Lang, Jens ; Schmitt, Bernhard A. (2022)
Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems.
In: Algorithms, 2022, 15 (9)
doi: 10.26083/tuprints-00022452
Artikel, Zweitveröffentlichung, Verlagsversion
Kurzbeschreibung (Abstract)
This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasize that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. In addition, as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.3°)-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A(84°)-stable method with nodes symmetric in [0, 1] to a common center that performs equally well. Numerical tests with four well established optimal control problems confirm the theoretical findings also concerning A-stability.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2022 |
| Autor(en): | Lang, Jens ; Schmitt, Bernhard A. |
| Art des Eintrags: | Zweitveröffentlichung |
| Titel: | Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems |
| Sprache: | Englisch |
| Publikationsjahr: | 2022 |
| Ort: | Darmstadt |
| Publikationsdatum der Erstveröffentlichung: | 2022 |
| Verlag: | MDPI |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | Algorithms |
| Jahrgang/Volume einer Zeitschrift: | 15 |
| (Heft-)Nummer: | 9 |
| Kollation: | 30 Seiten |
| DOI: | 10.26083/tuprints-00022452 |
| URL / URN: | https://tuprints.ulb.tu-darmstadt.de/22452 |
| Zugehörige Links: | |
| Herkunft: | Zweitveröffentlichung DeepGreen |
| Kurzbeschreibung (Abstract): | This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasize that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. In addition, as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.3°)-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A(84°)-stable method with nodes symmetric in [0, 1] to a common center that performs equally well. Numerical tests with four well established optimal control problems confirm the theoretical findings also concerning A-stability. |
| Freie Schlagworte: | implicit Peer two-step methods, BDF-methods, nonlinear optimal control, first-discretize-then-optimize, discrete adjoints |
| Status: | Verlagsversion |
| URN: | urn:nbn:de:tuda-tuprints-224527 |
| Zusätzliche Informationen: | This article belongs to the Section Analysis of Algorithms and Complexity Theory |
| Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Numerik und wissenschaftliches Rechnen |
| Hinterlegungsdatum: | 10 Okt 2022 12:47 |
| Letzte Änderung: | 11 Okt 2022 13:52 |
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