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Computational
Homology
Project
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Much of the fascination and challenge of studying nonlinear dynamical
systems arises from the complicated temporal and/or spatial behavior they
exhibit. On the level of mathematics this complicated behavior can occur
at all scales both in phase space and in parameter space. Somewhat
paradoxically,
from a scientific perspective, this points to the need for a coherent set
of mathematical techniques that is capable of extracting coarse but robust
information about the structure of these systems. Furthermore, most of
our understanding of specific systems comes from
experimental observation or
numerical simulations and thus it is important that these techniques
be computationally efficient.
Algebraic Topology, and in particular Homology, is the classical
mathematical tool for the global analysis of nonlinear spaces and
functions. The material described through these web pages represents
our ongoing effort to develop computationally efficient methods
to compute homology and to apply these methods to nonlinear problems.
Highlights of Research
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Phase separation processes in compound materials can produce intriguing and complicated patterns. Yet, characterizing the geometry of these patterns quantitatively can be quite challenging. Here we propose the use of computational homology to obtain such a characterization. Our method is illustrated for the complex microstructures observed during spinodal decomposition and early coarsening in both the deterministic Cahn-Hilliard theory, as well as in the stochastic Cahn-Hilliard-Cook model. While both models produce microstructures that are qualitatively similar to the ones observed experimentally, our topological characterization points to significant differences. One particular aspect of our method is its ability to quantify boundary effects in finite size systems. | |
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Computational Homology is used as a measure of the complexity of geometric structures in the weakly turbulent state of spiral defect chaos in experiments on Rayleigh-Bénard convection. Different attractors of spiral defect chaos are distinguished by their homology. This technique reveals asymmetric patterns. In addition global stochastic ergodicity is observed for system parameter values where locally chaotic dynamics has been previously reported. | |
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Coarse, topological measurements of dynamical systems defined by maps can be used to uncover information about invariant sets for the system. These techniques are based on the Conley index and have been used to detect invariant structures from fixed points and periodic orbits, to connecting orbits and sets which exhibit chaotic symbolic dynamics. More notably, these techniques have been used to study both finite-dimensional and infinite-dimensional systems and to prove the existence of unstable invariant sets. The method relies on first building a pair of compact sets called an index pair and then computing the relative homology of the pair. This information is then used to make conclusions about the associated invariant structures. | |
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It is well established both numerically and experimentally that nonlinear systems involving diffusion, chemotaxis, and/or convection mechanisms can generate complicated time-dependent patterns. Since this phenomenon is global in nature, obtaining a quantitative mathematical characterization that to some extent records or preserves the geometric structures of the complex patterns is difficult. Here we show that using algebraic topology, in particular homology, we can measure Lyapunov exponents that imply the existence of spatial-temporal chaos and suggest a tentative step towards the classification and/or identification of patterns within the FitzHugh-Nagumo system. | |
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A method based on the Conley index theory for proving the existence of periodic trajectories in a smooth dynamical system in Rn is developed. This method can be applied in the cases where a hyperbolic periodic orbit is numerically observed. The method uses cubical index pairs and requires the computation of the map induced in homology by the cubical index map. | |
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Within this project, quantitative characterizations for the thermal-elastic response of calcite-based polycrystals are developed. The characterization is based on topological measurements such as the number of components and the number of handles of a complex microstructure. These characterizations are then applied to characterize both the grain boundary misorientations in the polycrystal and the resulting elastic energy density and principal stress fields. It is demonstrated that the topological analysis can quantitatively distinguish between different types of grain boundary misorientations, as well as between the resulting differences in the response fields. | |
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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. | |
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This project is aimed at the development of efficient algorithms and software for the homology computation. The necessity of such algorithms appears evident as new applications of the homology computation arise in research. Currently available software, including the package that can be downloaded from the CHomP website, allows to compute homology of n-dimensional cubical sets and maps, including relative homology, but there is still a lot of room for improvement, and processing large data is often very time and memory consuming. This motivates the search for improved algorithms and for new ways of approach to the task of homology computation.
This is a joint project of the CHomP group in Atlanta and the CAPD group in Kraków. | |
Supported in part by grants from
DARPA,
DOE,
and NSF.
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