A previous knowledge of the domains of dependence of a Hamilton–Jacobi equation can be useful in its study and approximation. Information of this nature is, in general, difficult to obtain directly from the data of the problem. In this paper we formally introduce the concept of an independent sub-domain, discuss its main properties and provide a constructive implicit representation formula. Through these results, we propose an algorithm for the approximation of these sets that is shown to be relevant in the numerical resolution, via parallel computing.
Accepté le :
DOI : 10.1051/m2an/2015070
Mots clés : Hamilton–Jacobi equations, viscosity solutions, numerical approximation, parallel computing, domain decomposition
@article{M2AN_2016__50_4_1223_0,
author = {Festa, Adriano},
title = {Reconstruction of independent sub-domains for a class of {Hamilton{\textendash}Jacobi} equations and application to parallel computing},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1223--1240},
publisher = {EDP-Sciences},
volume = {50},
number = {4},
year = {2016},
doi = {10.1051/m2an/2015070},
mrnumber = {3535237},
zbl = {1347.49044},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2015070/}
}
TY - JOUR AU - Festa, Adriano TI - Reconstruction of independent sub-domains for a class of Hamilton–Jacobi equations and application to parallel computing JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1223 EP - 1240 VL - 50 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015070/ DO - 10.1051/m2an/2015070 LA - en ID - M2AN_2016__50_4_1223_0 ER -
%0 Journal Article %A Festa, Adriano %T Reconstruction of independent sub-domains for a class of Hamilton–Jacobi equations and application to parallel computing %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1223-1240 %V 50 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015070/ %R 10.1051/m2an/2015070 %G en %F M2AN_2016__50_4_1223_0
Festa, Adriano. Reconstruction of independent sub-domains for a class of Hamilton–Jacobi equations and application to parallel computing. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1223-1240. doi : 10.1051/m2an/2015070. http://archive.numdam.org/articles/10.1051/m2an/2015070/
and , Patchy vector fields and asymptotic stabilization. ESAIM: COCV 4 (1999) 445–471. | Numdam | MR | Zbl
and , An approximation scheme for the minimum time function. SIAM J. Control Optim. 28 (1990) 950–965. | DOI | MR | Zbl
M. Bardi and I.C.-Dolcetta, Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Systems and Control: Foundations and Applications. Birkhäuser, Boston (1997). | MR | Zbl
M. Bardi, M. Falcone and P. Soravia, Numerical methods for pursuit-evasion games via viscosity solutions. In Stochastic and differential games. Birkhauser, Boston (1999) 105–175. | MR | Zbl
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Vol. 17 of Math. Appl. Springer, Paris (1994). | MR | Zbl
S. Cacace, E. Cristiani and M. Falcone, A local ordered upwind method for Hamilton–Jacobi and Isaacs equations. In Proc. of 18th IFAC World Congress (2011) 6800–6805.
, , and , A patchy dynamic programming scheme for a class of Hamilton–Jacobi–Bellman equations. SIAM J. Sci. Comput. 34 (2012) A2625–A2649. | DOI | MR | Zbl
F. Camilli, M. Falcone, P. Lanucara and A. Seghini, A domain decomposition method for bellman equations. In Vol. 180, Domain Decomposition methods in Scientific and Engineering Computing. Contemp. Math. (1994) 477–483. | MR | Zbl
P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton–Jacobi equations, and optimal control. In Vol. 58 Progress Nonlin. Differ. Eq. Appl. Birkhäuser, Boston, MA (2004). | MR | Zbl
, , and , Convergence of a generalized fast-marching method for an eikonal equation with a velocity-changing sign. SIAM J. Numer. Anal. 46 (2008) 2920–2952. | DOI | MR | Zbl
, and , A parallel fast sweeping method for the eikonal equation. J. Comput. Phys. 237 (2013) 46–55. | DOI | MR
and , Approximate solutions of the Bellman equation of deterministic control theory. Appl. Math. Optim. 11 (1984) 161–181. | DOI | MR | Zbl
and , The existence of value in differential games, Bull. Amer. Math. Soc. 78 (1972) 427–431. | DOI | MR | Zbl
, A numerical approach to the infinite horizon problem of deterministic control theory. Appl. Math. Optim. 15 (1987) 1–13. | DOI | MR | Zbl
and , Convergence analysis for a class of high-order semi-Lagrangian advection schemes. SIAM J. Numer. Anal. 35 (1998) 909–940. | DOI | MR | Zbl
M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton–Jacobi Equations. Appl. Math. SIAM, Philadelphia (2014). | MR
A. Festa and R.B. Vinter, A decomposition technique for pursuit evasion games with many pursuers. 52nd IEEE Control and Decision Conference CDC (2013) 5797–5802.
and , Decomposition of differential games with multiple targets. J. Optim. Theory Appl. 169 (2016) 848–875. | DOI | MR | Zbl
C. Navasca and A.J. Krener, Patchy solutions of Hamilton–Jacobi-Bellman partial differential equations. In Vol. 364 of Modeling, Estimation and Control. Lect. Notes Control Inform. Sci., edited by A. Chiuso, A. Ferrante, and S. Pinzoni. Springer, Berlin (2007) 251–270. | MR | Zbl
, Estimates of convergence of fully discrete schemes for the Isaacs equation of pursuit-evasion differential games via maximum principle. SIAM J. Control Optim. 36 (1998) 1–11. | DOI | MR | Zbl
, Domain decomposition algorithms for solving hamilton–jacobi–bellman equations. Numer. Func. Anal. Optim. 14 (1993) 145–166. | DOI | MR | Zbl
A. Valli and A. Quarteroni, Domain decomposition methods for partial differential equations. Oxford University Press, Oxford (1999). | MR | Zbl
R.B. Vinter, Optimal Control Theory. Birkhäuser, Boston Heidelberg (2000). | MR
, Parallel implementations of the fast sweeping method. J. Comput. Math. 25 (2007) 421–429. | MR
and , A new domain decomposition method for an hjb equation. J. Comput. Appl. Math. 159 (2003) 195–204. | DOI | MR | Zbl
Cité par Sources :
