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Variable-stepsize implicit Peer triplets in ODE constrained optimal control

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 460卷

作者： Lang, Jens Schmitt, Bernhard A. Tech Univ Darmstadt Dept Math Dolivostr 15 D-64293 Darmstadt Germany Philipps Univ Marburg Dept Math Hans Meerwein Str 6 D-35043 Marburg Germany

This paper is concerned with the theory, construction and application of implicit Peer twostep 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 Lang and Schmitt (Algorithms 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.

关键词： Implicit Peer two-step methods Nonlinear optimal control first-discretize-then-optimize Discrete adjoints Variable stepsizes Super-convergence

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Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第1期203卷 985-1026页

作者： Lang, Jens Schmitt, Bernhard A. Tech Univ Darmstadt Dolivostr 15 D-64293 Darmstadt Germany Philipps Univ Marburg Hans Meerwein Str 6 D-35043 Marburg Germany

It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint variables. High accuracy with moderate computing time can be achieved by such time integration methods that satisfy a sufficiently large number of adjoint order conditions and supply gradients with higher orders of consistency. In this paper, we upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods use several stages of the same high stage order, a decisive advantage is their lack of order reduction as for semi-discretized PDE problems with boundary control. Additional order conditions for the control and certain positivity requirements now intensify the demands on the Peer triplet. We discuss the construction of 4-stage methods with order pairs (3, 3) and (4, 3) in detail and provide three Peer triplets of practical interest. We prove convergence of order s-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s-1$$\end{document}, at least, for s-stage methods if state, adjoint and control satisfy the corresponding order conditions. Numerical tests show the expected order of convergence for the new Peer triplets.

关键词： Implicit peer two-step methods Nonlinear optimal control Gradient-based optimization first-discretize-then-optimize Discrete adjoints

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Exact Discrete Solutions of Boundary Control Problems for the 1D Heat Equation

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2023年第3期196卷 1106-1118页

作者： Lang, Jens Schmitt, Bernhard A. A. Tech Univ Darmstadt Dept Math Dolivostr 15 D-64293 Darmstadt Germany Philipps Univ Marburg Dept Math Hans Meerwein Str 6 D-35043 Marburg Germany

Method-of-lines discretizations are demanding test problems for stiff integration methods. However, for PDE problems with known analytic solution, the presence of space discretization errors or the need to use codes to compute reference solutions may limit the validity of numerical test results. To overcome these drawbacks, we present in this short note a simple test problem with boundary control, a situation where one-step methods may suffer from order reduction. We derive exact formulas for the solution of an optimal boundary control problem governed by a one-dimensional discrete heat equation and an objective function that measures the distance of the final state from the target and the control costs. This analytical setting is used to compare the numerically observed convergence orders for selected implicit Runge-Kutta and Peer two-step methods of classical order four, which are suitable for optimal control problems.

关键词： Optimal control Implicit Peer two-step methods Symplectic Runge-Kutta methods first-discretize-then-optimize Discrete adjoints

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Maximizing the storage capacity of gas networks: a global MINLP approach

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OPTIMIZATION AND ENGINEERING 2019年第2期20卷 543-573页

作者： Burlacu, Robert Egger, Herbert Gross, Martin Martin, Alexander Pfetsch, Marc E. Schewe, Lars Sirvent, Mathias Skutella, Martin Friedrich Alexander Univ Erlangen Nurnberg FAU Discrete Optimizat Cauerstr 11 D-91058 Erlangen Germany Tech Univ Darmstadt Res Grp Numer & Sci Comp Dolivostr 15 D-64293 Darmstadt Germany Rhein Westfal TH Aachen Chair Management Sci Kackertstr 7 D-52072 Aachen Germany Tech Univ Darmstadt Res Grp Optimizat Dolivostr 15 D-64293 Darmstadt Germany Tech Univ Berlin Fac Math & Nat Sci 2 Str 17 Juni 136 D-10623 Berlin Germany

In this paper, we study the transient optimization of gas networks, focusing in particular on maximizing the storage capacity of the network. We include nonlinear gas physics and active elements such as valves and compressors, which due to their switching lead to discrete decisions. The former is described by a model derived from the Euler equations that is given by a coupled system of nonlinear parabolic partial differential equations (PDEs). We tackle the resulting mathematical optimization problem by a first-discretize-then-optimize approach. To this end, we introduce a new discretization of the underlying system of parabolic PDEs and prove well-posedness for the resulting nonlinear discretized system. Endowed with this discretization, we model the problem of maximizing the storage capacity as a non-convex mixed-integer nonlinear problem (MINLP). For the numerical solution of the MINLP, we algorithmically extend a well-known relaxation approach that has already been used very successfully in the field of stationary gas network optimization. This method allows us to solve the problem to global optimality by iteratively solving a series of mixed-integer problems. Finally, we present two case studies that illustrate the applicability of our approach.

关键词： Mixed-integer nonlinear programming Transient gas transport optimization Storage capacity maximization Power-to-gas first-discretize-then-optimize

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Discrete adjoint implicit Peer methods in optimal control

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2022年 416卷

作者： Lang, Jens Schmitt, Bernhard A. Tech Univ Darmstadt Dept Math Dolivostr 15 D-64293 Darmstadt Germany Philipps Univ Marburg Dept Math Hans Meerwein Str 6 D-35043 Marburg Germany

It is well known that in the first-discretize-then-optimize approach in the control of ordinary differential equations the discrete adjoint method may converge under additional order conditions only. For Peer two-step methods we derive such adjoint order conditions and pay special attention to different formulations and boundary steps. For s-stage methods, we prove convergence of order s for the state variables if the adjoint method satisfies the conditions for order s -1, at least. We remove some bottlenecks at the boundaries encountered in an earlier paper of Schroder et al. (2014) and discuss the construction of 3-stage methods for the order pair (3,2) in detail. The impact of nodes having equal differences is highlighted. It turns out that the most attractive methods are related to backward differentiation formulas. Three 3-stage methods are constructed, which show the expected orders in numerical tests.(c) 2022 Elsevier B.V. All rights reserved.

关键词： Implicit Peer two-step methods BDF-methods Nonlinear optimal control first-discretize-then-optimize Discrete adjoints

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Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems

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ALGORITHMS 2022年第9期15卷

作者： Lang, Jens Schmitt, Bernhard A. Tech Univ Darmstadt Dept Math Dolivostr 15 D-64293 Darmstadt Germany Philipps Univ Marburg Dept Math Hans Meerwein Str 6 D-35043 Marburg Germany

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(73.35 degrees)-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 degrees)-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.

关键词： implicit Peer two-step methods BDF-methods nonlinear optimal control first-discretize-then-optimize discrete adjoints

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