  In this paper, we present an adaptive framework for constructing a
pseudo-optimal reduced model for the frequency-limited H2-optimal model order
reduction problem. We show that the frequency-limited pseudo-optimal
reduced-order model has an inherent property of monotonic decay in error if the
interpolation points and tangential directions are selected appropriately. We
also show that this property can be used to make an automatic selection of the
order of the reduced model for an allowable tolerance in error. The proposed
algorithm adaptively increases the order of the reduced model such that the
frequency-limited H2-norm error decays monotonically irrespective of the choice
of interpolation points and tangential directions. The stability of the
reduced-order model is also guaranteed. Additionally, it also generates the
approximations of the frequency-limited system Gramians that monotonically
approach the original solution. Further, we show that the low-rank alternating
direction implicit iteration method for solving large-scale frequency-limited
Lyapunov equations implicitly performs frequency-limited pseudo-optimal model
order reduction. We consider two numerical examples to validate the theory
presented in the paper.
