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February  2016, 9(1): 109-123. doi: 10.3934/dcdss.2016.9.109

A dynamic domain decomposition for the eikonal-diffusion equation

Simone Cacace 1, and  Maurizio Falcone 1,

1. 

Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy, Italy

Received  September 2014 Revised  February 2015 Published  December 2015

We propose a parallel algorithm for the numerical solution of the eikonal-diffusion equation, by means of a dynamic domain decomposition technique. The new method is an extension of the patchy domain decomposition method presented in [5] for first order Hamilton-Jacobi-Bellman equations. Using the connection with stochastic optimal control theory, the semi-Lagrangian scheme underlying the original method is modified in order to deal with (possibly degenerate) diffusion. We show that under suitable relations between the discretization parameters and the diffusion coefficient, the parallel computation on the proposed dynamic decomposition can be faster than that on a static decomposition. Some numerical tests in dimension two are presented, in order to show the features of the proposed method.
Keywords: Hamilton-Jacobi equations, eikonal-diffusion equation, semi-Lagrangian schemes, Domain decomposition, stochastic optimal control..
Mathematics Subject Classification: Primary: 65N99, 65N55; Secondary: 49L2.
Citation: Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
References:
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S. Cacace and M. Falcone, A dynamic domain decomposition for a class of second order semi-linear equations,, in preparation, ().   Google Scholar

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show all references

References:
[1]

Birkhäuser, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[2]

SIAM Journal of Control and Optimization, 28 (1990), 950-965. doi: 10.1137/0328053.  Google Scholar

[3]

in Analysis and Optimization of Systems (eds. A. Bensoussan, J.-L. Lions), Lecture Notes in Control and Information Sciences, 144, Springer-Verlag, 1990, 103-112. doi: 10.1007/BFb0120033.  Google Scholar

[4]

in System Modeling and Optimization (eds. C. Pötzsche, et al.), 443, Springer, 2014, 74-84. doi: 10.1007/978-3-662-45504-3_7.  Google Scholar

[5]

SIAM Journal on Scientific Computing, 34 (2012), 2625-2649. doi: 10.1137/110841576.  Google Scholar

[6]

S. Cacace and M. Falcone, A dynamic domain decomposition for a class of second order semi-linear equations,, in preparation, ().   Google Scholar

[7]

Mathematical Modelling and Numerical Analysis, 29 (1995), 97-122.  Google Scholar

[8]

in Domain Decomposition methods in Scientific and Engineering Computing (eds. D. E. Keyes and J. Xu), Contemporary Mathematics, 180, AMS, 1994, 477-483. doi: 10.1090/conm/180/02008.  Google Scholar

[9]

SIAM, 2014.  Google Scholar

[10]

Applied Numerical Mathematics, 15 (1994), 207-218. doi: 10.1016/0168-9274(94)00017-4.  Google Scholar

[11]

Oxford University Press, 1999.  Google Scholar

[12]

Springer, 2008. Google Scholar

[13]

Proc. Natl. Acad. Sci. USA, 93 (1996), 1591-1595. doi: 10.1073/pnas.93.4.1591.  Google Scholar

[14]

Cambridge University Press, 1999.  Google Scholar

[15]

SIAM J. Numer. Anal., 41 (2003), 325-363. doi: 10.1137/S0036142901392742.  Google Scholar

[16]

SIAM J. Numer. Anal., 41 (2003), 673-694. doi: 10.1137/S0036142901396533.  Google Scholar

[17]

IEEE Trans. Autom. Control, 40 (1995), 1528-1538. doi: 10.1109/9.412624.  Google Scholar

[18]

Math. Comp., 74 (2005), 603-627. doi: 10.1090/S0025-5718-04-01678-3.  Google Scholar

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