Variable-Stepsize Implicit Peer Triplets in ODE Constrained Optimal Control
Abstract: This paper is concerned with the theory, construction and application of implicit Peer two-step methods that are super-convergent for variable stepsizes, i.e., preserve their classical order achieved for uniform stepsizes when applied to ODE constrained optimal control problems in a first-discretize-then-optimize setting. We upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to get ready for dynamical systems with varying time scales without loosing efficiency. Peer triplets consist of a standard Peer method for interior time steps supplemented by matching methods for the starting and end steps. A decisive advantage of Peer methods is their absence of order reduction since they use stages of the same high stage order. The consistency analysis of variable-stepsize implicit Peer methods results in additional order conditions and severe new difficulties for uniform zero-stability, which intensifies the demands on the Peer triplet. Further, we discuss the construction of 4-stage methods with order pairs (4,3) and (3,3) for state and adjoint variables in detail and provide four Peer triplets of practical interest. We rigorously prove convergence of order $s-1$ for $s$-stage Peer methods applied on grids with bounded or smoothly changing stepsize ratios. Numerical tests show the expected order of convergence for the new variable-stepsize Peer triplets.
- Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems. Applied Mathematics and Computation, 354:460–477, 2019.
- I. Almuslimani and G. Vilmart. Explicit stabilized integrators for stiff optimal control problems. SIAM J. Sci. Comput., 43:A721–A743, 2021.
- Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Society for Industrial and Applied Mathematics, 1995.
- Approximation of weak adjoints by reverse automatic differentiation of BDF methods. Numer. Math., 126:383–412, 2014.
- V.D. Blondel and Y. Nesterov. Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices. SIAM J. Matrix Anal. Appl., 31:865–876, 2010.
- F.J. Bonnans and J. Laurent-Varin. Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control. Numer. Math., 103:1–10, 2006.
- W.W. Hager. Runge-Kutta methods in optimal control and the transformed adjoint system. Numer. Math., 87:247–282, 2000.
- Solving Ordinary Differential Equations I. Springer-Verlag, New York, 2006.
- Implicit-explicit Runge-Kutta schemes for numerical discretization of optimal control problems. SIAM J. Numer. Anal., 51:1875–1899, 2013.
- W. Huang and R.D. Russell. Adaptive Moving Mesh Methods. Springer New York, Dordrecht, Heidelberg, London, 2011.
- J. Lang and B.A. Schmitt. Discrete adjoint implicit peer methods in optimal control. J. Comput. Appl. Math., 416:114596, 2022.
- J. Lang and B.A. Schmitt. Implicit A-stable peer triplets for ODE constrained optimal control problems. Algorithms, 15:310, 2022.
- J. Lang and B.A. Schmitt. Exact discrete solutions of boundary control problems for the 1D heat equation. J. Optim. Theory Appl., 196:1106–1118, 2023.
- X. Liu and J. Frank. Symplectic Runge-Kutta discretization of a regularized forward-backward sweep iteration for optimal control problems. J. Comput. Appl. Math., 383:113133, 2021.
- B.A. Schmitt. Algebraic criteria for A-stability of peer two-step methods. Technical Report arXiv:1506.05738, 2015.
- B.A. Schmitt and R. Weiner. Efficient A-stable peer two-step methods. J. Comput. Appl. Math., pages 319–32, 2017.
- Implicit parallel peer methods for stiff initial value problems. Appl. Numer. Math., 53:457–470, 2005.
- Stability and consistency of discrete adjoint implicit peer methods. J. Comput. Appl. Math., 262:73–86, 2014.
- J.L. Troutman. Variational Calculus and Optimal Control. Springer, New York, 1996.
- Y. Xu. A characterization of positive quadrature formulae. Math. Comp., 62:703–718, 1994.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.