  Within the reduced basis methods approach, an effective low-dimensional
subspace of a quantum many-body Hilbert space is constructed in order to
investigate, e.g., the ground-state phase diagram. The basis of this subspace
is built from solutions of snapshots, i.e., ground states corresponding to
particular and well-chosen parameter values. Here, we show how a greedy
strategy to assemble the reduced basis and thus to select the parameter points
can be implemented based on matrix-product-states (MPS) calculations. Once the
reduced basis has been obtained, observables required for the computation of
phase diagrams can be computed with a computational complexity independent of
the underlying Hilbert space for any parameter value. We illustrate the
efficiency and accuracy of this approach for different one-dimensional quantum
spin-1 models, including anisotropic as well as biquadratic exchange
interactions, leading to rich quantum phase diagrams.
