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An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs

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Abstract

We present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. Our approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume central-upwind scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem are presented.

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Acknowledgments

The work of A. Chertock was supported in part by the NSF Grants DMS-0712898 and DMS-1115682. The work of M. Herty was supported by the DAAD 54365630, 55866082, EXC128. The work of A. Kurganov was supported in part by the NSF Grant DMS-1115718 and the German Research Foundation DFG under the Grant No. INST 247/609-1. The authors also acknowledge the support by NSF RNMS Grant DMS-1107444.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Alina Chertock

  2. Department of Mathematics, RWTH Aachen University, Templergraben 55, 52056, Aachen, Germany

    Michael Herty

  3. Mathematics Department, Tulane University, New Orleans, LA, 70118, USA

    Alexander Kurganov

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  1. Alina Chertock
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  2. Michael Herty
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Correspondence to Michael Herty.

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Chertock, A., Herty, M. & Kurganov, A. An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs. Comput Optim Appl 59, 689–724 (2014). https://doi.org/10.1007/s10589-014-9655-y

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