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Inexactly constrained discrete adjoint approach for steepest descent-based optimization algorithms

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Abstract

The problem of constrained optimization via the gradient-based discrete adjoint steepest descent method is studied under the assumption that the constraint equations are solved inexactly. Error propagation from the constraint equations to the gradient is studied analytically, as is the convergence rate of the inexactly constrained algorithm as it relates to the exact algorithm. A method is developed for adapting the residual tolerance to which the constraint equations are solved. The adaptive tolerance method is applied to two simple test cases to demonstrate the potential gains in computational efficiency.

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References

  1. Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific, Belmont (1996)

    MATH  Google Scholar 

  2. Giles, M.B., Pierce, N.A.: An introduction to the adjoint approach to design. Flow Turbulence and Combustion 65, 393–415 (2000)

    Article  MATH  Google Scholar 

  3. Hicken, J.E., Zingg, D.W.: Aerodynamic optimization algorithm with integrated geometry parameterization and mesh movement. AIAA J. 48(2), 401–413 (2010)

    Article  Google Scholar 

  4. Hirsch, C.: Numerical Computation of Internal and External Flows. Wiley, New York (1988)

    MATH  Google Scholar 

  5. Optimum aerodynamic design using CFD and control. AIAA-95-1729 (1995)

  6. Nadarajah, S.K., Jameson, A.: A comparison of the continuous and the discrete adjoint approach to automatic aerodynamic optimization. AIAA-2000-0667 (2000)

  7. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  8. Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. SIAM (1970)

  9. Pironneau, O.: On optimum design in fluid mechanics. J. Fluid Mech. 64, 97–110 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pulliam, T.H., Zingg, D.W.: Fundamental algorithms in computational fluid dynamics. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  11. Reuther, J., Jameson, A.: Control theory based airfoil design for potential flow and a finite volume discretization. AIAA-94-0499 (1994)

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Acknowledgments

The authors gratefully acknowledge funding from the National Sciences and Engineering Research Council of Canada (NSERC) as well as Bombardier Aerospace.

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Authors and Affiliations

  1. McGill University, Montreal, QC, Canada

    David A. Brown & Siva Nadarajah

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  1. David A. Brown
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  2. Siva Nadarajah
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Correspondence to David A. Brown.

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Brown, D.A., Nadarajah, S. Inexactly constrained discrete adjoint approach for steepest descent-based optimization algorithms. Numer Algor 78, 983–1000 (2018). https://doi.org/10.1007/s11075-017-0409-7

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  • DOI: https://doi.org/10.1007/s11075-017-0409-7

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