Conservation law constrained optimization based upon front-tracking

Martin Gugat; Michaël Herty; Axel Klar; Gunter Leugering

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2006)

  • Volume: 40, Issue: 5, page 939-960
  • ISSN: 0764-583X

Abstract

top
We consider models based on conservation laws. For the optimization of such systems, a sensitivity analysis is essential to determine how changes in the decision variables influence the objective function. Here we study the sensitivity with respect to the initial data of objective functions that depend upon the solution of Riemann problems with piecewise linear flux functions. We present representations for the one–sided directional derivatives of the objective functions. The results can be used in the numerical method called Front-Tracking.

How to cite

top

Gugat, Martin, et al. "Conservation law constrained optimization based upon front-tracking." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 40.5 (2006): 939-960. <http://eudml.org/doc/245702>.

@article{Gugat2006,
abstract = {We consider models based on conservation laws. For the optimization of such systems, a sensitivity analysis is essential to determine how changes in the decision variables influence the objective function. Here we study the sensitivity with respect to the initial data of objective functions that depend upon the solution of Riemann problems with piecewise linear flux functions. We present representations for the one–sided directional derivatives of the objective functions. The results can be used in the numerical method called Front-Tracking.},
author = {Gugat, Martin, Herty, Michaël, Klar, Axel, Leugering, Gunter},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {sensitivity calculus; front-tracking; conservation laws; conservation law; sensitivity analysis; Riemann problem; Front-Tracking; optimization},
language = {eng},
number = {5},
pages = {939-960},
publisher = {EDP-Sciences},
title = {Conservation law constrained optimization based upon front-tracking},
url = {http://eudml.org/doc/245702},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Gugat, Martin
AU - Herty, Michaël
AU - Klar, Axel
AU - Leugering, Gunter
TI - Conservation law constrained optimization based upon front-tracking
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2006
PB - EDP-Sciences
VL - 40
IS - 5
SP - 939
EP - 960
AB - We consider models based on conservation laws. For the optimization of such systems, a sensitivity analysis is essential to determine how changes in the decision variables influence the objective function. Here we study the sensitivity with respect to the initial data of objective functions that depend upon the solution of Riemann problems with piecewise linear flux functions. We present representations for the one–sided directional derivatives of the objective functions. The results can be used in the numerical method called Front-Tracking.
LA - eng
KW - sensitivity calculus; front-tracking; conservation laws; conservation law; sensitivity analysis; Riemann problem; Front-Tracking; optimization
UR - http://eudml.org/doc/245702
ER -

References

top
  1. [1] A. Aw and M. Rascle, Resurrection of second order models of traffic flow? SIAM J. Appl. Math. 60 (2000) 916–938. Zbl0957.35086
  2. [2] A. Bressan, Hyperbolic Systems of Conservation Laws. Oxford University Press, Oxford (2000). Zbl0997.35002MR1816648
  3. [3] C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38 (1972) 33–41. Zbl0233.35014
  4. [4] K. Ehrhardt and M. Steinbach, Nonlinear optimization in gas networks, in Modeling, Simulation and Optimization of Complex Processes, H.G. Bock, E. Kostina, H.X. Phu, R. Ranacher Eds. (2005) 139–148. Zbl1069.90014
  5. [5] M. Gugat, Optimal nodal control of networked hyperbolic systems: Evaluation of derivatives. AMO Advanced Modeling and Optimization 7 (2005) 9–37. Zbl1165.49307
  6. [6] M. Gugat, G. Leugering, K. Schittkowski and E.J.P.G. Schmidt, Modelling, stabilization and control of flow in networks of open channels, in Online optimization of large scale systems, M. Grötschel, S.O. Krumke, J. Rambau Eds., Springer (2001) 251–270. Zbl0987.93056
  7. [7] M. Gugat, G. Leugering and E.J.P.G. Schmidt, Global controllability between steady supercritical flows in channel networks. Math. Meth. Appl. Sci. (2003) 781–802. Zbl1047.93028
  8. [8] M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks. J. Optim. Theory Appl. 126 (2005) 589–616. Zbl1079.49024
  9. [9] D. Helbing, Verkehrsdynamik. Springer-Verlag, Berlin, Heidelberg, New York (1997). 
  10. [10] R. Holdahl, H. Holden and K.-A. Lie, Unconditionally stable splitting methods for the shallow water equations. BIT 39 (1999) 451–472. Zbl0945.76059
  11. [11] H. Holden and L. Holden, On scalar conservation laws in one-dimension, in Ideas and Methods in Mathematical Analysis, Stochastics and Applications S. Albeverio, J. Fenstad, H. Holden, T. Lindstrøm Eds. (1992) 480–509. Zbl0851.65064
  12. [12] H. Holden and N.H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. 26 (1995) 999–1017. Zbl0833.35089
  13. [13] H. Holden and N.H. Risebro, Front tracking for hyperbolic conservation laws. Springer, New York, Berlin, Heidelberg (2002). Zbl1006.35002MR1912206
  14. [14] H. Holden, L. Holden and R. Hoegh-Krohn, A numerical method for first order nonlinear scalar conservation laws in one-dimension. Comput. Math. Anal. 15 (1988) 595–602. Zbl0658.65085
  15. [15] S.N. Kruzkov, First order quasi linear equations in several independent variables. Math. USSR Sbornik, 10 (1970) 217–243. 
  16. [16] R.J. LeVeque, Numerical methods for conservation laws. Birkhäuser Verlag, Basel, Boston, Berlin (1990). Zbl0723.65067MR1077828
  17. [17] M.J. Lighthill and J.B. Whitham, On kinematic waves. Proc. Roy. Soc. Lond. A229 (1955) 281–345. Zbl0064.20905
  18. [18] J. Smoller, Shock waves and reaction diffusion equations. Springer, New York, Berlin, Heidelberg (1994). Zbl0508.35002MR1301779
  19. [19] S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41 (2002) 740–797. Zbl1019.49026
  20. [20] S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Control Lett. 3 (2003) 309–324. Zbl1157.49306

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.