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

Recycling Krylov Subspaces for Efficient Partitioned Solution of Aerostructural Adjoint Systems

Robust and efficient solvers for coupled-adjoint linear systems are crucial to successful aerostructural optimization. Monolithic and partitioned strategies can be applied. The monolithic approach is expected to offer better robustness and efficiency for strong fluid-structure interactions. However, it requires a high implementation cost and convergence may depend on appropriate scaling and initialization strategies. On the other hand, the modularity of the partitioned method enables a straightforward implementation while its convergence may require relaxation. In addition, a partitioned solver leads to a higher number of iterations to get the same level of convergence as the monolithic one. The objective of this paper is to accelerate the fluid-structure coupled-adjoint partitioned solver by considering techniques borrowed from approximate invariant subspace recycling strategies adapted to sequences of linear systems with varying right-hand sides. Indeed, in a partitioned framework, the structural source term attached to the fluid block of equations affects the right-hand side with the nice property of quickly converging to a constant value. We also consider deflation of approximate eigenvectors in conjunction with advanced inner-outer Krylov solvers for the fluid block equations. We demonstrate the benefit of these techniques by computing the coupled derivatives of an aeroelastic configuration of the ONERA-M6 fixed wing in transonic flow. For this exercise the fluid grid was coupled to a structural model specifically designed to exhibit a high flexibility. All computations are performed using RANS flow modeling and a fully linearized one-equation Spalart-Allmaras turbulence model. Numerical simulations show up to 39% reduction in matrix-vector products for GCRO-DR and up to 19% for the nested FGCRO-DR solver.Comment: 42 pages, 21 figure

A General Algorithm for Reusing Krylov Subspace Information. I. Unsteady Navier-Stokes

A general algorithm is developed that reuses available information to accelerate the iterative convergence of linear systems with multiple right-hand sides A x = b (sup i), which are commonly encountered in steady or unsteady simulations of nonlinear equations. The algorithm is based on the classical GMRES algorithm with eigenvector enrichment but also includes a Galerkin projection preprocessing step and several novel Krylov subspace reuse strategies. The new approach is applied to a set of test problems, including an unsteady turbulent airfoil, and is shown in some cases to provide significant improvement in computational efficiency relative to baseline approaches

Spectral recycling strategies for the solution of nonlinear eigenproblems in thermoacoustics

International audienceIn this work we consider the numerical solution of large nonlinear eigenvalue problems that arise in thermoacoustic simulations involved in the stability analysis of large combustion devices. We briefly introduce the physical modeling that leads to a non-linear eigenvalue problem that is solved using a nonlinear fixed point iteration scheme. Each step of this nonlinear method requires the solution of a complex non-Hermitian linear eigenvalue problem. We review a set of state of the art eigensolvers and discuss strategies to recycle spectral informations from one nonlinear step to the next. More precisely, we consider the Implicitly Restarted Arnoldi method, the Krylov-Schur solver and its block-variant as well as the subspace iteration method with Chebyshev acceleration. On a small test example we study the relevance of the different approaches and illustrate on a large industrial test case the performance of the parallel solvers best suited to recycle spectral information

Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

International audienceContemporary large-scale Partial Differential Equation (PDE) simulations usually require the solution of large and sparse linear systems. Moreover, it is often needed to solve these linear systems with different or multiple Right-Hand Sides (RHSs). In this paper, various strategies will be presented to extend the scalability of existing linear solvers using appropriate recycling strategies or block methods—i.e., by treating multiple right-hand sides simultaneously. The scalability of this work is assessed by performing simulations on up to 8,192 cores for solving linear systems arising from various physical phenomena modeled by Poisson's equation, the system of linear elasticity, or Maxwell's equation. This work is shipped as part of on open-source software, readily available and usable in any C, C++, or Python code. In particular, some simulations are performed on top of a well-established library, PETSc, and it is shown how our approaches can be used to decrease time to solution down by 30%