Towards a mixed-precision ADI method for Lyapunov equations

TL;DR

The paper addresses large-scale Lyapunov equations with a low-rank right-hand side by representing the solution as $X \approx Z Y Z^*$ and applying a mixed-precision LR-ADI that stores outer factors in $\varepsilon_{Z}$, outer increments in $\varepsilon_{VR}$, and inner factors in $\varepsilon_{TY}$ to reduce memory. It systematically experiments with descriptor-system–generated problems, assessing implicit and explicit residuals and convergence across precision variants, and demonstrates that certain mixed-precision configurations can approach double-precision accuracy for some tasks (e.g., H$_2$ norm) while offering memory benefits. However, explicit residuals often reveal accuracy loss, and in many cases the speedup comes at the cost of solution quality, especially for known-solution recovery. The work highlights the potential of mixed-precision strategies for memory-efficient Lyapunov solvers, and outlines future directions such as adaptive precision, iterative refinement, and alternative factorizations to improve robustness and practical impact.

Abstract

We apply mixed-precision to the low-rank Lyapunov ADI (LR-ADI) by performing certain aspects of the algorithm in a lower working precision. Namely, we accumulate the overall solution, solve the linear systems comprising the ADI iteration, and store the inner low-rank factors of the residuals in various combinations of IEEE 754 single and double precision. We empirically test our implementation on Lyapunov equations arising from first- and second-order descriptor systems. For the first-order examples, accumulating the solution in single-precision yields an almost-as-small residual as for the double-precision solution. For certain applications, like computing the H2 norm of a descriptor system, low- or mixed-precision variants of the ADI can be quite competitive

Figures (3)

• Figure 1: Implicit (top) and explicit residuals (bottom) of computing the Observability Gramian \ref{['eq:gramian']} over iteration index $k\in\mathbb{N}$ for ADI($\varepsilon_{Z}$, $\varepsilon_{V\!R}$, $\varepsilon_{TY}$) as shown in \ref{['alg:adi']}; see \ref{['sec:num:gramian']}. The bottom row shows implicit residuals in the background ( ). The symbols S and D refer to IEEE single and double precision, respectively.
• Figure 2: Implicit (top) and explicit residuals (middle) as well as wall-clock times (bottom) of ALE \ref{['eq:intro']} over the rank $g\in\mathbb{N}$ of a random constant term for ADI($\varepsilon_{Z}$, $\varepsilon_{V\!R}$, $\varepsilon_{TY}$) as shown in \ref{['alg:adi']}; see \ref{['sec:num:random-rhs']}. The middle row shows implicit residuals in the background ( ). The symbols S and D refer to IEEE single and double precision, respectively.
• Figure 3: Errors (top), representation overhead (middle), and wall-clock times (bottom) over the rank $\hat{z}\in\mathbb{N}$ of a known solution ${\hat{X}}$ for ADI($\varepsilon_{Z}$, $\varepsilon_{V\!R}$, $\varepsilon_{TY}$) as shown in \ref{['alg:adi']}; see \ref{['sec:num:known-X']}. The symbols S and D refer to IEEE single and double precision, respectively. If visible, the dashed line denotes the system dimension $n\in\mathbb{N}$. The solid markers denote uncompressed ADI approximates $X$, opaque markers denote low-rank compressions of $X$.

Theorems & Definitions (7)

• remark 1
• example 1: Steel Profile morwiki_steelBenS05bSaaB21
• example 2: Chip morwiki_convectionmorMooRGetal04
• example 3: Triple Chain TruV09
• remark 2
• remark 3
• remark 4