Numerically Validated Homology Computation

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.



Probabilistic and numerical validation of homology computations for nodal domains, S. Day, W.D. Kalies, K. Mischaikow, T. Wanner, Electronic Research Announcements of the American Mathematical Society 13, 60-73, 2007.

Probabilistic validation of homology computations for nodal domains, K. Mischaikow, T. Wanner, Annals of Applied Probability 17(3), 980-1018, 2007.

Verified homology computations for nodal domains, S. Day, W.D. Kalies, T. Wanner, SIAM Journal on Multiscale Modeling & Simulation 7(4), 1695-1726, 2009.

Topology-guided sampling of nonhomogeneous random processes, K. Mischaikow, T. Wanner, Annals of Applied Probability, to appear.