Homology has long been accepted as an important computational tool for quantifying complex structures. In many applications these structures arise as nodal domains or excursion sets of real-valued functions, and are therefore amenable to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurately the resulting homology can be computed. In this talk we present a probabilistic algorithm for correctly determining the topology of two-dimensional excursion sets. The approach relies on constructing an appropriate cubical approximation for the nodal domains based on the behavior of the defining function at the vertices of an adaptively generated grid. The algorithm is probabilistic in nature in order to alleviate grid alignment issues, and the homology of the resulting nonuniform cubical complex is determined using coreductions. We illustrate this approach with applications to time-dependent patterns generated by models for phase separation and to Conley index theory.