Numerical solution of differential Riccati equations arising in optimal control for parabolic PDEs

Mena, Hermann; Benner, Peter

doi:10.1002/pamm.200700913

Cited by 12 publications

(21 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Thus, instead of a Sylvester equation, the solution of a Lyapunov equation in every stage of a Rosenbrock method is required. Particularly, for large scale RDEs arising from the discretization in space of the optimal control problems governed by partial differential equations of parabolic type an efficient implementation of the Rosenbrock methods is proposed in [9]. The key ingredient there is to find a low rank approximation of the solution and to rewrite the method for the these low rank factors of the solution.…”

Section: Algorithm Iii2 Rosenbrock Methods Of Second Ordermentioning

confidence: 99%

“…Hence, the computational cost for solving RDEs using the Rosenbrock methods is reduced by a factor of k, where k is the average number of Newton steps, compared with the one using the BDF methods. In a number of numerical experiments provided in [9], Newton's method typically converges in 3-5 iterations in this context. Alternatively, the Schur method [19] can be applied to solve AREs.…”

Section: Application To Rdesmentioning

confidence: 99%

“…Recall that in Rosenbrock methods, only Lyapunov equations need to be solved which is significantly cheaper than solving AREs as in BDF methods. Additional numerical examples can be found in [9].…”

Section: B Examplementioning

confidence: 99%

“…Particularly, matrix-valued algorithms for solving RDEs based on generalizations of the BDF methods have been considered, see [6], [7], [8]. These methods are also suitable for large scale RDEs arising in optimal control problems for parabolic partial differential equations [9]. In general, the BDF methods require fewer function evaluations per step than one step methods, and they allow a simpler streamlined method design from the point of view of order and error estimation.…”

Section: Introduction We Consider Time-varying Riccati Differentiamentioning

confidence: 99%

See 3 more Smart Citations

Rosenbrock Methods for Solving Riccati Differential Equations

Benner¹,

Mena

2013

IEEE Trans. Automat. Contr.

Self Cite

View full text Add to dashboard Cite

Abstract-The Riccati differential equation (RDE) arises in several fields like optimal control, optimal filtering, H∞ control of linear timevarying systems, differential games, etc. In the literature there is a large variety of approaches to compute its solution. Particularly for stiff RDEs, matrix-valued versions of the standard multi-step methods for solving ordinary differential equations have given good results. In this paper we discuss a particular class of one-step methods. These are the linearimplicit Runge-Kutta methods or Rosenbrock methods. We show that they offer a practical alternative for solving stiff RDEs. They can be implemented with good stability properties and allow for a cheap step size control. The matrix valued version of the Rosenbrock methods for RDEs requires the solution of one Sylvester equation in each stage of the method. For the case in which the coefficient matrices of the Sylvester equation are dense, the Bartels-Stewart method can be efficiently applied for solving the equations. The computational cost (computing time and memory requirements) is smaller than for the multi-step methods.

show abstract

Section: Algorithm Iii2 Rosenbrock Methods Of Second Ordermentioning

confidence: 99%

Section: Application To Rdesmentioning

confidence: 99%

Section: B Examplementioning

confidence: 99%

Section: Introduction We Consider Time-varying Riccati Differentiamentioning

confidence: 99%

See 2 more Smart Citations

Rosenbrock Methods for Solving Riccati Differential Equations

Benner¹,

Mena

2013

IEEE Trans. Automat. Contr.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The idea so far in the large-scale case has been to apply the matrix versions of common time-stepping methods, e.g. BDF methods [6], [25] or Rosenbrock methods [7], [25], and realise that in each step an ARE or a number of Lyapunov equations have to be solved. Low-rank algorithms for the solution of these equations exist and we refer to [9], [31] for recent surveys, see also [5].…”

Section: Introductionmentioning

confidence: 99%

Low-Rank Second-Order Splitting of Large-Scale Differential Riccati Equations

Stillfjord

2015

IEEE Trans. Automat. Contr.

View full text Add to dashboard Cite

Abstract-We apply first-and second-order splitting schemes to the differential Riccati equation. Such equations are very important in e.g. linear quadratic regulator (LQR) problems, where they provide a link between the state of the system and the optimal input. The methods can also be extended to generalized Riccati equations, e.g. arising from LQR problems given in implicit form. In contrast to previously proposed schemes such as BDF or Rosenbrock methods, the splitting schemes exploit the fact that the nonlinear and affine parts of the problem, when considered in isolation, have closed-form solutions. We show that if the solution possesses low-rank structure, which is frequently the case, then this is preserved by the method. This feature is used to implement the methods efficiently for large-scale problems. The proposed methods are expected to be competitive, as they at most require the solution of a small number of linear equation systems per time step. Finally, we apply our low-rank implementations to the Riccati equations arising from two LQR problems. The results show that the rank of the solutions stay low, and the expected orders of convergence are observed.

show abstract