  This work introduces a reduced order modeling (ROM) framework for the
solution of parameterized second-order linear elliptic partial differential
equations formulated on unfitted geometries. The goal is to construct efficient
projection-based ROMs, which rely on techniques such as the reduced basis
method and discrete empirical interpolation. The presence of geometrical
parameters in unfitted domain discretizations entails challenges for the
application of standard ROMs. Therefore, in this work we propose a methodology
based on i) extension of snapshots on the background mesh and ii) localization
strategies to decrease the number of reduced basis functions. The method we
obtain is computationally efficient and accurate, while it is agnostic with
respect to the underlying discretization choice. We test the applicability of
the proposed framework with numerical experiments on two model problems, namely
the Poisson and linear elasticity problems. In particular, we study several
benchmarks formulated on two-dimensional, trimmed domains discretized with
splines and we observe a significant reduction of the online computational cost
compared to standard ROMs for the same level of accuracy. Moreover, we show the
applicability of our methodology to a three-dimensional geometry of a linear
elastic problem.
