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In this contribution, we extend the concept of $\mathcal{H}_2$ inner productand $\mathcal{H}_2$ pseudo-optimality to dynamical systems modeled bydifferential-algebraic equations (DAEs). To this end, we derive projectedSylvester equations that characterize the $\mathcal{H}_2$ inner product interms of the matrices of the DAE realization. Using this result, we extend the$\mathcal{H}_2$ pseudo-optimal rational Krylov algorithm for ordinarydifferential equations to the DAE case. This algorithm computes the globallyoptimal reduced-order model for a given subspace of $\mathcal{H}_2$ defined bypoles and input residual directions. Necessary and sufficient conditions for$\mathcal{H}_2$ pseudo-optimality are derived using the new formulation of the$\mathcal{H}_2$ inner product in terms of tangential interpolation conditions.Based on these conditions, the cumulative reduction procedure combined with theadaptive rational Krylov algorithm, known as CUREd SPARK, is extended to DAEs.Important properties of this procedure are that it guarantees stabilitypreservation and adaptively selects interpolation frequencies and reducedorder. Numerical examples are used to illustrate the theoretical discussion.Even though the results apply in theory to general DAEs, special structureswill be exploited for numerically efficient implementations.

$\mathcal{H}_2$ Pseudo-Optimal Reduction of Structured DAEs by Rational Interpolation