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Weak and strong form shape hessians and their automatic generation. (English) Zbl 1448.49036


MSC:

49M15 Newton-type methods
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
68T35 Theory of languages and software systems (knowledge-based systems, expert systems, etc.) for artificial intelligence
68Q42 Grammars and rewriting systems
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
68U15 Computing methodologies for text processing; mathematical typography
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[1] T. Albring, M. Sagebaum, and N. R. Gauger, A consistent and robust discrete adjoint solver for the SU\(^2\) framework—validation and application, Notes Numer. Fluid Mech. Multidiscip. Design, 132 (2016), pp. 77–86.
[2] M. Aln\ae s, A Compiler Framework for Automatic Linearization and Efficient Discretization of Nonlinear Partial Differential Equations, Ph.D. thesis, University of Oslo, 2009.
[3] M. S. Aln\ae s, A. Logg, K. B. Ø lgaard, M. E. Rognes, and G. N. Wells, Unified form language: A domain-specific language for weak formulations of partial differential equations, ACM Trans. Math. Software, 40 (2014), pp. 9:1–9:37.
[4] M. Berggren, A unified discrete-continuous sensitivity analysis method for shape optimization, in Applied and Numerical Partial Differential Equations, Comput. Methods Appl. Sci. 15, Springer, New York, 2010, pp. 25–39. · Zbl 1186.65078
[5] M. Blatt and P. Bastian, The Iterative Solver Template Library, Springer, Berlin, Heidelberg, 2007, pp. 666–675.
[6] C. Brandenburg, F. Lindemann, M. Ulbrich, and S. Ulbrich, Advanced Numerical Methods for PDE Constrained Optimization with Application to Optimal Design in Navier Stokes Flow, Springer, Basel, 2012, pp. 257–275. · Zbl 1356.49019
[7] A. Carnarius, F. Thiele, E. Özkaya, and N. Gauger, Adjoint Approaches for Optimal Flow Control, Report 2010-5088, AIAA, 2010.
[8] B. Christianson, Reverse accumulation and attractive fixed points, Optim. Methods Softw., 3 (1994), pp. 311–326.
[9] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd ed., Adv. Design Control 22, SIAM, Philadelphia, 2011, . · Zbl 1251.49001
[10] G. Doğan and R. H. Nochetto, First variation of the general curvature-dependent surface energy, ESAIM Math. Model. Numer. Anal., 46 (2012), pp. 59–79. · Zbl 1270.49042
[11] P. E. Farrell, D. A. Ham, S. W. Funke, and M. E. Rognes, Automated derivation of the adjoint of high-level transient finite element programs, SIAM J. Sci. Comput., 35 (2013), pp. C369–C393, . · Zbl 1362.65103
[12] H. Garcke, C. Hecht, M. Hinze, and C. Kahle, Numerical approximation of phase field based shape and topology optimization for fluids, SIAM J. Sci. Comput., 37 (2015), pp. A1846–A1871, . · Zbl 1322.35113
[13] N. Gauger, A. Walther, C. Moldenhauer, and M. Widhalm, Automatic Differentiation of an Entire Design Chain for Aerodynamic Shape Optimization, Springer, Berlin, Heidelberg, 2008, pp. 454–461.
[14] A. Griewank and G. F. Corliss, eds., Automatic Differentiation of Algorithms: Theory, Implementation, and Applications, SIAM, Philadelphia, 1991.
[15] A. Griewank and A. Walther, Algorithm \textup799: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation, ACM Trans. Math. Software, 26 (2000), pp. 19–45. · Zbl 1137.65330
[16] A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, SIAM, Philadelphia, 2008, . · Zbl 1159.65026
[17] M. Hintermüller and W. Ring, A second order shape optimization approach for image segmentation, SIAM J. Appl. Math., 64 (2003), pp. 442–467, . · Zbl 1073.68095
[18] A. Jameson, Aerodynamic design via control theory, J. Sci. Comput., 3 (1988), pp. 233–260. · Zbl 0676.76055
[19] A. Kowarz and A. Walther, Optimal checkpointing for time-stepping procedures in ADOL-C, in Computational Science – ICCS 2006, V. N. Alexandrov, G. D. van Albada, P. M. A. Sloot, and J. Dongarra, eds., Lecture Notes in Comput. Sci. 3994, Springer, New York, 2006, pp. 541–549.
[20] A. Logg, K.-A. Mardal, and G. Wells, eds., Automated Solution of Differential Equations by the Finite Element Method, Lect. Notes Comput. Sci. Eng. 84, Springer, Berlin, Heidelberg, 2012.
[21] C. Lozano, Discrete surprises in the computation of sensitivities from boundary integrals in the continuous adjoint approach to inviscid aerodynamic shape optimization, Comput. & Fluids, 56 (2012), pp. 118–127. · Zbl 1365.76100
[22] B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids, Numerical Mathematics and Scientific Computation, Clarendon Press, Oxford, 2001. · Zbl 0970.76003
[23] O. Pironneau, On optimum profiles in Stokes flow, J. Fluid Mech., 59 (1973), pp. 117–128. · Zbl 0274.76022
[24] O. Pironneau, On optimum design in fluid mechanics, J. Fluid Mech., 64 (1974), pp. 97–110. · Zbl 0281.76020
[25] F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange, F. Luporini, A. T. T. McRae, G. Bercea, G. R. Markall, and P. H. J. Kelly, Firedrake: Automating the Finite Element Method by Composing Abstractions, preprint, , 2015. · Zbl 1396.65144
[26] S. Schmidt, Efficient Large Scale Aerodynamic Design Based on Shape Calculus, Ph.D. thesis, University of Trier, Germany, 2010.
[27] S. Schmidt, C. Ilic, V. Schulz, and N. Gauger, Three dimensional large scale aerodynamic shape optimization based on shape calculus, AIAA J., 51 (2013), pp. 2615–2627.
[28] S. Schmidt and V. Schulz, Impulse response approximations of discrete shape Hessians with application in CFD, SIAM J. Control Optim., 48 (2009), pp. 2562–2580, . · Zbl 1387.49064
[29] S. Schmidt and V. Schulz, Shape derivatives for general objective functions and the incompressible Navier-Stokes equations, Control Cybernet., 39 (2010), pp. 677–713. · Zbl 1282.49033
[30] S. Schmidt, E. Wadbro, and M. Berggren, Large-scale three-dimensional acoustic horn optimization, SIAM J. Sci. Comput., 38 (2016), pp. B917–B940, . · Zbl 06646630
[31] V. Schulz and M. Siebenborn, Computational comparison of surface metrics for PDE constrained shape optimization, Comput. Methods Appl. Math., 16 (2016), pp. 485–497. · Zbl 1342.49065
[32] V. H. Schulz, A Riemannian view on shape optimization, Found. Comput. Math., 14 (2014), pp. 483–501. · Zbl 1296.49037
[33] J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin, Heidelberg, 1992.
[34] M. Sonntag, S. Schmidt, and N. Gauger, Shape derivatives for the compressible Navier–Stokes equations in variational form, J. Comput. Appl. Math., 296 (2016), pp. 334–351. · Zbl 1329.49093
[35] A. Walther and A. Griewank, Advantages of binomial checkpointing for memory-reduced adjoint calculations, in Numerical Mathematics and Advanced Applications – Proceedings of ENUMATH 2003, M. Feistauer, V. Dolejš\' \i, P. Knobloch, and K. Najzar, eds., Springer, New York, 2004, pp. 834–843, · Zbl 1056.65064
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