[Paper Review] Peer Methods for the Solution of Large-Scale Differential Matrix Equations

Peter Benner, Norman Lang|arXiv (Cornell University)|Jul 23, 2018

Matrix Theory and Algorithms30 references4 citations

TL;DR

This paper proposes implicit and Rosenbrock-type peer methods for solving large-scale differential matrix equations, particularly the differential Riccati equation (DRE). By leveraging low-rank symmetric indefinite factorizations and a reformulated Jacobian-free approach, the method reduces computational cost while maintaining high accuracy, outperforming traditional BDF and Rosenbrock schemes in efficiency and stability for large-scale problems.

ABSTRACT

We consider the application of implicit and linearly implicit (Rosenbrock-type) peer methods to matrix-valued ordinary differential equations. In particular the differential Riccati equation (DRE) is investigated. For the Rosenbrock-type schemes, a reformulation capable of avoiding a number of Jacobian applications is developed that, in the autonomous case, reduces the computational complexity of the algorithms. Dealing with large-scale problems, an efficient implementation based on low-rank symmetric indefinite factorizations is presented. The performance of both peer approaches up to order 4 is compared to existing implicit time integration schemes for matrix-valued differential equations.

Motivation & Objective

• Address the challenge of efficiently solving large-scale differential Riccati equations (DREs) arising in optimal control and model reduction.
• Overcome the high computational cost and storage demands of classical implicit time integration schemes for matrix-valued ODEs.
• Develop a low-rank formulation that exploits the numerical rank structure of DRE solutions to reduce complexity.
• Introduce a reformulation of Rosenbrock-type peer methods to minimize Jacobian evaluations, improving computational efficiency.
• Demonstrate superior performance of peer methods—especially Rosenbrock-type variants—compared to classical BDF and implicit schemes on large-scale problems.

Proposed method

• Apply peer methods, a class of general linear methods with peer stages sharing identical accuracy and stability properties, to matrix-valued ODEs.
• Adapt implicit and linearly implicit (Rosenbrock-type) peer schemes for the differential Riccati equation (DRE) in the form $ \dot{X}(t) = A(t)^T X(t) + X(t)A(t) - X(t)S(t)X(t) + W(t) $.
• Develop a reformulation of Rosenbrock-type peer methods that avoids redundant Jacobian applications by reusing stage evaluations, reducing computational overhead.
• Implement an efficient low-rank symmetric indefinite factorization (LRSIF) to solve the algebraic Riccati equations (AREs) arising in each time step, exploiting the low-rank structure of DRE solutions.
• Use low-rank Cholesky-type factorization $ X = ZZ^T $ with $ Z \in \mathbb{R}^{n \times k} $, $ k \ll n $, to represent the solution and reduce storage and computational costs.
• Integrate the peer schemes with Krylov subspace methods to maintain efficiency in large-scale settings without full matrix assembly.

Experimental results

Research questions

• RQ1 Can peer methods be effectively adapted to solve large-scale differential matrix equations, particularly the DRE, with improved efficiency and stability?
• RQ2 How does the proposed reformulation of Rosenbrock-type peer methods reduce Jacobian evaluations and computational cost in the autonomous case?
• RQ3 To what extent does low-rank symmetric indefinite factorization (LRSIF) enhance the scalability of time integration for large-scale DREs?
• RQ4 How do peer methods compare in accuracy and computational efficiency to classical BDF and Rosenbrock schemes for large-scale DREs?
• RQ5 Does the low-rank structure of DRE solutions enable significant performance gains when combined with peer integration schemes?

Key findings

• The Rosenbrock-type peer methods, especially the reformulated variants (mRosPeer), outperform classical BDF and Rosenbrock schemes in computational efficiency for large-scale DREs.
• For the convection-diffusion LTV problem with 800 time steps, the mRosPeer(1) scheme achieved a relative error of 2.09e-02 in 11.16 seconds, outperforming BDF(1) in speed despite similar accuracy.
• In the large-scale LTI convection-diffusion problem with $ n = 2025 $, the mRosPeer(2) scheme required only 543.07 seconds, significantly less than BDF(3) (870.76 s) and BDF(4) (813.07 s), while maintaining high accuracy.
• Peer(4) and RosPeer(4) showed higher computational times (2652.84 s and 913.28 s, respectively) due to increased stage number and order, indicating a trade-off between order and efficiency.
• The reformulated Rosenbrock-type peer schemes (mRosPeer) consistently outperformed their original counterparts (RosPeer) in computational time, especially for higher-order schemes.
• The low-rank symmetric indefinite factorization (LRSIF) enabled efficient solution of AREs in each time step, making large-scale DREs tractable without full matrix storage or operations.