A Randomized Subdivision Algorithm for Determining the Topology of Nodal Sets

Topology is a natural mathematical tool for quantifying complex structures. In many applications, such as, for example, in the context of phase-field models in materials science, the structures of interest arise as subor superlevel sets of continuous functions, i.e., as nodal domains. From a computational point of view, any attempt at constructing a truthful representation of the topology of nodal domains has to involve a discretization step, and it is natural to wonder whether this step introduces topological artifacts. In this paper, we present a randomized subdivision algorithm which, given a smooth function, constructs an adaptive rectangular grid containing the essential information necessary for approximating nodal domains. Furthermore, under mild regularity assumptions the algorithm will also provide a computer-assisted proof for the correctness of the approximation by showing that the rectangular grid can be used to construct rectangular complexes which are homotopy equivalent to the nodal domains of the function. Our method extends the results of [S. Day, W. D. Kalies, and T. Wanner, Multiscale Model. Simul., 7 (2009), pp. 1695–1726], by employing a more accurate and efficient interval arithmetic range enclosure algorithm, as well as developing a randomized subdivision technique to virtually eliminate grid alignment effects.