Ippei Obayashi

Combinatorial-topological framework for the analysis of global dynamics

We discuss an algorithmic framework based on efficient graph algorithms and algebraic-topological computational tools. The framework is aimed at automatic computation of a database of global dynamics of a given m-parameter semidynamical system with discrete time on a bounded subset of the n-dimensional phase space. We introduce the mathematical background, which is based upon Conleys topological approach to dynamics, describe the algorithms for the analysis of the dynamics using rectangular grids both in phase space and parameter space, and show two sample applications.

Formation mechanism of a basin of attraction for passive dynamic walking induced by intrinsic hyperbolicity.

Passive dynamic walking is a useful model for investigating the mechanical functions of the body that produce energy-efficient walking. The basin of attraction is very small and thin, and it has a fractal-like shape; this explains the difficulty in producing stable passive dynamic walking. The underlying mechanism that produces these geometric characteristics was not known. In this paper, we consider this from the viewpoint of dynamical systems theory, and we use the simplest walking model to clarify the mechanism that forms the basin of attraction for passive dynamic walking. We show that the intrinsic saddle-type hyperbolicity of the upright equilibrium point in the governing dynamics plays an important role in the geometrical characteristics of the basin of attraction; this contributes to our understanding of the stability mechanism of bipedal walking.

Persistent homology analysis of craze formation

We apply a persistent homology analysis to investigate the behavior of nanovoids during the crazing process of glassy polymers. We carry out a coarse-grained molecular dynamics simulation of the uniaxial deformation of an amorphous polymer and analyze the results with persistent homology. Persistent homology reveals the void coalescence during craze formation, and the results suggest that the yielding process is regarded as the percolation of nanovoids created by deformation.

Non-empirical identification of trigger sites in heterogeneous processes using persistent homology

Macroscopic phenomena, such as fracture, corrosion, and degradation of materials, are associated with various reactions which progress heterogeneously. Thus, material properties are generally determined not by their averaged characteristics but by specific features in heterogeneity (or ‘trigger sites’) of phases, chemical states, etc., where the key reactions that dictate macroscopic properties initiate and propagate. Therefore, the identification of trigger sites is crucial for controlling macroscopic properties. However, this is a challenging task. Previous studies have attempted to identify trigger sites based on the knowledge of materials science derived from experimental data (‘empirical approach’). However, this approach becomes impractical when little is known about the reaction or when large multi-dimensional datasets, such as those with multiscale heterogeneities in time and/or space, are considered. Here, we introduce a new persistent homology approach for identifying trigger sites and apply it to the heterogeneous reduction of iron ore sinters. Four types of trigger sites, ‘hourglass’-shaped calcium ferrites and ‘island’- shaped iron oxides, were determined to initiate crack formation using only mapping data depicting the heterogeneities of phases and cracks without prior mechanistic information. The identification of these trigger sites can provide a design rule for reducing mechanical degradation during reduction.

Persistence diagrams with linear machine learning models

Persistence diagrams have been widely recognized as a compact descriptor for characterizing multiscale topological features in data. When many datasets are available, statistical features embedded in those persistence diagrams can be extracted by applying machine learnings. In particular, the ability for explicitly analyzing the inverse in the original data space from those statistical features of persistence diagrams is significantly important for practical applications. In this paper, we propose a unified method for the inverse analysis by combining linear machine learning models with persistence images. The method is applied to point clouds and cubical sets, showing the ability of the statistical inverse analysis and its advantages.

Persistent Homology and Materials Informatics

This paper provides an introduction to persistent homology and a survey of its applications to materials science. Mathematical prerequisites are limited to elementary linear algebra. Important concepts in topological data analysis such as persistent homology and persistence diagram are explained in a self-contained manner with several examples. These tools are applied to glass structural analysis, crystallization of granular systems, and craze formation of polymers.

Continuation of Point Clouds via Persistence Diagrams

Abstract In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton–Raphson continuation method in this setting. Given an original point cloud P , its persistence diagram D , and a target persistence diagram D ′ , we gradually move from D to D ′ , by successively computing intermediate point clouds until we finally find a point cloud P ′ having D ′ as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.

Chemical state mapping of heterogeneous reduction of iron ore sinter

Iron ore sinter constitutes the major component of the iron-bearing burden in blast furnaces, and its reduction mechanism is one of the keys to improving the productivity of ironmaking. Iron ore sinter is composed of multiple iron oxide phases and calcium ferrites (CFs), and their heterogeneous reduction was investigated in terms of changes in iron chemical state: FeIII, FeII, and Fe0 were examined macroscopically by 2D X-ray absorption and microscopically by 3D transmission X-ray microscopy (TXM). It was shown that the reduction starts at iron oxide grains rather than at calcium ferrite (CF) grains, especially those located near micropores. The heterogeneous reduction causes crack formation and deteriorates the mechanical strength of the sinter. These results help us to understand the fundamental aspects of heterogeneous reduction schemes in iron ore sinter.

Capturing the Global Behavior of Dynamical Systems with Conley-Morse Graphs

We present a computational machinery for describing and capturing the global qualitative behavior of dynamical systems (Arai et al. SIAM J Appl Dyn Syst 8:757–789, 2009). Given a dynamical system, by subdividing the phase space into a finite number of blocks, we construct a directed graph which represents the topological behavior of the system. Then we apply fast graph algorithms for the automatic analysis of the dynamics. In particular, the dynamics can be easily decomposed into recurrent and gradient-like parts which allows further analysis of asymptotic dynamics. The automatization of this process allows one to scan large sets of parameters of a given dynamical system to determine changes in dynamics automatically and to search for “interesting” regions of parameters worth further attention. We also discuss an application of the method to time series analysis. The method presented in Sects. 1 –4 below is given in [1] for the first time, which is based on and combines a number of theoretical results as well as computational software packages developed earlier. For the details, see the original paper [1].