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Approximate linearization of fixed-point iterations

Error analysis of tangent and adjoint problems linearized about non-stationary points

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Abstract

Previous papers have shown the impact of partial convergence of discretized PDEs on the accuracy of tangent and adjoint linearizations. A series of papers suggested linearization of the fixed-point iteration used in the solution process as a means of computing the sensitivities rather than linearizing the discretized PDE, as the lack of convergence of the nonlinear problem indicates that the discretized form of the governing equations has not been satisfied. These works showed that the accuracy of an approximate linearization depends in part on the convergence of the nonlinear system. This work shows an error analysis of the impact of the approximate linearization and the convergence of the nonlinear problem for both the tangent and adjoint modes and provides a series of results for an exact Newton solver, an inexact Newton solver, and a low-storage explicit Runge-Kutta scheme to confirm the error analyses.

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Acknowledgements

Computing time was provided by ARCC on the Teton supercomputer.

Funding

This work was supported in part by NASA Grant NNX16AT23H and the NASA Graduate Aeronautics Scholars Program.

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Authors and Affiliations

  1. National Institute of Aerospace, 100 Exploration Way, Hampton, VA, 23666, USA

    Emmett Padway

  2. Department of Mechanical Engineering, University of Wyoming, 1000 E. University Ave, Laramie, WY, 82071, USA

    Dimitri Mavriplis

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  1. Emmett Padway
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  2. Dimitri Mavriplis
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Correspondence to Emmett Padway.

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Padway, E., Mavriplis, D. Approximate linearization of fixed-point iterations. Numer Algor 91, 583–614 (2022). https://doi.org/10.1007/s11075-022-01274-2

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