Validation of Homology Computations
Homology is 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 the papers cited below, we present a
probabilistic approach to quantifying the validity of homology
computations for nodal domains of random Fourier series in one and two
space dimensions, which furnishes explicit probabilistic a-priori
bounds for the suitability of certain discretization sizes. In
addition, we introduce a numerical method for verifying the homology
computation using interval arithmetic.