Optimal model reduction for sparse linear systems

A novel $H^2$ optimal model reduction problem is formulated for large-scale sparse linear systems as a nonconvex optimization problem. The analysis on the gradient of the objective function shows that the nonconvex optimization problem can be simplified to solve a linear equation in multi-input single-output (MISO) or single-input multi-input (SIMO) cases. Thus, a simple and efficient model reduction algorithm based on the simplified problem is proposed for huge-scale systems. Moreover, an additional algorithm with guaranteed global convergence is developed for multi-input multi-output (MIMO) cases by focusing on the convexity of the objective function in terms of each variable based on the proximal alternating projection method. Both the algorithms guarantee that all the eigenvalues of the state matrix of a generated reduced system with the state dimension $r$ completely coincide with the $r$ largest eigenvalues of the original state matrix. The numerical experiments demonstrate that the algorithm proposed for MISO or SIMO cases and the algorithm developed for MIMO cases can reduce sparse systems having original dimensions larger than $10^7$ and $10^6$ to a practical time period, respectively. Furthermore, it is shown that the proposed algorithms of this study deliver superior performance to an existing method for large-scale systems in terms of the objective function, eigenvalues of the reduced state matrix, and computational time