Advanced Krylov subspace methods are investigated for the solution of large sparse linear systems arising from stiff adjoint-based aerodynamic shape optimization problems. A special attention is paid to the flexible inner-outer GMRES strategy combined with most relevant preconditioning and deflation techniques. The choice of this specific class of Krylov solvers for challenging problems is based on its outstanding convergence properties. Typically in our implementation the efficiency of the preconditioner is enhanced with a domain decomposition method with overlapping. However, maintaining the performance of the preconditioner may be challenging since scalability and efficiency of a preconditioning technique are properties often antagonistic to each other. In this paper we demonstrate how flexible inner-outer Krylov methods are able to overcome this critical issue. A numerical study is performed considering either a Finite Volume (FV), or a high-order Discontinuous Galerkin (DG) discretization which affect the arithmetic intensity and memory-bandwith of the algebraic operations. We consider test cases of transonic turbulent flows with RANS modelling over the two-dimensional supercritical OAT15A airfoil and the three-dimensional ONERA M6 wing. Benefits in terms of robustness and convergence compared to standard GMRES solvers are obtained. Strong scalability analysis shows satisfactory results. Based on these representative problems a discussion of the recommended numerical practices is proposed.

翻译：针对基于伴随的刚性气动形状优化问题中产生的大型稀疏线性系统，本文研究了高级Krylov子空间方法。特别关注了柔性内外GMRES策略与最相关的预条件和去flation技术的结合。基于该类Krylov求解器在挑战性问题中的出色收敛特性，选择了这一特定求解器族。在实际实现中，通常采用重叠区域分解方法增强预条件器的效率。然而，由于预条件技术的可扩展性与效率往往相互矛盾，保持预条件器性能可能具有挑战性。本文论证了柔性内外Krylov方法如何克服这一关键问题。通过采用有限体积法（FV）或高阶间断伽辽金法（DG）离散格式（二者分别影响代数运算的计算强度与内存带宽），开展了数值研究。测试案例包含基于RANS模型的跨声速湍流流动，涉及二维超临界OAT15A翼型与三维ONERA M6机翼。与标准GMRES求解器相比，该方法在稳健性与收敛性方面获得显著提升。强可扩展性分析显示结果令人满意。基于这些代表性问题的分析，提出了推荐的数值实践方案。