Numerical aspects of large-time optimal control of Burgers equation

Allahverdi, Navid; Pozo, Alejandro; Zuazua, Enrique

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 5, pp. 1371-1401.

In this paper, we discuss the efficiency of various numerical methods for the inverse design of the Burgers equation, both in the viscous and in the inviscid case, in long time-horizons. Roughly, the problem consists in, given a final desired target, to identify the initial datum that leads to it along the Burgers dynamics. This constitutes an ill-posed backward problem. We highlight the importance of employing a proper discretization scheme in the numerical approximation of the equation under consideration to obtain an accurate approximation of the optimal control problem. Convergence in the classical sense of numerical analysis does not suffice since numerical schemes can alter the dynamics of the underlying continuous system in long time intervals. As we shall see, this may end up affecting the efficiency on the numerical approximation of the inverse design, that could be polluted by spurious high frequency numerical oscillations. To illustrate this, two well-known numerical schemes are employed: the modified Lax−Friedrichs scheme (MLF) and the Engquist−Osher (EO) one. It is by now well-known that the MLF scheme, as time tends to infinity, leads to asymptotic profiles with an excess of viscosity, while EO captures the correct asymptotic dynamics. We solve the inverse design problem by means of a gradient descent method and show that EO performs robustly, reaching efficiently a good approximation of the minimizer, while MLF shows a very strong sensitivity to the selection of cell and time-step sizes, due to excess of numerical viscosity. The achieved numerical results are confirmed by numerical experiments run with the open source nonlinear optimization package (IPOPT).

Classification : 49M, 35Q35, 65M06
Mots clés : Burgers equation, inverse design, optimization, numerics, descent method

P.-A. Absil, R. Mahony and B. Andrews, Convergence of the iterates of descent methods for analytic functions. SIAM J. Optim. 16 (2005) 531–547. | DOI | MR | Zbl

J.J. Alonso and M.R. Colonno, Multidisciplinary optimization with applications to sonic-boom minimization. Ann. Rev. Fluid Mech. 44 (2012) 505–526. | DOI | MR | Zbl

H. Attouch, J. Bolte and B.F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Progr. 137 (2013) 91–129. | DOI | MR | Zbl

C. Bardos and O. Pironneau, Data assimilation for conservation laws. Methods Appl. Anal. 12 (2005) 103–134. | DOI | MR | Zbl

F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal.: Theory, Methods Appl. 32 (1998) 891–933. | DOI | MR | Zbl

F. Bouchut and F. James, Differentiability with respect to initial data for a scalar conservation law. In Hyperbolic Problems: Theory, Numerics, Applications, edited by M. Fey and R. Jeltsch. Vol. 129 of International Series of Numerical Mathematics. Springer-Verlag (1999) 113–118. | MR | Zbl

Y. Brenier and S. Osher, The discrete one-sided Lipschitz condition for convex scalar conservation laws. SIAM J. Numerical Anal. 25 (1988) 8–23. | DOI | MR | Zbl

A. Bressan and A. Marson, A variational calculus for discontinuous solutions of systems of conservation laws. Commun. Partial Differ. Eq. 20 (1995) 1491–1552. | DOI | MR | Zbl

C. Castro, F. Palacios and E. Zuazua, An alternating descent method for the optimal control of the inviscid Burgers’ equation in the presence of shocks. Math. Models Methods Appl. Sci. 18 (2008) 369–416. | DOI | MR | Zbl

C. Castro, F. Palacios and E. Zuazua, Optimal control and vanishing viscosity for the Burgers equation. In Chapter 7 of Integral Methods in Science and Engineering, edited by C. Costanda and M.E. Pérez. Birkhäuser Verlag 2 (2010) 65–90. | MR | Zbl

Ph.G. Ciarlet, Introduction to numerical linear algebra and optimisation. Vol. 2 of Cambridge Texts in Applied Mathematics. Cambridge University Press (1989). | MR | Zbl

R.O. Cleveland, Propagation of sonic booms through a real, stratified atmosphere. Ph.D. thesis, University of Texas at Austin (1995).

S. Ervedoza and E. Zuazua, Numerical Approximation of Exact Controls for Waves. Springer Briefs in Mathematics. Springer-Verlag (2013). | MR | Zbl

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case. Adv. Differ. Equ. 5 (2000) 465–514. | MR | Zbl

R. Fourer, D.M. Gay and B.W. Kernighan, A modeling language for mathematical programming. Manag. Sci. 36 (1990) 519–554. | DOI | Zbl

M. Ghil and P. Malanotte-Rizzoli, Data assimilation in meteorology and oceanography. Vol. 33 of Advances in Geophysics. Academic Press (1991).

M.B. Giles, Discrete adjoint approximations with shocks. In Hyperbolic Problems: Theory, Numerics, Applications, edited by Th.Y. Hou and E. Tadmor. Springer-Verlag (2003) 185–194. | MR | Zbl

M.B. Giles and N.A. Pierce, An introduction to the adjoint approach to design. Turbulence and Combustion 65 (2000) 393–415. | DOI | Zbl

M.B. Giles and S. Ulbrich, Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 1: linearized approximations and linearized output functionals. SIAM J. Numer. Anal. 48 (2010) 882–904. | DOI | MR | Zbl

M.B. Giles and S. Ulbrich, Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions. SIAM J. Numer. Anal. 48 (2010) 905–921. | DOI | MR | Zbl

R. Glowinski, J.-Louis Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach. Vol. 117 of Encyclopedia of Mathematics and its Applications. Cambridge University Press (2008). | MR | Zbl

E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Number 3 in Mathematiques & Applications. Ellipses (1991). | MR | Zbl

L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients. Math. Comput. 69 (2000) 987–1015. | DOI | MR | Zbl

A. Haraux, Some applications of the Łojasiewicz gradient inequality. Commun. Pure Appl. Anal. 11 (2012) 2417–2427. | DOI | MR | Zbl

E. Hopf, The partial differential equation $u_t+u{u}_x=\mu u_{xx}$. Commun. Pure Appl. Math. 3 (1950) 201–230. | DOI | MR | Zbl

L.I. Ignat, A. Pozo and E. Zuazua, Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws. Math. Comput. 84 (2015) 1633–1662. | DOI | MR | Zbl

F. James and M. Sepúlveda, Convergence results for the flux identification in a scalar conservation law. SIAM J. Control Optim. 37 (1999) 869–891. | DOI | MR | Zbl

F. James and N. Vauchelet, A remark on duality solutions for some weakly nonlinear scalar conservation laws. C. R. Acad. Sci. 349 (2011) 657–661. | MR | Zbl

Y.-J. Kim and A.E. Tzavaras, Diffusive N-waves and metastability in the Burgers equation. SIAM J. Math. Anal. 33 (2001) 607–633. | DOI | MR | Zbl

Y. Li, S. Osher and R. Tsai, Heat source identification based on $l_1$ constrained minimization. Inverse Probl. Imaging 8 (2014) 199–221. | DOI | MR | Zbl

J.-Louis Lions and B. Malgrange, Sur l’unicité rétrograde dans les problèmes mixtes paraboliques. Math. Scand. 8 (1960) 277–286. | DOI | MR | Zbl

T.-P. Liu and M. Pierre, Source-solutions and asymptotic behavior in conservation laws. J. Diff. Equ. 51 (1984) 419–441. | DOI | MR | Zbl

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications. Commun. Pure Appl. Anal. 8 (2010) 685–702. | DOI | MR | Zbl

J. Nocedal and S.J. Wright, Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2nd edition. Springer-Verlag (2006). | MR | Zbl

S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms. Habilitation Thesis, Fakultät für Mathematik, Technische Universität München (2001).

S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41 (2002) 740–797. | DOI | MR | Zbl

A. Wächter and L.T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Progr. 106 (2006) 25–57. | DOI | MR | Zbl

G.B. Whitham, Linear and nonlinear waves. John Wiley & Sons (1974). | MR | Zbl

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243. | DOI | MR | Zbl