Free Access
Issue
ESAIM: M2AN
Volume 50, Number 5, September-October 2016
Page(s) 1371 - 1401
DOI https://doi.org/10.1051/m2an/2015076
Published online 04 August 2016
  1. 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. [CrossRef] [MathSciNet] [Google Scholar]
  2. J.J. Alonso and M.R. Colonno, Multidisciplinary optimization with applications to sonic-boom minimization. Ann. Rev. Fluid Mech. 44 (2012) 505–526. [CrossRef] [Google Scholar]
  3. 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. [Google Scholar]
  4. C. Bardos and O. Pironneau, Data assimilation for conservation laws. Methods Appl. Anal. 12 (2005) 103–134. [CrossRef] [MathSciNet] [Google Scholar]
  5. F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal.: Theory, Methods Appl. 32 (1998) 891–933. [CrossRef] [MathSciNet] [Google Scholar]
  6. 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. [Google Scholar]
  7. Y. Brenier and S. Osher, The discrete one-sided Lipschitz condition for convex scalar conservation laws. SIAM J. Numerical Anal. 25 (1988) 8–23. [CrossRef] [Google Scholar]
  8. A. Bressan and A. Marson, A variational calculus for discontinuous solutions of systems of conservation laws. Commun. Partial Differ. Eq. 20 (1995) 1491–1552. [CrossRef] [Google Scholar]
  9. 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. [CrossRef] [Google Scholar]
  10. 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. [Google Scholar]
  11. Ph.G. Ciarlet, Introduction to numerical linear algebra and optimisation. Vol. 2 of Cambridge Texts in Applied Mathematics. Cambridge University Press (1989). [Google Scholar]
  12. R.O. Cleveland, Propagation of sonic booms through a real, stratified atmosphere. Ph.D. thesis, University of Texas at Austin (1995). [Google Scholar]
  13. S. Ervedoza and E. Zuazua, Numerical Approximation of Exact Controls for Waves. Springer Briefs in Mathematics. Springer-Verlag (2013). [Google Scholar]
  14. E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case. Adv. Differ. Equ. 5 (2000) 465–514. [Google Scholar]
  15. R. Fourer, D.M. Gay and B.W. Kernighan, A modeling language for mathematical programming. Manag. Sci. 36 (1990) 519–554. [CrossRef] [Google Scholar]
  16. M. Ghil and P. Malanotte-Rizzoli, Data assimilation in meteorology and oceanography. Vol. 33 of Advances in Geophysics. Academic Press (1991). [Google Scholar]
  17. 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. [Google Scholar]
  18. M.B. Giles and N.A. Pierce, An introduction to the adjoint approach to design. Turbulence and Combustion 65 (2000) 393–415. [Google Scholar]
  19. 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. [CrossRef] [Google Scholar]
  20. 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. [CrossRef] [Google Scholar]
  21. 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). [Google Scholar]
  22. E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Number 3 in Mathematiques & Applications. Ellipses (1991). [Google Scholar]
  23. L. Gosse and F. James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients. Math. Comput. 69 (2000) 987–1015. [CrossRef] [MathSciNet] [Google Scholar]
  24. A. Haraux, Some applications of the Łojasiewicz gradient inequality. Commun. Pure Appl. Anal. 11 (2012) 2417–2427. [CrossRef] [MathSciNet] [Google Scholar]
  25. E. Hopf, The partial differential equation ut + uux = μuxx. Commun. Pure Appl. Math. 3 (1950) 201–230. [Google Scholar]
  26. 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. [CrossRef] [Google Scholar]
  27. F. James and M. Sepúlveda, Convergence results for the flux identification in a scalar conservation law. SIAM J. Control Optim. 37 (1999) 869–891. [CrossRef] [MathSciNet] [Google Scholar]
  28. 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. [CrossRef] [Google Scholar]
  29. Y.-J. Kim and A.E. Tzavaras, Diffusive N-waves and metastability in the Burgers equation. SIAM J. Math. Anal. 33 (2001) 607–633. [CrossRef] [MathSciNet] [Google Scholar]
  30. Y. Li, S. Osher and R. Tsai, Heat source identification based on l1 constrained minimization. Inverse Probl. Imaging 8 (2014) 199–221. [CrossRef] [MathSciNet] [Google Scholar]
  31. J.-Louis Lions and B. Malgrange, Sur l’unicité rétrograde dans les problèmes mixtes paraboliques. Math. Scand. 8 (1960) 277–286. [CrossRef] [MathSciNet] [Google Scholar]
  32. T.-P. Liu and M. Pierre, Source-solutions and asymptotic behavior in conservation laws. J. Diff. Equ. 51 (1984) 419–441. [Google Scholar]
  33. B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications. Commun. Pure Appl. Anal. 8 (2010) 685–702. [CrossRef] [MathSciNet] [Google Scholar]
  34. J. Nocedal and S.J. Wright, Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2nd edition. Springer-Verlag (2006). [Google Scholar]
  35. S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms. Habilitation Thesis, Fakultät für Mathematik, Technische Universität München (2001). [Google Scholar]
  36. 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. [CrossRef] [MathSciNet] [Google Scholar]
  37. 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. [CrossRef] [MathSciNet] [Google Scholar]
  38. G.B. Whitham, Linear and nonlinear waves. John Wiley & Sons (1974). [Google Scholar]
  39. E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you