Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group

Yves Achdou; Italo Capuzzo-Dolcetta

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 4, page 565-591
  • ISSN: 0764-583X

Abstract

top
We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like h where h is the mesh step. Such an estimate is similar to those available in the Euclidean geometrical setting. The theoretical results are tested numerically on some examples for which semi-analytical formulas for the computation of geodesics are known. Other simulations are presented, for both steady and unsteady problems.

How to cite

top

Achdou, Yves, and Capuzzo-Dolcetta, Italo. "Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group." ESAIM: Mathematical Modelling and Numerical Analysis 42.4 (2008): 565-591. <http://eudml.org/doc/250427>.

@article{Achdou2008,
abstract = { We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like $\sqrt\{h\}$ where h is the mesh step. Such an estimate is similar to those available in the Euclidean geometrical setting. The theoretical results are tested numerically on some examples for which semi-analytical formulas for the computation of geodesics are known. Other simulations are presented, for both steady and unsteady problems. },
author = {Achdou, Yves, Capuzzo-Dolcetta, Italo},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Degenerate Hamilton-Jacobi equation; Heisenberg group; finite difference schemes; error estimates.; degenerate Hamilton Jacobi equation; error estimates; numerical examples; viscosity solution; geodesics},
language = {eng},
month = {5},
number = {4},
pages = {565-591},
publisher = {EDP Sciences},
title = {Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group},
url = {http://eudml.org/doc/250427},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Achdou, Yves
AU - Capuzzo-Dolcetta, Italo
TI - Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/5//
PB - EDP Sciences
VL - 42
IS - 4
SP - 565
EP - 591
AB - We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like $\sqrt{h}$ where h is the mesh step. Such an estimate is similar to those available in the Euclidean geometrical setting. The theoretical results are tested numerically on some examples for which semi-analytical formulas for the computation of geodesics are known. Other simulations are presented, for both steady and unsteady problems.
LA - eng
KW - Degenerate Hamilton-Jacobi equation; Heisenberg group; finite difference schemes; error estimates.; degenerate Hamilton Jacobi equation; error estimates; numerical examples; viscosity solution; geodesics
UR - http://eudml.org/doc/250427
ER -

References

top
  1. Y. Achdou and N. Tchou, A finite difference scheme on a non commutative group. Numer. Math.89 (2001) 401–424.  Zbl1008.65074
  2. M. Bardi, A boundary value problem for the minimum-time function. SIAM J. Control Optim.27 (1989) 776–785.  Zbl0682.49034
  3. M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA (1997). With appendices by M. Falcone and P. Soravia.  Zbl0890.49011
  4. G. Barles and E.R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal.43 (2005) 540–558 (electronic).  Zbl1092.65077
  5. R. Beals, B. Gaveau and P.C. Greiner, Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. J. Math. Pures Appl.79 (2000) 633–689.  Zbl0959.35035
  6. A. Bellaïche and J.-J. Risler, Eds., Sub-Riemannian Geometry, Progress in Mathematics144. Birkhäuser Verlag, Basel (1996).  
  7. I. Birindelli and J. Wigniolle, Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Commun. Pure Appl. Anal.2 (2003) 461–479.  Zbl1043.35025
  8. R.W. Brockett, Control theory and singular Riemannian geometry, in New directions in applied mathematics (Cleveland, Ohio, 1980), Springer, New York (1982) 11–27.  
  9. I. Capuzzo Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl. Math. Optim.10 (1983) 367–377.  Zbl0582.49019
  10. I. Capuzzo Dolcetta, The Hopf-Lax solution for state dependent Hamilton-Jacobi equations (Viscosity solutions of differential equations and related topics) (Japanese). Sūrikaisekikenkyūsho Kōkyūroku1287 (2002) 143–154.  
  11. I. Capuzzo Dolcetta, The Hopf solution of Hamilton-Jacobi equations, in Elliptic and parabolic problems (Rolduc/Gaeta, 2001), World Sci. Publishing, River Edge, NJ (2002) 343–351.  
  12. I. Capuzzo Dolcetta, A generalized Hopf-Lax formula: analytical and approximations aspects, in Geometric Control and Nonsmooth Analysis, F. Ancona, A. Bressan, P. Cannarsa, F. Clarkeă and P.R. Wolenski Eds., Series on Advances in Mathematics for Applied Sciences76, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2008).  Zbl1195.35105
  13. I. Capuzzo Dolcetta and H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory. Appl. Math. Optim.11 (1984) 161–181.  
  14. M.G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations. Math. Comp.43 (1984) 1–19.  Zbl0556.65076
  15. A. Cutrí and F. Da Lio, Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations. ESAIM: COCV13 (2007) 484–502.  Zbl1125.70013
  16. B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws. Math. Comp.36 (1981) 321–351.  Zbl0469.65067
  17. M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory. Appl. Math. Optim.15 (1987) 1–13.  Zbl0715.49023
  18. M. Falcone and R. Ferretti, Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations. Numer. Math.67 (1994) 315–344.  Zbl0791.65046
  19. S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev.43 (2001) 89–112 (electronic).  Zbl0967.65098
  20. A. Harten, B. Engquist, S. Osher and S.R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes. III. J. Comput. Phys.71 (1987) 231–303.  Zbl0652.65067
  21. A. Korányi and H.M. Reimann, Quasiconformal mappings on the Heisenberg group. Invent. Math.80 (1985) 309–338.  Zbl0567.30017
  22. N.V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients. Probab. Theory Relat. Fields117 (2000) 1–16.  Zbl0971.65081
  23. N.V. Krylov, The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients. Appl. Math. Optim.52 (2005) 365–399.  Zbl1087.65100
  24. J.J. Manfredi and B. Stroffolini, A version of the Hopf-Lax formula in the Heisenberg group. Comm. Partial Diff. Eq.27 (2002) 1139–1159.  Zbl1080.49023
  25. S. Osher and J.A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys.79 (1988) 12–49.  Zbl0659.65132
  26. S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal.28 (1991) 907–922.  Zbl0736.65066
  27. J.A. Sethian, Level set methods and fast marching methods, Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, Cambridge Monographs on Applied and Computational Mathematics3. Cambridge University Press, Cambridge, 2nd edition (1999).  Zbl0973.76003

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.