Variable-Stepsize Implicit Peer Triplets in ODE Constrained Optimal Control
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866907876014948352 |
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| author | Lang, Jens Schmitt, Bernhard A. |
| author_facet | Lang, Jens Schmitt, Bernhard A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_13716 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Variable-Stepsize Implicit Peer Triplets in ODE Constrained Optimal Control Lang, Jens Schmitt, Bernhard A. Optimization and Control 34H05, 49J15, 65L05, 65L06 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. |
| title | Variable-Stepsize Implicit Peer Triplets in ODE Constrained Optimal Control |
| topic | Optimization and Control 34H05, 49J15, 65L05, 65L06 |
| url | https://arxiv.org/abs/2404.13716 |