On the numerical discretization of optimal control problems for conservation laws

• Autorzy: Schäfer Aguilar Paloma , Schmitt Johann Michael , Ulbrich Stefan , Moos Michael
• Języki publikacji: EN

Abstrakty

EN

We analyze the convergence of discretization schemes for the adjoint equation arising in the adjoint-based derivative computation for optimal control problems governed by entropy solutions of conservation laws. The difficulties arise from the fact that the correct adjoint state is the reversible solution of a transport equation with discontinuous coefficient and discontinuous end data. We derive the discrete adjoint scheme for monotone difference schemes in conservation form. It is known that convergence of the discrete adjoint can only be expected if the numerical scheme has viscosity of order O(h<supα</sup>) with appropriate 0 < α < 1, which leads to quite viscous shock profiles. We show that by a slight modification of the end data of the discrete adjoint scheme, convergence to the correct reversible solution can be obtained also for numerical schemes with viscosity of order O(h) and with sharp shock resolution. The theoretical findings are confirmed by numerical results.

Słowa kluczowe

EN

conservation laws; adjoint state; linear transport equation; discontinuous coefficients; finite difference schemes

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2019
• Tom: Vol. 48, No. 2
• Strony: 345--376
• Opis fizyczny: Bibliogr. 21 poz., rys., tab.

Twórcy

autor

Schäfer Aguilar Paloma

• Department of Mathematics, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany

autor

Schmitt Johann Michael

• Department of Mathematics, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany

autor

Ulbrich Stefan

• Department of Mathematics, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany

autor

Moos Michael

• Department of Mathematics, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany

Bibliografia

• Bardos, C. and Pironneau, O. (2005) Data assimilation for conservation laws. Methods Appl. Anal. 12(2), 103-134.
• Bouchut, F. and James, F. (1998) One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal. 32(7), 891-933. DOI 10.1016/S0362-546X(97)00536-1.
• Brenier, Y. and Osher, S. (1988) The discrete one-sided Lipschitz condition for convex scalar conservation laws. SIAM J. Numer. Anal. 25(1), 8-23.
• Castro, C., Palacios, F. and Zuazua, E. (2008) An alternating descent method for the optimal control ofthe inviscid Burgers equation in the presence of shocks. Math. Models Methods Appl. Sci. 18(3), 369-416.
• Crandall, M.G. and Majda, A. (1980) Monotone difference approximations for scalar conservation laws. Math. Comp. 34(149), 1-21.
• Engquist, B. and Osher, S. (1981) One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36(154), 321-351.
• Giles, M. and Ulbrich, S. (2010a) 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(3), 882-904. DOI10.1137/080727464.
• Giles, M. and Ulbrich, S. (2010b) Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: Adjoint approximations and extensions. SIAM J. Numer. Anal. 48(3), 905-921. DOI 10.1137/09078078X.
• Gosse, L. and James, F. (2000) Numerical approximations of onedimensional linear conservation equations with discontinuous coefficients. Math. Comp. 69(231), 987-1015.
• Hajian, S., Hintermuller, M. and Ulbrich, S. (2019) Total variation diminishing schemes in optimal control of scalar conservation laws. IMA J. Numer. Anal. 39(1), 105-140.
• Homescu, C. and Navon, I.M. (2003) Optimal control of flow with discontinuities. J. Comput. Phys. 187(2), 660-682.
• Kruzkov, S.N. (1970) First order quasilinear equations in several independent variables. Math. USSR Sb. 10(2), 217-243.
• Kuznecov, N.N. (1976) The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation. Z. Vycisl. Mat. i Mat. Fiz. 16(6), 1489-1502, 1627.
• Malek, J., Necas, J., Rokyta, M. and Ruzicka, M. (1996) Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London.
• Nessyahu, H. and Tadmor, E. (1992) The convergence rate of approximate solutions for nonlinear scalar conservation laws. SIAM J. Numer. Anal. 29(6), 1505-1519.
• Oleinik, O.A. (1963) Discontinuous solutions of non-linear differential equations. Amer. Math. Soc. Transl. (2) 26, 95-172.
• Pfaff, S. and Ulbrich, S. (2015) Optimal boundary control of nonlinear hyperbolic conservation laws with switched boundary data. SIAM J. Control and Optimization 53(3), 1250-1277. DOI 1137/140995799.
• Teng, Z.H. and Zhang, P. (1997) Optimal L1-rate of convergence for the viscosity method and monotone scheme to piecewise constant solutions with shocks. SIAM J. Numer. Anal. 34(3), 959-978.
• Ulbrich, S. (2001) Optimal control of nonlinear hyperbolic conservation laws with source terms. Habilitation, Zentrum Mathematik, Technische Universitat München, Germany.
• Ulbrich, S. (2002) A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41(3), 740-797. DOI 10.1137/S0363012900370764. URL http://dx.doi.org/10.1137/S0363012900370764
• Ulbrich, S. (2003) Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Systems Control Lett. 48(3-4), 313-328. DOI 10.1016/S0167- 6911(02)00275-X.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).