Abstract
In many applications one wishes to optimize designs on the basis of an established simulation tool. We consider the situation where “simulation” means solving a system of state equations by a fixed point iteration. “Optimization” may then be performed by appending an adjoint solver and an iteration step on the design variables. The main mathematical goal of this chapter is to quantify and estimate the retardation factor, i.e., the complexity of an optimization run compared to that of a single simulation, measured in terms of contraction rates. It is generally believed that the retardation factor should be bounded by a reasonably small number irrespective of discretization widths and other incidental quantities. We show that this is indeed the case for a simple elliptic control problem, when the state equations are solved by Jacobi or a multigrid V-cycle. Moreover, there is strong dependence on a regularization term. This is also shown to be true when the state equation is solved by Newton’s method and the projected Hessian is explicitly available
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
A. Brandt, Multi-Level Adaptive Solutions to Boundary-Value Problems, Math. Comp. 31 p. 333–390, (1977).
A. Brandt, Multigrid Techniques: Guide with Applications to Fluid Dynamics, GMD-Studien. no 85, St. Augustin, Germany, (1984).
A. Brandt and N. Dinar, Multigrid Solutions to Elliptic Flow Problems In: S.V. Parter (ed.), Numerical Methods for Partial Differential Equations, Academic Press, New York, (1979)
A. Brandt, S. McCormick and J. Ruge, Multigrid Methods for Differential Eigenproblems, SIAM J. Sci. Stat. Comput. 4(2):244–260, (1983).
G.H. Golub and J. M. Ortega, Scientific Computing And Differential Equations: An Introduction To Numerical Methods, Academic Press, Boston, (1991).
A. Griewank, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Society for Industrial and Applied Mathematics, Philadelphia-USA, (2000)
A. Griewank, Projected Hessians for Preconditioning in One-Step One-Shot Design Optimization, Large Scale Nonlinear Optimization, p. 151–171,(2006).
A. Hamdi and A. Griewank, Properties of an Augmented Lagrangian for Design Optimization, Optimization Methods and Software, 25(4):645–664, (2009).
A. Hamdi and A. Griewank, Reduced Quasi-Newton Method for Simultaneous Design and Optimization, Computational Optimization and Applications, Springer Netherlands, (2009).
M. Hinze, M. Köster, S. Turek: A Space-Time Multigrid Solver for Distributed Control of the Time-dependent Navier-Stokes System, Priority Programme 1253, Preprint-Nr.: SPP1253-16-02 (2008).
R.H.W. Hoppe, Multilevel Based All-at-once Methods in PDE constrai Optimization with Applications to Shape Optimization of Active Microfluidic Biochips DFG SPP 1253 Annual Meeting, Kloster Banz, 21–23.09.2008, (2008).
K. Ito, K. Kunisch, I. Gherman, V. Schulz, Approximate Nullspace Iterations for KKT Systems in Model Based Optimization, SIAM Journal on Matrix Analysis and Applications, 31:1835–1847 (2010)
A. Jameson. Multigrid Algorithms for Compressible Flow Calculations. In W. Hackbusch and U. Trottenberg, editors, Multigrid Methods II, volume 1228 of Lecture Notes in Mathematics, pages 166–201. Springer, 1986.
A. Jameson, W. Schmidt, and E. Turkel, Numerical solutions of the Euler Equation by Fnite Volume Methods Using Runge-Kutta time-step** schemes. AIAA 81–1259, 1981.
J. A. Nelder and R. A. Mead. A Simplex Method for Function Minimisation. Comput. J., 7:308–313, (1964).
R. C. Swanson and E. Turkel, Multistage Scheme with Multigrid for Euler and Navier-Stokes Equations (components and analysis), Technical Report 3631, NASA, (1997).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Griewank, A., Hamdi, A., Özkaya, E. (2013). Quantifying Retardation in Simulation Based Optimization. In: Chinchuluun, A., Pardalos, P., Enkhbat, R., Pistikopoulos, E. (eds) Optimization, Simulation, and Control. Springer Optimization and Its Applications, vol 76. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5131-0_6
Download citation
DOI: https://doi.org/10.1007/978-1-4614-5131-0_6
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-5130-3
Online ISBN: 978-1-4614-5131-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)