Umair Zulfiqar

Umair Zulfiqar

This paper considers large-scale T-Sylvester equations of the form $AX - X^\top E^\top + B_1B_2^\top = 0$, which admit a low-rank solution. It is shown that when the unique solution of the T-Sylvester equation is low-rank, the problem naturally reduces to a tangential interpolation problem via oblique projection. The specific interpolation points and tangential directions needed to obtain the low-rank solution are not known a priori, thus requiring an iterative approach. An iterative interpolatory projection algorithm is proposed based on these interpolation conditions, which iteratively refines the interpolation data as the projection matrices expand in the number of columns. Numerical examples demonstrate that the proposed algorithm converges with projection matrices having significantly fewer columns compared to existing Krylov-subspace-based projection methods, confirming the superiority of the proposed algorithm over existing approaches.

Umair Zulfiqar

The low-rank alternating direction implicit (ADI) method is an efficient and effective solver for large-scale standard continuous-time algebraic Riccati equations that admit low-rank solutions. However, the existing low-rank ADI algorithm for Riccati equations (RADI) cannot be directly applied to general-form Riccati equations, such as those involving indefinite quadratic terms. This paper introduces a generalized RADI algorithm based on an $LDL^\top$ factorization, which efficiently handles the general Riccati equations arising in important applications like state estimation and controller design. An approach for automatically and efficiently generating ADI shifts is also discussed, along with a MATLAB implementation of the generalized RADI method. Numerical examples solving several Riccati equations of order $10^6$ accurately and efficiently are presented, demonstrating the effectiveness of the proposed algorithm.

Umair Zulfiqar, Qiu-Yan Song, Zhi-Hua Xiao et al.

Quadrature-based approximation of Gramians in standard balanced truncation yields a non-intrusive, data-driven implementation that requires only transfer function samples on the imaginary axis, which can be measured experimentally. This idea has recently been extended to several generalizations of balanced truncation, including positive-real balanced truncation, bounded-real balanced truncation, and balanced stochastic truncation. However, these extensions require samples of some spectral factorizations on the imaginary axis, and no practical method exists to obtain such data experimentally. As a result, these non-intrusive implementations are mainly of theoretical interest at present. This paper shows that if the Gramians in these generalizations are approximated via rational interpolation rather than numerical integration, the resulting non-intrusive implementations do not require spectral factorization samples. Instead, they rely only on transfer function samples. Based on this idea, non-intrusive implementations are first developed for several variants of balanced truncation, wherein the Gramians are approximated implicitly using low-rank Alternating Direction Implicit (ADI) methods for Lyapunov and Riccati equations. These formulations require transfer function samples in the right half of the \(s\)-plane, which cannot be measured experimentally. Next, building on these results, novel data-driven non-intrusive implementations are proposed that require only transfer function samples on the imaginary axis. Hence, unlike the quadrature-based and ADI-based approaches, these non-intrusive formulations can be implemented using practically measurable data. Numerical results are presented for benchmark problems in model order reduction, which show that the proposed non-intrusive implementations achieve accuracy comparable to their intrusive counterparts.

Umair Zulfiqar

This paper considers large-scale nonsymmetric continuous-time algebraic Riccati equations (NAREs) that admit low-rank solutions. Low-rank alternating direction implicit (ADI) methods have proven to be an efficient approach for solving several matrix equations, including Lyapunov equations, Sylvester equations, and symmetric Riccati equations. Although a low-rank algorithm for the Sylvester equation has been used as an inner loop in computing low-rank solutions of NAREs, no low-rank ADI algorithm currently exists for NAREs themselves. This paper fills this gap by developing a low-rank ADI algorithm for large-scale NAREs that admit a low-rank solution. Since Lyapunov equations, Sylvester equations, and symmetric Riccati equations are special cases of the NARE, the existing low-rank ADI methods in the literature are special cases of the more general low-rank ADI method proposed here. An automatic and computationally efficient method for shift generation is also discussed, and a subspace-accelerated projection approach is presented to generate shifts for subsequent iterations without user intervention. Once initialized with arbitrary shifts, the proposed algorithm solves large-scale NAREs autonomously, generating its own shifts. Numerical results are presented using benchmark example of order $10^6$, demonstrating the computational efficiency and accuracy of the proposed algorithm.