Efficient Solution of Adjoint Linear Systems for High-Fidelity Aerodynamic Shape Optimization

Core Concepts

Flexible inner-outer Krylov subspace methods combined with spectral deflation and advanced preconditioning techniques can efficiently solve large, sparse and ill-conditioned adjoint linear systems arising from high-fidelity computational fluid dynamics models.

Abstract

The content presents a comparative study of inner-outer Krylov solvers for the efficient solution of large, sparse and ill-conditioned linear systems that arise in adjoint-based aerodynamic shape optimization problems. Key highlights: The authors investigate the use of flexible inner-outer GMRES (FGMRES) solvers, which allow for variable preconditioning, in conjunction with spectral deflation techniques to address the challenges posed by the stiff adjoint systems. Two different discretization schemes are considered - a finite volume (FV) method and a high-order discontinuous Galerkin (DG) method. This affects the arithmetic intensity and memory-bandwidth requirements of the linear algebra operations. For the FV case, the authors compare the performance of FGMRES preconditioned by LU-SGS and BILU(0) applied to either an approximate or an exact Jacobian matrix. Deflation is shown to be crucial to overcome the stagnation of the standard GMRES solver. For the DG case, the BILU(0) preconditioner combined with a Restricted Additive Schwarz (RAS) domain decomposition method is used. Deflation provides a slight improvement in convergence. Strong scalability analysis demonstrates satisfactory parallel efficiency for both the FV and DG cases, with the DG case exhibiting some memory-bandwidth limitations. The authors discuss the recommended numerical practices based on the representative test cases.

Stats

The number of nonzero entries (NNZ) of the preconditioner matrices are: For the FV case: JEX_O1: 28 million NNZ J_APP_O1: 21 million NNZ For the DG case: J_EX_O3: 288 million NNZ

Quotes

"Typically in our implementation the efficiency of the preconditioner is enhanced with a domain decomposition method with overlapping." "Maintaining the performance of the preconditioner may be challenging since scalability and efficiency of a preconditioning technique are properties often antagonistic to each other." "We demonstrate how flexible inner-outer Krylov methods are able to overcome this critical issue."

Key Insights Distilled From

Comparative study of inner-outer Krylov solvers for linear systems in structured and high-order unstructured CFD problems

by Mehd... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.17870.pdf

Deeper Inquiries

How would the performance of the FGMRES-DR solver be affected if the turbulence model was not fully linearized

The performance of the FGMRES-DR solver would be affected if the turbulence model was not fully linearized. When the turbulence model is not fully linearized, the resulting adjoint system of equations becomes more ill-conditioned, leading to a more challenging numerical problem. In such cases, the convergence of the solver may be slower, and the number of iterations required to reach a solution may increase. The deflation strategy employed in the FGMRES-DR solver may not be as effective in accelerating convergence when dealing with highly ill-conditioned systems. Additionally, the accuracy of the gradient computation, which depends on the accuracy of the solution of the linear system, may be compromised, affecting the overall optimization process.

What other preconditioning techniques could be explored to further improve the scalability of the solver, especially for the memory-bandwidth limited DG case

To further improve the scalability of the solver, especially for the memory-bandwidth limited DG case, other preconditioning techniques could be explored. One approach could be to investigate domain decomposition methods with more advanced overlapping strategies to enhance the preconditioning effect on the entire fluid domain. Additionally, exploring hybrid preconditioning techniques that combine different types of preconditioners, such as algebraic and geometric methods, could help improve the scalability of the solver. Adaptive preconditioning techniques that dynamically adjust the preconditioner based on the characteristics of the problem could also be beneficial in optimizing the solver's performance for memory-bandwidth limited cases.

What are the potential applications of this efficient adjoint solver beyond aerodynamic shape optimization, such as in other areas of computational physics or engineering

The efficient adjoint solver, such as the FGMRES-DR solver, has potential applications beyond aerodynamic shape optimization in various areas of computational physics and engineering. Some potential applications include: Structural Optimization: The solver can be used in structural optimization problems to minimize material usage while maintaining structural integrity. Heat Transfer Optimization: In thermal engineering, the solver can be applied to optimize heat transfer systems for improved efficiency. Acoustic Optimization: In acoustics, the solver can help optimize designs to reduce noise levels in various engineering applications. Electromagnetic Optimization: The solver can be utilized in electromagnetic simulations to optimize antenna designs, electromagnetic shielding, and other applications. Biomedical Engineering: In biomedical engineering, the solver can aid in optimizing medical devices, prosthetics, and other healthcare technologies for better performance and patient outcomes.