Optimal control of flow with discontinuities

Abstract

7Optimal control of the 1-D Riemann problem of Euler equations is studied, with the initial values for pressure and 8 density as control parameters. The least-squares type cost functional employs either distributed observations in time or 9 observations calculated at the end of the assimilation window. Existence of solutions for the optimal control problem is 10 proven. Smooth and nonsmooth optimization methods employ the numerical gradient (respectively, a subgradient) of 11 the cost functional, obtained from the adjoint of the discrete forward model. The numerical flow obtained with the 12 optimal initial conditions obtained from the nonsmooth minimization matches very well with the observations. The 13 algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the 14 longer time horizon, except when the observations corresponding to shocks are detected and removed. 20Optimal control methods are presently employed for various applications in different fields: aerody-21 namics, meteorology, acoustics, financial mathematics and chemistry to mention but a few. Since the vast 22 majority of applications consist of the minimization of a cost functional derived from continuous models, 23 we have solved an optimization problem involving a cost functional for a discontinuous model. Our results 24 show that, for the example considered, nonsmooth optimization methods provide very good results in 25 combination with the adjoint method for subgradient computation. 26Nondifferentiable optimization algorithms employing subgradients were introduced following the sem-27 inal work of Lemarechal [45] (e.g. [11,46,53,55,66] to cite but a few).

Analysis of Discrete Adjoints for Upwind Numerical Schemes

“…Giles studied the construction and properties of discrete adjoints for hyperbolic systems with shocks [4,5]. Homescu and Navon [6] discuss the optimal control of flows with discontinuities.…”

Relaxation approaches to the optimal control of the Euler equations

“…The solution of the optimal control problem (4) poses problems in practice due to shock waves that can occur in the solution of the flow equation (1) [30,26]. To overcome these problems, two semi-linear approximations of the Euler equations (1) namely the Lattice Boltzmann approximation and the relaxation approximation are applied.…”

“…One can derive, formally, using standard techniques, the adjoint system related to the optimization of Euler flows. The result is a backward linear system of conservation laws in the adjoint variables [26,30].…”

“…Given a set of measurements on some actual flow at time T, the best estimate for the initial state that leads to the observed flow is determined. This problem has been explored before by many authors [30,28,29], but here, the linearization of the flow equation as described above and the underlying derivation of the adjoint variables for the computation of the gradient of the objective functional is used. An example taken from [34] is considered.…”

Impact of non‐smooth observation operators on variational and sequential data assimilation for a limited‐area shallow‐water equation model

“…MLEF has been tested with the shallow-water equations in , Uzunoglu et al (2007) and Fletcher and Zupanski (2008) and non-smooth variational data assimilation problems have been investigated in Makela and Neittaanmaki (1992), Homescu and Navon (2003), Zhang et al (2000), Zhu et al (2002) and Bardos and Pironneau (2005) for highly simplified problems. However, the existing optimization algorithms at the time those studies were conducted were not suitable for large-scale non-convex optimization (Haarala et al, 2004).…”