  Homology has long been accepted as an important computable tool for
quantifying complex structures. In many applications, these structures arise as
nodal domains of real-valued functions and are therefore amenable only to a
numerical study based on suitable discretizations. Such an approach immediately
raises the question of how accurate the resulting homology computations are. In
this paper, we present a probabilistic approach to quantifying the validity of
homology computations for nodal domains of random fields in one and two space
dimensions, which furnishes explicit probabilistic a priori bounds for the
suitability of certain discretization sizes. We illustrate our results for the
special cases of random periodic fields and random trigonometric polynomials.
