Year: 2007
Type: article
Abstract: 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 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.
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: bronze
Awards: G222768804, Complex Transient Patterns in Phase-Field Models, MULTISCALE ANALYSIS OF NONLINEAR SYSTEMS USING COMPUTATIONAL TOPOLOGY, Dispersion in Spacio-Temporally Heterogeneous Environments, MULTISCALE ANALYSIS OF NONLINEAR SYSTEMS USING COMPUTATIONAL TOPOLOGY, MULTISCALE ANALYSIS OF NONLINEAR SYSTEMS USING COMPUTATIONAL TOPOLOGY, INITITIATIVE FOR THE ENHANCEMENT OF MATHEMATICAL RESEARCH AND EDUCATION AT CORNELL, MULTISCALE ANALYSIS OF NONLINEAR SYSTEMS USING COMPUTATIONAL TOPOLOGY, Collaborative Research: Topological Methods for the Study of Nonlinear Infinite Dimensional Systems, Collaborative Research: Topological Methods for the Study of Nonlinear Infinite Dimensional Systems, Mathematical Sciences Research Institute 5 Year Proposal
OpenAlex ID: W2006214482