A New Low-Rank Cholesky-Factor ADI Algorithm Allowing Shifts Anywhere in the Complex Plane with Applications to Data-Driven Model Reduction

Abstract

The low-rank Cholesky factor alternating direction implicit (LRCF-ADI) iteration method is an effective and efficient approach for computing low-rank solutions to large-scale Lyapunov equations in the form \(P\approx ZZ^\top\). This form is useful for balanced truncation, as the square-root algorithm requires computing the controllability and observability Gramians in this form. The standard LRCF-ADI method requires all ADI shifts to have negative real parts, which can be restrictive for applications like frequency-limited and data-driven balanced truncation, where purely imaginary ADI shifts are a more suitable choice. This paper proposes a new LRCF-ADI method where the ADI shifts can be located anywhere in the complex plane, including on the imaginary axis. The proposed generalized LRCF-ADI algorithm reduces to the standard LRCF-ADI algorithm as a special case. The new method is also extended to solve frequency-limited Lyapunov equations, time-limited Lyapunov equations, and Riccati equations. Approximations of matrix logarithm and matrix exponential products using the proposed method are also discussed. LRCF-ADI-based reduced models for balanced truncation can be constructed non-intrusively from transfer function samples at the mirror images of the ADI shifts, without accessing the state-space realization of the original model. Since the standard LRCF-ADI method requires all ADI shifts to have negative real parts, its non-intrusive implementation requires samples in the right half of the complex plane, which cannot be measured in an experimental setting. However, the ADI shifts in the proposed method can lie on the imaginary axis. Exploiting this property, we also propose a data-driven low-rank balanced truncation algorithm that requires only transfer function samples on the imaginary axis, which can be measured experimentally.