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

• Published in: Computational optimization and applications Vol. 59; no. 3; pp. 689 - 724
• Main Authors: Chertock, Alina, Herty, Michael, Kurganov, Alexander
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.12.2014; Springer Nature B.V
• Subjects: Adjoints; Balancing; Calculus; Computer science; Control theory; Convergence; Convex and Discrete Geometry; Eulers equations; Lagrange multiplier; Law; Management Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Methods; Nonlinearity; Numerical analysis; Operations Research; Operations Research/Decision Theory; Optimization; Ordinary differential equations; Partial differential equations; Statistics; Studies; Multidimensional hyperbolic partial differential equations; Optimal control; Numerical methods
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-014-9655-y

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.