recent advancements in low synchronization Gram-Schmidt and generalized minimal residual algorithms, Swirydowicz et al.~\cite{2020-swirydowicz-nlawa}, Carson et al. \cite{Carson2022}, and Lund \cite{Lund2022}, can be incorporated, thereby mitigating the loss of orthogonality of the basis vectors. An augmented Arnoldi formulation of recycling leads to a matrix decomposition and the associated algorithm can also be viewed as a {\it block} Krylov method. Generalizations of both classical and modified block Gram-Schmidt algorithms have been proposed, Carson et al.~\cite{Carson2022}. Here, an inverse compact $WY$ modified Gram-Schmidt algorithm is applied for the inter-block orthogonalization scheme with a block lower triangular correction matrix $T_k$ at iteration $k$. When combined with a weighted (oblique inner product) projection step, the inverse compact $WY$ scheme leads to significant (over 10$\times$ in certain cases) reductions in the number of solver iterations per linear system. The weight is also interpreted in terms of the angle between restart residuals in LGMRES, as defined by Baker et al.\cite{Baker2005}. In many cases, the recycle subspace eigen-spectrum can substitute for a preconditioner

Deflation and augmentation techniques in Krylov linear solvers

Preliminary version of the book chapter entitled "Deflation and augmentation techniques in Krylov linear solvers" published in "Developments in Parallel, Distributed, Grid and Cloud Computing for Engineering", ed. Topping, B.H.V and Ivanyi, P., Saxe-Coburg Publications, Kippen, Stirlingshire, United Kingdom, ISBN 978-1-874672-62-3, p. 249-275, 2013In this paper we present deflation and augmentation techniques that have been designed to accelerate the convergence of Krylov subspace methods for the solution of linear systems of equations. We review numerical approaches both for linear systems with a non-Hermitian coefficient matrix, mainly within the Arnoldi framework, and for Hermitian positive definite problems with the conjugate gradient method.Dans ce rapport nous présentons des techniques de déflation et d'augmentation qui ont été développées pour accélérer la convergence des méthodes de Krylov pour la solution de systémes d'équations linéaires. Nous passons en revue des approches pour des systémes linéaires dont les matrices sont non-hermitiennes, principalement dans le contexte de la méthode d'Arnoldi, et pour des matrices hermitiennes définies positives avec la méthode du gradient conjugué