We address the problem of computing a low-rank estimate Y of the solution X of the Lyapunov equation AX + XA' + Q = O without computing the matrix X itself. This problem has applications in both the reduced-order modeling and the control of large dimensional systems as well as in a hybrid algorithm for the rapid numerical solution of the Lyapunov equation via the alternating direction implicit method. While no known methods for low-rank approximate solution provide the two-norm optimal rank k estimate Xk of the exact solution X of the Lyapunov equation, our iterative algorithms provide an effective method for estimating the matrix X(k) by minimizing the error AY + YA'+ Q(F).
Hodel, A. S.,
& Misra, P.
(1997). Least-Squares Approximate Solution of Overdetermined Sylvester Equations. SIAM Journal on Matrix Analysis and Applications, 18 (2), 279-290.