Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

Albi, Giacomo; Herty, Michaël; Pareschi, Lorenzo

doi:10.1016/j.amc.2019.02.021

Cited by 15 publications

(18 citation statements)

References 34 publications

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“…In contrast, multistep methods including Peer two-step methods avoid order reductions and allow a simple implementation [5,6]. However, the discrete adjoint schemes of linear multistep methods are in general not consistent or show a significant decrease of their approximation order [1,20]. Recently, we have developed implicit Peer two-step methods [12] with three stages to solve ODE constrained optimal control problems of the form minimize C y(T ) (1) subject to y (t) = f y(t), u(t) , u(t) ∈ U ad , t ∈ (0, T ],…”

Section: Introductionmentioning

confidence: 99%

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

Lang¹,

Schmitt²

2022

Preprint

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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 emphasizes that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. Also as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.35 o )-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 o )-stable method with nodes symmetric in [0, 1] to a common center that performs equally well. Numerical tests with three well established optimal control problems confirm the theoretical findings also concerning A-stability.

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Section: Introductionmentioning

confidence: 99%

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

Lang¹,

Schmitt²

2022

Preprint

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“…This result is related to the order of symplectic partitioned Runge-Kutta methods, and it implies in particular that applying naively a Runge-Kutta method to (1) yields in general an order reduction phenomenon. This analysis was then extended to other classes of Runge-Kutta type schemes in [17,18,14], see also [15,8] in the context of hyperbolic problems and multistep methods. The use of symplectic integrators is motivated by the recent publication [20] which proves the convergence of forward-backward iterative algorithm (Algorithm 2.3 in the present paper), to implement discretized optimal control problems, when using a symplectic Runge-Kutta method.…”

Section: Introductionmentioning

confidence: 99%

Explicit Stabilized Integrators for Stiff Optimal Control Problems

Almuslimani¹,

Vilmart²

2021

SIAM J. Sci. Comput.

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Explicit stabilized methods are an efficient alternative to implicit schemes for the time integration of stiff systems of differential equations in large dimension. In this paper we derive explicit stabilized integrators of orders one and two for the optimal control of stiff systems. We analyze their favorable stability properties based on the continuous optimality conditions. Furthermore, we study their order of convergence taking advantage of the symplecticity of the corresponding partitioned Runge-Kutta method involved for the adjoint equations. Numerical experiments including the optimal control of a nonlinear diffusion-advection PDE illustrate the efficiency of the new approach.

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“…A large literature in this direction has been devoted to the construction of IMEX Runge-Kutta schemes satisfying the asymptotic-preserving (AP) property in the case of hyperbolic problems [10,12,24,27] and for kinetic equations [11,14,16,17]. For the case of IMEX linear multistep methods, we refer to [1,2,4,6,18,20,21,30,31] for results on the construction and properties…”

Section: Introductionmentioning

confidence: 99%

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

Albi

Pareschi

2021

Commun. Appl. Math. Comput.

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We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

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Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

Cited by 15 publications

References 34 publications

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

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

Explicit Stabilized Integrators for Stiff Optimal Control Problems

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

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