CVODES
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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References in zbMATH (referenced in 29 articles )
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