# zbMATH — the first resource for mathematics

Automated calculation of higher order partial differential equation constrained derivative information. (English) Zbl 07105504

##### MSC:
 65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 68N20 Theory of compilers and interpreters 35 Partial differential equations
Full Text:
##### References:
 [1] M. S. Aln\aes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N. Wells, The FEniCS project version 1.5, Arch. Numer. Software, 3 (2015), pp. 9–23. [2] M. S. Aln\aes, A. Logg, K. B. Ølgaard, 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. · Zbl 1308.65175 [3] N. L. Baker and R. Daley, Observation and background adjoint sensitivity in the adaptive observation-targeting problem, Quart. J. Roy. Meteorological Soc., 126 (2000), pp. 1431–1454. [4] S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. Gropp, D. Karpeyev, D. Kaushik, M. Knepley, D. May, L. Curfman McInnes, R. Mills, T. Munson, K. Rupp, P. Sanan, B. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc Users Manual, Tech. Report ANL-95/11 Rev 3.9, Argonne National Laboratory, 2018. [5] S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, Efficient management of parallelism in object-oriented numerical software libraries, in Modern Software Tools for Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, eds., Birkhäuser, Boston, MA, 1997, pp. 163–202. · Zbl 0882.65154 [6] C. Bischof, A. Carle, G. Corliss, A. Griewank, and P. Hovland, ADIFOR—Generating derivative codes from Fortran programs, Scientific Programming, 1 (1992), pp. 1–29. [7] I. Charpentier and J. Utke, Fast higher-order derivative tensors with Rapsodia, Optim. Methods Softw., 24 (2009), pp. 1–14. · Zbl 1166.65009 [8] B. Christianson, Reverse accumulation and attractive fixed points, Optim. Methods Softw., 3 (1994), pp. 311–326. [9] A. Cioaca, A. Sandu, and E. de Sturler, Efficient methods for computing observation impact in $$4$$D-Var data assimilation, Comput. Geosci., 17 (2013), pp. 975–990. · Zbl 1393.86018 [10] K. Cuffey and W. S. B. Paterson, The Physics of Glaciers, 4th ed., Butterworth Heinemann, Oxford, UK, 2010. [11] D. N. Daescu, On the sensitivity equations of four-dimensional variational (4D-Var) data assimilation, Monthly Weather Review, 136 (2008), pp. 3050–3065. [12] L. D. Dalcin, R. R. Paz, P. A. Kler, and A. Cosimo, Parallel distributed computing using Python, Advances in Water Resources, 34 (2011), pp. 1124–1139. [13] T. A. Davis, Algorithm 832: UMFPACK V 4.3 — an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30 (2004), pp. 196–199. · Zbl 1072.65037 [14] R. D. Falgout and U. M. Yang, HYPRE: A library of high performance preconditioners, in Computational Science — ICCS 2002: International Conference, Part III, P. M. A. Sloot, C. J. K. Tan, J. J. Dongarra, and A. G. Hoekstra, eds., Lecture Notes in Comput. Sci. 2331, Springer-Verlag, Berlin, 2002, pp. 632–641. · Zbl 1056.65046 [15] P. E. Farrell and S. W. Funke, Libadjoint Manual, Version 1.6, Applied Modelling & Computation Group, Department of Earth Science and Engineering, Royal School of Mines, Imperial College London, 2018. [16] 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. [17] C. Geuzaine and J.-F. Remacle, GMSH: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79 (2009), pp. 1309–1331. · Zbl 1176.74181 [18] R. Giering and T. Kaminski, Recipes for adjoint code construction, ACM Trans. Math. Software, 24 (1998), pp. 437–474. · Zbl 0934.65027 [19] R. Giering and T. Kaminski, Applying TAF to generate efficient derivative code of Fortran 77–95 programs, Proc. Appl. Math. Mech., 2 (2003), pp. 54–57. [20] R. Giering, T. Kaminski, and T. Slawig, Generating efficient derivative code with TAF: Adjoint and tangent linear Euler flow around an airfoil, Future Generation Computer Systems, 21 (2005), pp. 1345–1355. [21] J. G. Gilbert, Automatic differentiation and iterative processes, Optim. Methods Softw., 1 (1992), pp. 13–21. [22] B. D. Goddard, A. Nold, and S. Kalliadasis, Dynamical density functional theory with hydrodynamic interactions in confined geometries, J. Chemical Phys., 145 (2016), 214106. [23] B. D. Goddard, A. Nold, N. Savva, P. Yatsyshin, and S. Kalliadasis, Unification of dynamic density functional theory for colloidal fluids to include inertia and hydrodynamic interactions: Derivation and numerical experiments, J. Phys. Condensed Matter, 25 (2013), 035101. [24] D. N. Goldberg, A variationally derived, depth-integrated approximation to a higher-order glaciologial flow model, J. Glaciology, 57 (2011), pp. 157–170. [25] D. N. Goldberg and P. Heimbach, Parameter and state estimation with a time-dependent adjoint marine ice sheet model, Crysosphere, 7 (2013), pp. 1659–1678. [26] D. N. Goldberg, S. H. K. Narayanan, L. Hascoet, and J. Utke, An optimized treatment for algorithmic differentiation of an important glaciological fixed-point problem, Geoscientific Model Development, 9 (2016), pp. 1891–1904. [27] A. Griewank, On automatic differentiation, in Mathematical Programming: Recent Developments and Applications, M. Iri and K. Tanabe, eds., Kluwer, Dordrecht, 1989, pp. 83–108. [28] A. Griewank, Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optim. Methods Softw., 1 (1992), pp. 35–54. [29] A. Griewank and A. Walther, Algorithm 799: 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 [30] A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed., SIAM, Philadelphia, 2008. · Zbl 1159.65026 [31] M. D. Gunzburger, Perspectives in Flow Control and Optimization, Adv. Design Control 5, SIAM, Philadelphia, 2003. [32] L. Hascoet and V. Pascual, The Tapenade automatic differentiation tool: Principles, model, and specification, ACM Trans. Math. Software, 39 (2013), pp. 20:1–20:43. · Zbl 1295.65026 [33] V. E. Henson and U. M. Yang, BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41 (2002), pp. 155–177. · Zbl 0995.65128 [34] V. Hernandez, J. E. Roman, and V. Vidal, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Software, 31 (2005), pp. 351–362. · Zbl 1136.65315 [35] V. Heuveline and A. Walther, Online checkpointing for parallel adjoint computation in PDEs: Application to goal-oriented adaptivity and flow control, in Euro-Par 2006 Parallel Processing: 12th International Euro-Par Conference, W. E. Nagel, W. V. Walter, and W. Lehner, eds., Lecture Notes in Comput. Sci. 4128, Springer-Verlag, Berlin, 2006, pp. 689–699. [36] T. Isaac, N. Petra, G. Stadler, and O. Ghattas, Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet, J. Comput. Phys., 296 (2015), pp. 348–368. · Zbl 1352.86017 [37] A. G. Kalmikov and P. Heimbach, A Hessian-based method for uncertainty quantification in global ocean state estimation, SIAM J. Sci. Comput., 36 (2014), pp. S267–S295. · Zbl 1311.35321 [38] N. Kukreja, J. Hückelheim, M. Lange, M. Louboutin, A. Walther, S. W. Funke, and G. Gorman, High-Level Python Abstractions for Optimal Checkpointing in Inversion Problems, , 2018. [39] F.-X. Le Dimet, H.-E. Ngodock, and B. Luong, Sensitivity analysis in variational data assimilation, J. Meteorological Soc. Japan, 75 (1997), pp. 245–255. [40] A. Logg, K.-A. Mardal, and G. N. Wells, eds., Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Lect. Notes Comput. Sci. Eng. 84, Springer-Verlag, Berlin, 2012. · Zbl 1247.65105 [41] D. R. MacAyeal, Large-scale ice flow over a viscous basal sediment: Theory and application to ice stream B, Antarctica, J. Geophys. Res. Solid Earth, 94 (1989), pp. 4071–4087. [42] J. R. Maddison and P. E. Farrell, Rapid development and adjoining of transient finite element models, Comput. Methods Appl. Mech. Engrg., 276 (2014), pp. 95–121. · Zbl 1423.76257 [43] S. K. Mitusch, An Algorithmic Differentiation Tool for FEniCS, Master’s thesis, University of Oslo, 2018. [44] L. W. Morland, Unconfined ice-shelf flow, in Dynamics of the West Antarctic Ice Sheet, C. J. V. der Veen and J. Oerlemans, eds., Reidel, Dordrecht, 1987, pp. 99–116. [45] U. Naumann, The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation, Software Environ. Tools 24, SIAM, Philadelphia, 2012. · Zbl 1275.65015 [46] T. E. Oliphant, Python for scientific computing, Comput. Sci. Eng., 9 (2007), pp. 10–20. [47] F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange, F. Luporini, A. T. T. Mcrae, G.-T. Bercea, G. R. Markall, and P. H. J. Kelly, Firedrake: Automating the finite element method by composing abstractions, ACM Trans. Math. Software, 43 (2016), pp. 24:1–24:27. · Zbl 1396.65144 [48] J. M. Restrepo, G. K. Leaf, and A. Griewank, Circumventing storage limitations in variational data assimilation studies, SIAM J. Sci. Comput., 19 (1998), pp. 1586–1605. · Zbl 0956.76078 [49] J. E. Roman, C. Campos, E. Romero, and A. Tomás, SLEPc Users Manual, Tech. Report DSIC-II/24/02, Departamento de Sistemas Informáticos y Computación, Universitat Politècnica de València, 2018. [50] P. Stumm and A. Walther, MultiStage approaches for optimal offline checkpointing, SIAM J. Sci. Comput., 31 (2009), pp. 1946–1967. · Zbl 1194.65084 [51] P. Stumm and A. Walther, New algorithms for optimal online checkpointing, SIAM J. Sci. Comput., 32 (2010), pp. 836–854. · Zbl 1214.65038 [52] Heirarchical Data Format, Version 5, The HDF Group, 1997, http://www.hdfgroup.org/HDF5/. [53] J. Utke, U. Naumann, M. Fagan, N. Tallent, M. Strout, P. Heimbach, C. Hill, and C. Wunsch, OpenAD/F: A modular open-source tool for automatic differentiation of Fortran codes, ACM Trans. Math. Software, 34 (2008), pp. 18:1–18:36. · Zbl 1291.65140 [54] Q. Wang, P. Moin, and G. Iaccarino, Minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation, SIAM J. Sci. Comput., 31 (2009), pp. 2549–2567. · Zbl 1196.65050 [55] Z. Wang, I. M. Navon, F. X. Le Dimet, and X. Zou, The second order adjoint analysis: Theory and applications, Meteorology Atmospheric Physics, 50 (1992), pp. 3–20.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.