Solving parameter estimation problems with discrete adjoint exponential integrators

References

• M. Alexe and A. Sandu, Forward and adjoint sensitivity analysis with continuous explicit Runge–Kutta schemes, Appl. Math. Comput. 208 (2009), pp. 328–34. Available at https://dx.doi.org/10.1016/j.amc.2008.11.035.
   Web of Science ®Google Scholar
• M. Alexe and A. Sandu, On the discrete adjoints of variable step time integrators, J. Comput. Appl. Math. 233 (2009), pp. 1005–1020. Available at https://dx.doi.org/10.1016/j.cam.2009.08.109.
   Web of Science ®Google Scholar
• M.S. Alnæs, 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. Soft. 3 (2015), pp. 9–23.
   Google Scholar
• J.L. Anderson, Spatially and temporally varying adaptive covariance inflation for ensemble filters, Tellus A 61 (2009), pp. 72–83.
   Google Scholar
• A. Augustine and A. Sandu, MATLODE: A MATLAB ODE solver and sensitivity analysis toolbox, ACM TOMS in preparation, 2018.
   Google Scholar
• F. Bachinger, U. Langer, and J. Schöberl, Numerical analysis of nonlinear multiharmonic eddy current problems, Numer. Math. 100 (2005), pp. 593–616.
   Web of Science ®Google Scholar
• J.T. Betts, Practical methods for optimal control and estimation using nonlinear programming, SIAM, Philadelphia, 2010.
   Google Scholar
• J.T. Betts and S.L. Campbell, Discretize then optimize, Proceedings of the SIAM Conference on Mathematics for Industry, Challenges and Frontiers, Toronto, Ontario, October 13–15, 2003.
   Google Scholar
• J.T. Betts, S.L. Campbell, and A. Engelsone, Direct Transcription Solution of Inequality Constrained Optimal Control Problems, in American Control Conference, 2004. Proceedings of the 2004, Vol. 2, IEEE, 2004, pp. 1622–1626.
   Google Scholar
• O. Bíró, Edge element formulations of eddy current problems, Comput. Methods Appl. Mech. Eng. 169 (1999), pp. 391–405.
   Web of Science ®Google Scholar
• J.F. Bonnans and J. Laurent-Varin, Computation of order conditions for symplectic partitioned Runge–Kutta schemes with application to optimal control, Numer. Math. 103 (2006), pp. 1–10.
   Web of Science ®Google Scholar
• R.H. Byrd, P. Lu, J. Nocedal, and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Comput. 16 (1995), pp. 1190–1208.
   Web of Science ®Google Scholar
• C.M. Campos, S. Ober-Blöbaum, and E. Trélat, High order variational integrators in the optimal control of mechanical systems, preprint (2015). Available at arXiv:1502.00325.
   Google Scholar
• H. Cheng, M. Jardak, M. Alexe, and A. Sandu, A hybrid approach to estimating error covariances in variational data assimilation, Tellus A 62 (2010), pp. 288–297. Available at https://dx.doi.org/10.1111/j.1600-0870.2010.00442.x.
   Google Scholar
• D. Daescu, G. Carmichael, and A. Sandu, Adjoint implementation of Rosenbrock methods applied to variational data assimilation, in Air Pollution Modeling and its Application XIV, S. Gryning and F. Schiermeier, eds., Springer, 2000, pp. 361–369. Available at https://dx.doi.org/10.1111/j.1600-0870.2010.00442.x.
   Google Scholar
• D. Daescu, G. Carmichael, and A. Sandu, Adjoint implementation of Rosenbrock methods applied to variational data assimilation problems, J. Comput. Phys. 165 (2000), pp. 496–510. Available at https://dx.doi.org/10.1006/jcph.2000.6622.
   Web of Science ®Google Scholar
• D. Daescu, A. Sandu, and G. Carmichael, Direct and adjoint sensitivity analysis of chemical kinetic systems with KPP: II – Numerical validation and applications, Atmos. Environ. 37 (2003), pp. 5097–5114. Available at https://dx.doi.org/10.1016/j.atmosenv.2003.08.020.
   Web of Science ®Google Scholar
• E.J. Fertig, J. Harlim, and B.R. Hunt, A comparative study of 4D-VAR and a 4D Ensemble Kalman Filter: perfect model simulations with Lorenz-96, Tellus A: Dynamic Meteorology and Oceanography 59 (2007), pp. 96–100.
   Web of Science ®Google Scholar
• C. Geuzaine and J.F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities, Int. J. Numer. Methods Eng. 79 (2009), pp. 1309–1331.
   Web of Science ®Google Scholar
• T. Gou and A. Sandu, Continuous versus discrete advection adjoints in chemical data assimilation with CMAQ, Atmos. Environ. 45 (2011), pp. 4868–4881. Available at http://dx.doi.org/10.1016/j.atmosenv.2011.06.015.
   Web of Science ®Google Scholar
• A. Griewank and A. Walther, Algorithm 799: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation, ACM Trans. Math. Softw. 26 (2000), pp. 19–45.
   Web of Science ®Google Scholar
• A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, SIAM, Philadelphia, 2008.
   Google Scholar
• W.W. Hager, Runge–Kutta methods in optimal control and the transformed adjoint system, Numer. Math. 87 (2000), pp. 247–282.
   Web of Science ®Google Scholar
• E. Hairer and G. Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, 2nd ed., no. 14 in Springer Series in Computational Mathematics, Springer-Verlag, Berlin, Heidelberg, 1996.
   Google Scholar
• L. Hascoët and V. Pascual, The Tapenade automatic differentiation tool: Principles, model, and specification, ACM Trans. Math. Softw. 39 (2013). Available at http://dx.doi.org/10.1145/2450153.2450158.
   Web of Science ®Google Scholar
• M. Hochbruck and C. Lubich, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal. 34 (1997), pp. 1911–1925.
   Web of Science ®Google Scholar
• E. Jones, T. Oliphant, and P. Peterson, SciPy: Open source scientific tools for Python, 2001. Available at http://www.scipy.org/.
   Google Scholar
• E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, Cambridge, 2003.
   Google Scholar
• J. Lang and J.G. Verwer, W-methods in optimal control, Numer. Math. 124 (2013), pp. 337–360.
   Web of Science ®Google Scholar
• J. Lei and P. Bickel, A moment matching ensemble filter for nonlinear Non-Gaussian data assimilation, Mon. Weather Rev. 139 (2011), pp. 3964–3973.
   Web of Science ®Google Scholar
• H. Li, E. Kalnay, and T. Miyoshi, Simultaneous estimation of covariance inflation and observation errors within an ensemble Kalman filter, Q. J. R. Meteorol. Soc. 135 (2009), pp. 523–533.
   Web of Science ®Google Scholar
• A. Logg, K.A. Mardal, and G.N. Wells, Automated Solution of Differential Equations by the Finite Element Method, Springer-Verlag, Berlin, 2012.
   Google Scholar
• E.N. Lorenz, Predictability – a problem partly solved, in Predictability of Weather and Climate, Cambridge University Press (CUP), 1996, pp. 40–58. Available at http://dx.doi.org/10.1017/cbo9780511617652.004.
   Google Scholar
• D.C. Meeker, Finite element method magnetics, version 4.0.1 (03dec2006 build). Available at http://www.femm.info.
   Google Scholar
• P. Miehe and A. Sandu, Forward, Tangent Linear, and Adjoint Runge Kutta Methods in KPP-2.2, in International Conference on Computational Science (ICCS 2006), Lecture Notes in Computer Science, Vol. 3993, Reading, UK, Springer, Berlin, Heidelberg, 2006, pp. 120–127. Available at 10.1007/11758532_18.
   Google Scholar
• M. Narayanamurthi, P. Tranquilli, A. Sandu, and M. Tokman, EPIRK-W and EPIRK-K time discretization methods, CoRR abs/1701.06528 (2017). Available at http://arxiv.org/abs/1701.06528.
   Google Scholar
• U. Naumann, The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation, SIAM, Philadelphia 2011.
   Google Scholar
• R.D. Neidinger, Introduction to automatic differentiation and MATLAB object-oriented programming, SIAM Rev. 52 (2010), pp. 545–563.
   Web of Science ®Google Scholar
• E. Ott, B.R. Hunt, I. Szunyogh, A.V. Zimin, E.J. Kostelich, M. Corazza, E. Kalnay, D.J. Patil, and J.A. Yorke, A local ensemble Kalman filter for atmospheric data assimilation, Tellus A: Dynamic Meteorology and Oceanography 56 (2004), pp. 415–428.
   Web of Science ®Google Scholar
• P. Peterson, F2py: A tool for connecting Fortran and Python programs, Int. J. Comput. Sci. Eng. 4 (2009), pp. 296–305.
   Google Scholar
• V. Rao and A. Sandu, An adjoint-based scalable algorithm for time-parallel integration, J. Comput. Sci. 5 (2014), pp. 76–84. Available at http://www.sciencedirect.com/science/article/pii/S1877750313000380, empowering Science through Computing + BioInspired Computing.
   Web of Science ®Google Scholar
• V. Rao and A. Sandu, A posteriori error estimates for the solution of variational inverse problems, SIAM/ASA J. Uncertain. Quant. 3 (2015), pp. 737–761. Available at http://dx.doi.org/10.1137/140990036.
   Web of Science ®Google Scholar
• E. Rosseel, H. De Gersem, and S. Vandewalle, Nonlinear stochastic Galerkin and collocation methods: Application to a ferromagnetic cylinder rotating at high speed, Commun. Comput. Phys. 8 (2010), pp. 947–975.
   Web of Science ®Google Scholar
• K. Rothauge, E. Haber, and U. Ascher, The discrete adjoint method for exponential integration, preprint (2016). Available at arXiv:1610.02596.
   Google Scholar
• A. Sandu, On the properties of Runge-Kutta discrete adjoints, in Lecture Notes in Computer Science, Vol. LNCS 3994, Part IV, Springer, Berlin, Heidelberg, 2006, pp. 550–557. Available at http://dx.doi.org/10.1007/11758549_76.
   Google Scholar
• A. Sandu, On consistency properties of discrete adjoint linear multistep methods, Tech. Rep. CS-TR-07-40, Computer Science Department, Virginia Tech, 2007. Available at http://vtechworks.lib.vt.edu/handle/10919/19773.
   Google Scholar
• A. Sandu, Solution of inverse ODE problems using discrete adjoints, in Large Scale Inverse Problems and Quantification of Uncertainty, L. Biegler, G. Biros, O. Ghattas, M. Heinkenschloss, D. Keyes, B. Mallick, L. Tenorio, B. van Bloemen Waanders, and K. Willcox, eds., Chapter 12, John Wiley & Sons, 2010, pp. 345–364. Available at http://dx.doi.org/10.1002/9780470685853.ch16.
   Google Scholar
• A. Sandu and P. Miehe, Forward, tangent linear, and adjoint Runge Kutta methods in KPP–2.2 for efficient chemical kinetic simulations, Int. J. Comput. Math. 87 (2010), pp. 2458–2479. Available at http://dl.acm.org/citation.cfm?id=1868060.
   Web of Science ®Google Scholar
• A. Sandu, D. Daescu, and G. Carmichael, Direct and adjoint sensitivity analysis of chemical kinetic systems with KPP: I – Theory and software tools, Atmos. Environ. 37 (2003), pp. 5083–5096. Available at http://dx.doi.org/10.1016/j.atmosenv.2003.08.019.
   Web of Science ®Google Scholar
• A. Sandu, D. Daescu, G. Carmichael, and T. Chai, Adjoint sensitivity analysis of regional air quality models, J. Comput. Phys. 204 (2005), pp. 222–252. Available at http://dx.doi.org/10.1016/j.jcp.2004.10.011.
   Web of Science ®Google Scholar
• J.M. Sanz-Serna, Symplectic Runge–Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more, SIAM Rev. 58 (2016), pp. 3–33.
   Web of Science ®Google Scholar
• R.B. Sidje, Expokit: A software package for computing matrix exponentials, ACM Trans. Math. Softw. 24 (1998), pp. 130–156. Available at http://dx.doi.org/10.1145/285861.285868.
   Web of Science ®Google Scholar
• K. Singh, P. Eller, A. Sandu, D. Henze, K. Bowman, M. Kopacz, and M. Lee, Towards the Construction of a Standard Adjoint GEOS-Chem Model, in Proceedings of the 2009 Spring Simulation Multiconference (SpringSim'09), High Performance Computing Symposium (HPC-2009), March, 2009, p. 8. Available at http://dl.acm.org/citation.cfm?id=1639809.1639925.
   Google Scholar
• K. Singh, A. Sandu, M. Jardak, K.W. Bowman and M. Lee, A practical method to estimate information content in the context of 4D-VAR data assimilation, SIAM/ASA J. Uncertain. Quant. 1 (2013), pp. 106–138.
   Web of Science ®Google Scholar
• T. Steihaug and A. Wolfbrandt, An attempt to avoid Jacobian and nonlinear equations in the numerical solution of stiff differential equations, Math. Comput. 33 (1979), pp. 521–521. Available at http://dx.doi.org/10.1090/s0025-5718-1979-0521273-8.
   Web of Science ®Google Scholar
• M. Tokman, Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods, J. Comput. Phys. 213 (2006), pp. 748–776.
   Web of Science ®Google Scholar
• M. Tokman, A new class of exponential propagation iterative methods of Runge–Kutta type (EPIRK), J. Comput. Phys. 230 (2011), pp. 8762–8778.
   Web of Science ®Google Scholar
• H. Zhang and A. Sandu, FATODE: A Library for Forward, Adjoint, and Tangent Linear Integration of ODEs, SIAM J. Sci. Comput. 36 (2014), pp. C504–C523. Available at http://dx.doi.org/10.1137/130912335.
   Web of Science ®Google Scholar