Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++
ACM Transactions on Mathematical Software (TOMS)
Numerical methods and software for sensitivity analysis of differential-algebraic systems
Applied Numerical Mathematics
Efficient sensitivity analysis of large-scale differential-algebraic systems
Applied Numerical Mathematics
A Supernodal Approach to Sparse Partial Pivoting
SIAM Journal on Matrix Analysis and Applications
Parallel Newton-Krylov methods for PDE-constrained optimization
SC '99 Proceedings of the 1999 ACM/IEEE conference on Supercomputing
Computational Methods for Inverse Problems
Computational Methods for Inverse Problems
SIAM Journal on Scientific Computing
Cheap Second Order Directional Derivatives of Stiff ODE Embedded Functionals
SIAM Journal on Scientific Computing
SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers
ACM Transactions on Mathematical Software (TOMS) - Special issue on the Advanced CompuTational Software (ACTS) Collection
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We present an efficient method for computing, in a parallel distributed environment, gradients of functionals depending on the solution sensitivities of dynamical systems described by ordinary differential equations. This work was motivated by the need to compute cost function gradients for dynamically constrained optimization for robustness, i.e. the problem of finding the model parameters of a given dynamical system that lead to minimum solution sensitivity. The proposed approach for this particular second-order sensi-tivity problem falls into the class of the so-called 'adjoint-over-forward' methods and is based on solving the continuous forward sensitivity equations (for evaluating the cost functional) and their continuous adjoint systems (for evaluation of the gradient of the cost functional). Efficiency is ensured by: (i) a suitable distribution over processors of the individual sensitivity systems, both forward and adjoint so that inter-process communication is minimized; and (ii) use of an iterative solver combined with a block-diagonal preconditioner for the solution of the linear systems arising in the implicit integration of the resulting ordinary differential equations (ODE) systems. The proposed algorithm was implemented as an extension of the cvodes solver in sundials [P. Brown, K. Grant, A. Hindmarsh, S. Lee, R.S.D. Shumaker, and C. Woodward. SUNDIALS: SUite of Nonlinear and DIfferential/ALgebraic equation Solvers, ACM Trans. Math. Software, 31(3) (2005), pp. 363-396], but it can be used in conjunction with any sensitivity-enabled ODE integrator that provides adjoint sensitivity capabilities and (if based on implicit integration methods) support for iterative linear algebra.