A tangential low-rank ADI method for solving indefinite Lyapunov equations

TL;DR

This paper extends the ADI framework for large-scale Lyapunov equations to the indefinite RHS setting by introducing a tangential, low-rank reformulation that compresses updates via tangential directions. It develops a residual-friendly theory and adaptive algorithms for selecting shifts and tangential directions, including eigenvector-based directions of the center matrix and projection-based shifts. The proposed tangential ADI method (TADI) achieves comparable convergence to classical block ADI while greatly reducing the size of the solution factors, with demonstrated benefits on heat-transfer and vibrational-structure benchmarks. Overall, the approach offers a scalable, adaptable tool for efficiently solving indefinite Lyapunov equations in large-scale applications.

Abstract

Continuous-time algebraic Lyapunov equations have become an essential tool in various applications. In the case of large-scale sparse coefficient matrices and indefinite constant terms, indefinite low-rank factorizations have successfully been used to allow methods like the alternating direction implicit (ADI) iteration to efficiently compute accurate approximations to the solution of the Lyapunov equation. However, classical block-type approaches quickly increase in computational costs when the rank of the constant term grows. In this paper, we propose a novel tangential reformulation of the ADI iteration that allows for the efficient construction of low-rank approximations to the solution of Lyapunov equations with indefinite right-hand sides even in the case of constant terms with higher ranks. We provide adaptive methods for the selection of the corresponding ADI parameters, namely shifts and tangential directions, which allow for the automatic application of the method to any relevant problem setting. The effectiveness of the developed algorithms is illustrated by several numerical examples.

Figures (6)

• Figure 1: Convergence behavior of the classical block ADI method (ADI), the tangential ADI with random directions (TADI(rand)) and the tangential ADI with eigenvector directions (TADI(eig)): The eigenvector-based method behaves similarly to the classical approach and converges to a small normalized residual, while the choice of random directions leads to divergence of the tangential ADI.
• Figure 2: Comparison of the different heuristic selection strategies for tangential direction selection: The full heuristic selection based on \ref{['eqn:fullheur']} performs best closely followed by the projected variant \ref{['eqn:romheur']}. The residual-based heuristic selection \ref{['eqn:resheur']} performs slightly worse than the other heuristics but still provides reasonably fast convergence.
• Figure 3: Convergence plots for ADI methods on synthetic examples: The tangential ADI methods converge similarly to the classical block ADI methods in these examples to a normalized residual of about $10^{-12}$. For all shown methods, the implicit residual update formulas align with the true explicit Lyapunov residuals.
• Figure 4: Convergence plot for the linear steel profile example: Both ADI methods converge fast to an accurate solution approximation. The tangential approach yields smaller solution factors than the block approach for the same level of accuracy due to the effective compression via the tangential directions.
• Figure 5: Convergence plot for the bilinear steel profile example: The tangential approach quickly converges to a solution approximation of dimension about $500$, while the classical block approach gives back a solution approximation of dimension $16\,000$. Thus, the tangential approximation is $32\times$ smaller than the results of the classical block method.

Theorems & Definitions (13)

• Lemma 1: ADI iteration errors
• proof
• Theorem 1: Implicit residual factorization
• proof
• Theorem 2: Conjugate double ADI update
• proof
• Theorem 3: Tangential ADI update
• proof
• Theorem 4: Tangential conjugate double ADI update
• proof