Publication: Application of AD-based quasi-Newton-Methods to stiff ODEs
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Application of AD-based quasi-Newton-Methods to stiff ODEs

- incollection -
 

Area
Ordinary Differential Equations

Author(s)
Sebastian Schlenkrich , Andrea Walther , Andreas Griewank

Published in
Automatic Differentiation: Applications, Theory, and Implementations

Editor(s)
H. M. Bücker, G. Corliss, P. Hovland, U. Naumann, B. Norris

Year
2005

Publisher
Springer

Abstract
Systems of stiff ordinary differential equations (ODEs) can be integrated properly only by implicit methods. For that purpose, one usually has to solve a system of nonlinear equations at each time step. This system of equations may be solved by variants of Newton's method. The main computing effort lies in forming and factoring the Jacobian or a suitable approximation to it. We examine a new approach of constructing an appropriate quasi-Newton approximation for solving stiff ODEs. The method makes explicit use of tangent and adjoint information that can be obtained using the forward and the reverse modes of algorithmic differentiation (ad). We elaborate the conditions for invariance with respect to linear transformations of the state space and thus similarity transformations of the Jacobian. We present one new updating variant that yields such an invariant method. Numerical results for Runge-Kutta methods and linear multi-step methods are discussed.

Cross-References
Bucker2005ADA

AD Tools
ADOL-C

BibTeX
@INCOLLECTION{
         Schlenkrich2005AoA,
       author = "Sebastian Schlenkrich and Andrea Walther and Andreas Griewank",
       title = "Application of {AD}-based quasi-{N}ewton-Methods to stiff {ODE}s",
       editor = "H. M. B{\"u}cker and G. Corliss and P. Hovland and U. Naumann and B.
         Norris",
       booktitle = "Automatic Differentiation: {A}pplications, Theory, and Implementations",
       series = "Lecture Notes in Computational Science and Engineering",
       publisher = "Springer",
       year = "2005",
       abstract = "Systems of stiff ordinary differential equations (ODEs) can be integrated properly
         only by implicit methods. For that purpose, one usually has to solve a system of nonlinear equations
         at each time step. This system of equations may be solved by variants of Newton's method. The
         main computing effort lies in forming and factoring the Jacobian or a suitable approximation to it.
         We examine a new approach of constructing an appropriate quasi-Newton approximation for solving
         stiff ODEs. The method makes explicit use of tangent and adjoint information that can be obtained
         using the forward and the reverse modes of algorithmic differentiation (AD). We elaborate the
         conditions for invariance with respect to linear transformations of the state space and thus
         similarity transformations of the Jacobian. We present one new updating variant that yields such an
         invariant method. Numerical results for Runge-Kutta methods and linear multi-step methods are
         discussed.",
       crossref = "Bucker2005ADA",
       ad_area = "Ordinary Differential Equations",
       ad_tools = "ADOL-C",
       pages = "89--98",
       doi = "10.1007/3-540-28438-9_8"
}


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