Peer Two-Step Methods with Embedded Sensitivity Approximation for Parameter-Dependent ODEs

@article{Schmitt2012PeerTM,
  title={Peer Two-Step Methods with Embedded Sensitivity Approximation for Parameter-Dependent ODEs},
  author={Bernhard A. Schmitt and Ekaterina A. Kostina},
  journal={SIAM J. Numer. Anal.},
  year={2012},
  volume={50},
  pages={2182-2207}
}
Peer two-step methods have been successfully applied to initial value problems for stiff and nonstiff ordinary differential equations (ODEs) both on parallel and sequential computers. Their essential property is the use of several stages per time step with the same accuracy. As a new application area these methods are now used for parameter-dependent ODEs where the peer stages approximate the solution also at different places in the parameter space. The main interest here is sensitivity data… 

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The derivation of the algorithm is based on a reformulation of the parameterization as a linear parameter-free ordinary differential equation and on approximating the product of the matrix exponential and a vector with a Krylov method.
Peer methods with improved embedded sensitivities for parameter-dependent ODEs
  • B. A. Schmitt
  • Mathematics, Computer Science
    J. Comput. Appl. Math.
  • 2014

References

SHOWING 1-10 OF 20 REFERENCES
Implicit peer methods for large stiff ODE systems
TLDR
Numerical tests in Matlab for several semi-discretized partial differential equations show the efficiency of the implicit two-step peer methods compared to other Krylov codes.
Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration
TLDR
A subclass having order s−1 is introduced where optimal damping for stiff problems is obtained by using different system parameters in different stages and a superconvergence property is proved using a polynomial collocation formulation.
Implicit parallel peer methods for stiff initial value problems
Parallel Two-Step W-Methods with Peer Variables
TLDR
A new class of methods for the solution of stiff initial value problems is introduced that is parallel by design and propagates different "peer" solution variables with essentially identical characteristics from step to step.
Explicit two-step peer methods
An Adaptive Newton-Picard Algorithm with Subspace Iteration for Computing Periodic Solutions
TLDR
A hybrid Newton--Picard scheme based on the shooting method is derived, which in its simplest form is the recursive projection method (RPM) of Shroff and Keller and is used to compute and determine the stability of both stable and unstable periodic orbits.
CVODES: The Sensitivity-Enabled ODE Solver in SUNDIALS
TLDR
The current capabilities of CVODES, its design principles, and its user interface are described, and an example problem is provided to illustrate the performance ofCVODES.
Parameter optimization for explicit parallel peer two-step methods
On the Role of Natural Level Functions to Achieve Global Convergence for Damped Newton Methods
TLDR
A new view on globalization techniques for Newton’s method based on strategies based on “natural level functions” and a “restrictive mono-tonicity test” are discussed.
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