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All issues Volume 30 (2024) ESAIM: COCV, 30 (2024) 16 References
Open Access
Issue
ESAIM: COCV
Volume 30, 2024
Article Number 16
Number of page(s) 53
DOI https://doi.org/10.1051/cocv/2023091
Published online 08 March 2024
  1. C. Aarset, M. Holler and T. T. N. Nguyen, Learning-informed parameter identification in nonlinear time-dependent PDEs. Appl. Math. Optim. 88 (2023) 53. [CrossRef] [Google Scholar]
  2. S.L. Brunton, J.L. Proctor and J.N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Nat. Acad. Sci. U.S.A. 113 (2016) 3932–3937. [Google Scholar]
  3. S. Court and K. Kunisch, Design of the monodomain model by artificial neural networks. Discrete Contin. Dyn. Syst. 42 (2022) 6031–6061. [CrossRef] [MathSciNet] [Google Scholar]
  4. G. Dong, M. Hintermüller and K. Papafitsoros, A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives. arXiv:2210.07900v1, 2022. [Google Scholar]
  5. G. Dong, M. Hintermüller and K. Papafitsoros, Optimization with learning-informed differential equation constraints and its applications. ESAIM Control Optim. Calc. Var. 28 (2022) 3. [CrossRef] [EDP Sciences] [Google Scholar]
  6. G. Dong, M. Hintermüller, K. Papafitsoros and K. Völkner, First-order conditions for the optimal control of learning-informed nonsmooth PDEs. arXiv:2206.00297v2, 2022. [Google Scholar]
  7. B. Kaltenbacher and T.T.N. Nguyen, Discretization of parameter identification in PDEs using neural networks. Inverse Problems 38 (2022) 124007. [CrossRef] [Google Scholar]
  8. T. Qin, K. Wu and D. Xiu, Data driven governing equations approximation using deep neural networks. J. Comput. Phys. 395 (2019) 620–635. [CrossRef] [MathSciNet] [Google Scholar]
  9. S.H. Rudy, S.L. Brunton, J.L. Proctor and J.N. Kutz, Data-driven discovery of partial differential equations. Sci. Adv. 3 (2017) e1602614. [CrossRef] [Google Scholar]
  10. C. Christof and J. Kowalczyk, On the omnipresence of spurious local minima in certain neural network training problems. Constr. Approx (2023) to appear. [Google Scholar]
  11. D.N. Hào, B.V. Huong, P.X. Thanh and D. Lesnic, Identification of nonlinear heat transfer laws from boundary observations. Appl. Anal. 94 (2015) 1784–1799. [CrossRef] [MathSciNet] [Google Scholar]
  12. A. Rösch, Identification of nonlinear heat transfer laws by optimal control. Num. Funct. Anal. Optim. 15 (1994) 417–434. [CrossRef] [Google Scholar]
  13. A. Rösch, Fréchet differentiability of the solution of the heat equation with respect to a nonlinear boundary condition. Z. Anal. Anwend. 15 (1996) 603–618. [CrossRef] [Google Scholar]
  14. A. Rösch, Second order optimality conditions and stability estimates for the identification of nonlinear heat transfer laws, in Control and Estimation of Distributed Parameter Systems, edited by W. Desch, F. Kappel, and K. Kunisch. Birkhäuser Basel, Basel (1998) 237–246. [CrossRef] [Google Scholar]
  15. A. Rösch, A Gauss–Newton method for the identification of nonlinear heat transfer laws, in Optimal Control of Complex Structures, edited by K.-H. Hoffmann, I. Lasiecka, G. Leugering, J. Sprekels, and F. Tröltzsch. Birkhäuser Basel, Basel (2002) 217–230. [Google Scholar]
  16. A. Rösch and F. Tröltzsch, An optimal control problem arising from the identification of nonlinear heat transfer laws. Arch. Control Sci. 1 (1992) 4–183. [Google Scholar]
  17. A. Rösch, Stability estimates for the identification of nonlinear heat transfer laws. Inverse Problems 12 (1996) 743–756. [CrossRef] [MathSciNet] [Google Scholar]
  18. S. Grützner and A. Muntean, Identifying processes governing damage evolution in quasi-static elasticity, part 1 – analysis. Adv. Math. Sci. Appl. 30 (2021) 305–334. [MathSciNet] [Google Scholar]
  19. V. Barbu and K. Kunisch, Identification of nonlinear elliptic equations. Appl. Math. Optim,. 33 (1996) 139–167. [CrossRef] [Google Scholar]
  20. V. Barbu, P. Neittaanmäki and A. Niemistö, A penalty method for the identification of nonlinear elliptic differential operator. Numer. Funct. Anal. Optim. 15 (1994) 503–530. [CrossRef] [MathSciNet] [Google Scholar]
  21. M. Goebel, Smooth and nonsmooth optimal Lipschitz control – a model problem, in Variational Calculus, Optimal Control and Applications, edited by W. H. Schmidt, K. Heier, L. Bittner, and R. Bulirsch. Birkhäuser Basel (1998) 53–60. [CrossRef] [Google Scholar]
  22. B. Kaltenbacher and W. Rundell, The inverse problem of reconstructing reaction-diffusion systems. Inverse Problems 36 (2020) 065011. [CrossRef] [MathSciNet] [Google Scholar]
  23. Y. Kian, Lipschitz and Hölder stable determination of nonlinear terms for elliptic equations. Nonlinearity 36 (2023) 1302. [CrossRef] [MathSciNet] [Google Scholar]
  24. C. Christof, C. Meyer, S. Walther and C. Clason, Optimal control of a non-smooth semilinear elliptic equation. Math. Control Relat. Fields 8 (2018) 247–276. [Google Scholar]
  25. J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000). [Google Scholar]
  26. B. Schweizer, Partielle Differentialgleichungen. Springer-Verlag, Berlin/Heidelberg (2013). [CrossRef] [Google Scholar]
  27. V.I. Bogachev, Measure Theory. Springer (2007). [CrossRef] [Google Scholar]
  28. H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces. SIAM, Philadelphia (2006). [Google Scholar]
  29. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer (2001). [CrossRef] [Google Scholar]
  30. G.A. Monteiro, A. Slavík and M. Tvrdý, Kurzweil-Stieltjes Integral: Theory and Applications. Vol. 15 of Series in Real Analysis. World Scientific, Singapore (2019). [Google Scholar]
  31. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford & New York (2000). [Google Scholar]
  32. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985). [Google Scholar]
  33. F. Tröltzsch, Optimal Control of Partial Differential Equations. AMS (2010). [Google Scholar]
  34. W.P. Ziemer, Weakly Differentiable Functions. Springer Verlag, New York (1989). [CrossRef] [Google Scholar]
  35. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Vol. 31 of Classics in Applied Mathematics. SIAM (2000). [Google Scholar]
  36. V. Barbu, Optimal Control of Variational Inequalities. Research Notes in Mathematics. Pitman (1984). [Google Scholar]
  37. M. Josephy, Composing functions of bounded variation. Proc. Amer. Math. Soc 83 (1981) 354–356. [CrossRef] [MathSciNet] [Google Scholar]
  38. C. Christof, Sensitivity Analysis of Elliptic Variational Inequalities of the First and the Second Kind. Ph.D. thesis, Technische Universität Dortmund (2018). [Google Scholar]
  39. B.S. Mityagin, The zero set of a real analytic function. Math. Notes 107 (2020) 529–530. [CrossRef] [MathSciNet] [Google Scholar]
  40. C. Christof and G. Müller, Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation. ESAIM Control Optim. Calc. Var. 27 (2021) Art. S13. [Google Scholar]
  41. C. Clason, V.H. Nhu and A. Rösch, No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation. ESAIM Control Optim. Calc. Var. 27 (2021) Art. 62. [CrossRef] [EDP Sciences] [Google Scholar]
  42. P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations. Birkhauser Verlag (2007). [Google Scholar]
  43. A. Beck, Introduction to Nonlinear Optimization. MOS/SIAM Series on Optimization. SIAM (2014). [CrossRef] [Google Scholar]
  44. C. Christof, Gradient-based solution algorithms for a class of bilevel optimization and optimal control problems with a nonsmooth lower level. SIAM J. Optim. 30 (2020) 290–318. [CrossRef] [MathSciNet] [Google Scholar]
  45. M.B. Cohen, J. Diakonikolas and L. Orecchia, On acceleration with noise-corrupted gradients, in Proceedings of the 35th International Conference on Machine Learning (PMLR 80). Stockholm, Sweden (2018) 1019–1028. [Google Scholar]
  46. L. Hertlein and M. Ulbrich, An inexact bundle algorithm for nonconvex nonsmooth minimization in Hilbert space. SIAM J. Control Optim,. 57 (2019) 3137–3165. [CrossRef] [Google Scholar]
  47. M. Hinze and A. Rösch, Discretization of optimal control problems, in Constrained Optimization and Optimal Control for Partial Differential Equations, edited by G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich and S. Ulbrich. Springer Basel, Basel (2012) 391–430. [CrossRef] [Google Scholar]
  48. G. Stadler, Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009). [Google Scholar]

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