Topology-guided sampling of nonhomogeneous random processes

Topological measurements are increasingly being accepted as an important tool for quantifying complex structures. In many applications, these structures can be expressed as nodal domains of real-valued functions and are obtained only through experimental observation or numerical simulations. In both cases, the data on which the topological measurements are based are derived via some form of finite sampling or discretization. In this paper, we present a probabilistic approach to quantifying the number of components of generalized nodal domains of nonhomogeneous random processes on the real line via finite discretizations, that is, we consider excursion sets of a random process relative to a nonconstant deterministic threshold function. Our results furnish explicit probabilistic a priori bounds for the suitability of certain discretization sizes and also provide information for the choice of location of the sampling points in order to minimize the error probability. We illustrate our results for a variety of random processes, demonstrate how they can be used to sample the classical nodal domains of deterministic functions perturbed by additive noise and discuss their relation to the density of zeros.Comment: Published in at http://dx.doi.org/10.1214/09-AAP652 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Probabilistic validation of homology computations for nodal domains

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.Comment: Published at http://dx.doi.org/10.1214/105051607000000050 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Discretization strategies for computing Conley indices and Morse decompositions of flows

Conley indices and Morse decompositions of flows can be found by using algorithms which rigorously analyze discrete dynamical systems. This usually involves integrating a time discretization of the flow using interval arithmetic. We compare the old idea of fixing a time step as a parameters to a time step continuously varying in phase space. We present an example where this second strategy necessarily yields better numerical outputs and prove that our outputs yield a valid Morse decomposition of the given flow

Conley: Computing connection matrices in Maple

In this work we announce the Maple package conley to compute connection and C-connection matrices. conley is based on our abstract homological algebra package homalg. We emphasize that the notion of braids is irrelevant for the definition and for the computation of such matrices. We introduce the notion of triangles that suffices to state the definition of (C)-connection matrices. The notion of octahedra, which is equivalent to that of braids is also introduced.Comment: conley is based on the package homalg: math.AC/0701146, corrected the false "counter example

<Contributed Talk 40>A combinatorial framework for nonlinear dynamics

[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPA