Document Type : Research Article

Author

• Yaprak Güldoğan Dericioğlu

Bitlis Eren University, Department of Mathematics, Bitlis, Türkiye

Abstract

The Extended Block Arnoldi–Backward Differentiation Formula (EBA–BDF) is a projection-based integrator for solving large-scale Matrix Differential Riccati Equations (DREs). Like many Krylov subspace methods, its performance depends on the choice of the subspace dimension. In practice, this parameter is often determined through empirical tuning. In this work, we introduce a lightweight, data-driven pre-solver to estimate this dimension \emph{a priori}. The approach uses a Random Forest model trained on spectral norms and discretization parameters, and predicts the required subspace size without modifying the numerical core or stability properties of the original method. Numerical experiments show that the proposed approach can automate parameter selection and reduce the need for manual tuning. The effect is more noticeable in diffusion-dominated regimes, where spectral properties lead to more regular Krylov convergence. By simplifying the initialization stage, the approach supports the practical use of EBA–BDF solvers in large-scale problems.

Highlights

• A data-driven pre-solver estimates the Krylov subspace dimension before integration begins.
• Random Forest regressor trained on spectral norms, conditioning, and discretization parameters.
• Achieves R² > 0.97 in diffusion-dominated regimes and R² ≈ 0.79 in convection–reaction cases.
• Reduces EBA–BDF(2) solver runtime by 40–50% without modifying the numerical core.
• Feature importance rankings align with classical Krylov convergence theory, validating the approach.

Keywords

• Low-rank differential Riccati equation
• Krylov subspace projection
• Backward differentiation formula
• Learning-assisted solvers
• Learning-assisted numerical solvers
• Spectral descriptors
• Machine learning

Main Subjects

• Numerical Methods