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We develop here a computationally effective approach for producinghigh-quality $\mathcal{H}_\infty$-approximations to large scale lineardynamical systems having multiple inputs and multiple outputs (MIMO). We extendan approach for $\mathcal{H}_\infty$ model reduction introduced by Flagg,Beattie, and Gugercin for the single-input/single-output (SISO) setting, whichcombined ideas originating in interpolatory $\mathcal{H}_2$-optimal modelreduction with complex Chebyshev approximation. Retaining this framework, ourapproach to the MIMO problem has its principal computational cost dominated by(sparse) linear solves, and so it can remain an effective strategy in manylarge-scale settings. We are able to avoid computationally demanding$\mathcal{H}_\infty$ norm calculations that are normally required to monitorprogress within each optimization cycle through the use of "data-driven"rational approximations that are built upon previously computed functionsamples. Numerical examples are included that illustrate our approach. Weproduce high fidelity reduced models having consistently better$\mathcal{H}_\infty$ performance than models produced via balanced truncation;these models often are as good as (and occasionally better than) modelsproduced using optimal Hankel norm approximation as well. In all casesconsidered, the method described here produces reduced models at far lower costthan is possible with either balanced truncation or optimal Hankel normapproximation.

Interpolatory methods for $\mathcal{H}_\infty$ model reduction of multi-input/multi-output systems