zbMATH — the first resource for mathematics

Reverse automatic differentiation for optimum design: from adjoint state assembly to gradient computation. (English) Zbl 1142.90524
Summary: Gradient descent is a key technique in optimal design problems. We describe a method to compute the gradient of an optimization criterion with respect to design parameters. This method is hybrid, using automatic differentiation to compute the residual of the adjoint system, and using this residual in a handwritten solver that computes the adjoint state and then the gradient. Automatic differentiation is used here in its so-called reverse mode, with a special refinement for gather-scatter loops. The handwritten solver uses a matrix-free algorithm, preconditioned by the first-order derivative of the flux function. This method was tested on a typical optimal design problem, for which we give validation and performance results.

90C90 Applications of mathematical programming
65K99 Numerical methods for mathematical programming, optimization and variational techniques
76N25 Flow control and optimization for compressible fluids and gas dynamics
Full Text: DOI
[1] Beux F., Notes on Numerical Fluid Mechanics 55 pp 115– (1994) · doi:10.1007/978-3-322-86570-0_8
[2] Corliss G., Automatic Differentiation of Algorithms. from Simulation to Optimization (2001)
[3] Dadone A., Computer and Fluids 29 pp 1– (2000) · Zbl 0955.76081 · doi:10.1016/S0045-7930(99)00002-X
[4] Desideri J.-A., SIAM J. Sci. Comput. pp 88– (1995) · Zbl 0821.65061 · doi:10.1137/0916007
[5] Francescatto J., International Journal for Numerical Methods in Fluids 26 pp 927– (1998) · Zbl 0928.76063 · doi:10.1002/(SICI)1097-0363(19980430)26:8<927::AID-FLD679>3.0.CO;2-0
[6] Giering R., Technical report, 1997, in: Tangent linear and adjoint model compiler, users manual (1997)
[7] Gilbert J.C., Optimization Methods and Software 1 pp 13– (1992) · doi:10.1080/10556789208805503
[8] Giles, M.B. Adjoint methods for aeronautical design. Proceedings of the ECCOMAS CFD Conference. 2001. In
[9] Griewank A., Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation (2000) · Zbl 0958.65028
[10] Griewank A., Numerical Algorithms 30 (2) pp 113– (2002) · Zbl 1005.65025 · doi:10.1023/A:1016051717120
[11] Hascoet L., Research report 4167, in: The data-dependence graph of adjoint programs (2001)
[12] Hascoet L., Adjoining independent computations 2 pp 185– (2001)
[13] Hovland P., Technical Report MCS-P687-0997, in: Automatic differentiation of Navicr-stokes compulations (1997)
[14] Technical report, in: On-line documentation of the Tapenade AD tool
[15] Jameson A., Report 1824 MAE, in: Aerodynamic design via control theory (1988) · Zbl 0676.76055
[16] Marco N., Numerical optimizers for aerodynamic design using transonic finite-element solvers (1995)
[17] Mohammadi B., Von Karman Lecture Series, in: Practical application to fluid flows of automatic differentiation for design problems (1997)
[18] Saad Y., J. Sci. Comput 7 pp 856– (1986)
[19] Sevin C., Optimisation de formes en mécanique des fluides numérique (1999)
[20] Taásan S., Aerodynamic design and optimization in one shot (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.