CVODES is a solver for stiff and nonstiff ODE systems (initial value problem) given in explicit form y’ = f(t,y,p) with sensitivity analysis capabilities (both forward and adjoint modes). CVODES is a superset of CVODE and hence all options available to CVODE (with the exception of the FCVODE interface module) are also available for CVODES. Both integration methods (Adams-Moulton and BDF) and the corresponding nonlinear iteration methods, as well as all linear solver and preconditioner modules are available for the integration of the original ODEs, the sensitivity systems, or the adjoint system. Depending on the number of model parameters and the number of functional outputs, one of two sensitivity methods is more appropriate. The forward sensitivity analysis (FSA) method is mostly suitable when the gradients of many outputs (for example the entire solution vector) with respect to relatively few parameters are needed. In this approach, the model is differentiated with respect to each parameter in turn to yield an additional system of the same size as the original one, the result of which is the solution sensitivity. The gradient of any output function depending on the solution can then be directly obtained from these sensitivities by applying the chain rule of differentiation. The adjoint sensitivity analysis (ASA) method is more practical than the forward approach when the number of parameters is large and the gradients of only few output functionals are needed. In this approach, the solution sensitivities need not be computed explicitly. Instead, for each output functional of interest, an additional system, adjoint to the original one, is formed and solved. The solution of the adjoint system can then be used to evaluate the gradient of the output functional with respect to any set of model parameters. The FSA module in CVODES implements a simultaneous corrector method as well as two flavors of staggered corrector methods -- for the case when sensitivity right hand sides are generated all at once or separated for each model parameter. The ASA module provides the infrastructure required for the backward integration in time of systems of differential equations dependent on the solution of the original ODEs. It employs a checkpointing scheme for efficient interpolation of forward solutions during the backward integration.

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  1. Rozhdestvensky, Kirill; Ryzhov, Vladimir; Fedorova, Tatiana; Safronov, Kirill; Tryaskin, Nikita; Sulaiman, Shaharin Anwar; Ovinis, Mark; Hassan, Suhaimi: Computer modeling and simulation of dynamic systems using Wolfram SystemModeler (2020)
  2. Chris Rackauckas, Mike Innes, Yingbo Ma, Jesse Bettencourt, Lyndon White, Vaibhav Dixit: DiffEqFlux.jl - A Julia Library for Neural Differential Equations (2019) arXiv
  3. Farrell, P. E.; Hake, J. E.; Funke, S. W.; Rognes, M. E.: Automated adjoints of coupled PDE-ODE systems (2019)
  4. Sanguinetti, Guido (ed.); Huynh-Thu, Vân Anh (ed.): Gene regulatory networks. Methods and protocols (2019)
  5. Melicher, Valdemar; Haber, Tom; Vanroose, Wim: Fast derivatives of likelihood functionals for ODE based models using adjoint-state method (2017)
  6. Perić, Nikola D.; Villanueva, Mario E.; Chachuat, Benoît: Sensitivity analysis of uncertain dynamic systems using set-valued integration (2017)
  7. Gul, R.; Bernhard, S.: Parametric uncertainty and global sensitivity analysis in a model of the carotid bifurcation: identification and ranking of most sensitive model parameters (2015)
  8. Janka, Dennis: Sequential quadratic programming with indefinite Hessian approximations for nonlinear optimum experimental design for parameter estimation in differential-algebraic equations (2015)
  9. Quirynen, Rien; Vukov, Milan; Diehl, Moritz: Multiple shooting in a microsecond (2015)
  10. Quirynen, R.; Vukov, M.; Zanon, M.; Diehl, M.: Autogenerating microsecond solvers for nonlinear MPC: a tutorial using ACADO integrators (2015)
  11. Schmitt, Bernhard A.: Peer methods with improved embedded sensitivities for parameter-dependent ODEs (2014)
  12. Schmitt, Bernhard A.: Reprint of: Peer methods with improved embedded sensitivities for parameter-dependent ODEs (2014)
  13. Zeng, X.; Anitescu, M.: Sequential Monte Carlo sampling in hidden Markov models of nonlinear dynamical systems (2014)
  14. Hug, S.; Raue, A.; Hasenauer, J.; Bachmann, J.; Klingmüller, U.; Timmer, J.; Theis, F. J.: High-dimensional Bayesian parameter estimation: case study for a model of JAK2/STAT5 signaling (2013)
  15. Zivari-Piran, Hossein; Enright, Wayne H.: Accurate first-order sensitivity analysis for delay differential equations (2012)
  16. Lunacek, Monte; Nag, Ambarish; Alber, David M.; Gruchalla, Kenny; Chang, Christopher H.; Graf, Peter A.: Simulation, characterization, and optimization of metabolic models with the high performance systems biology toolkit (2011)
  17. Koutsawa, Yao; Belouettar, Salim; Makradi, Ahmed; Nasser, Houssein: Sensitivities of effective properties computed using micromechanics differential schemes and high-order Taylor series: application to piezo-polymer composites (2010)
  18. Stumm, Philipp; Walther, Andrea: New algorithms for optimal online checkpointing (2010)
  19. Alexe, Mihai; Sandu, Adrian: Forward and adjoint sensitivity analysis with continuous explicit Runge-Kutta schemes (2009)
  20. Alexe, Mihai; Sandu, Adrian: On the discrete adjoints of adaptive time stepping algorithms (2009)

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