Comprehensible Presentation of Topological Information

Comprehensible Presentation of Topological Information Gunther H. Weber, Kenes Beketayev, Peer-Timo Bremer, Bernd Hamann, Maciej Haranczyk, Mario Hlawitschka and Valerio Pascucci Abstract Topological information has proven very valuable in the analysis of scientific data. An important challenge that remains is presenting this highly abstract information in a way that it is comprehensible even if one does not have an in-depth background in topology. Furthermore, it is often desirable to combine the structural insight gained by topological analysis with complementary information, such as geometric information. We present an overview over methods that use metaphors to make topological information more accessible to non-expert users, and we demonstrate their applicability to a range of scientific data sets. With the increasingly complex output of exascale simulations, the importance of having effective means of providing a comprehensible, abstract overview over data will grow. The techniques that we present will serve as an important foundation for this purpose. Introduction Topology-based data analysis is applicable to a wide range of scientific data understanding problems, including mixing of fluids [10] and the analysis of combustion simulations [4]. The ability of topology-based method to provide a compact, abstract overview over complex data makes them an important tool for the analysis of exascale simulations. However, the information provided by topological structures is usually in the form of abstract graphs, and it is difficult for users without fundamental background in topology to interpret these graphs. Thus, its necessary to find new ways to present information so that it is more easily comprehensible without having an in-depth knowledge of topology. In this context, the use of metaphors, such as the toporrery [11], or topological landscapes [13] shows promising results. In this paper, we describe two ways to present topological information to users in an intuitive fashion. The first method is aimed at the analysis of optimization problems, where minima or maxima of a function are entities of main interest. Here, we focus on a problem from computational chemistry where the main interest focuses on transitions between states—characterized by minima. We have developed a method that uses the information encoded in the Morse complex to present a map-like representation of the system that highlights probable state transition [2]. While currently focused on chemistry, this type of representation can be useful for many optimization problems. Another limitation of current presentation methods is that they focus on structural information. To alleviate this problem we have developed topological cacti. Topological cacti present the contour tree in an intuitive fashion, in- spired by the toporrery representation. In addition to this structural perspective, topological cacti support displaying corresponding quantitative information. Both methods are useful tools in presenting topological information to an in- creasing audience of researchers interested in applying topology-based methods to their data and simplify the analysis of complex simulation results. Map-based Representations for Analyzing Optimization Solution Spaces Understanding the solution space plays an important role in a wide range of optimization applications. Using the topology-based analysis, it is possible to extract relevant information from high-dimensional cost functions. The Morse complex, e.g., encodes information about minima (or maxima) and relationships between them. We have developed a map-based [2] representation that shows this information in an intuitive fashion. While we utilized this representation in the context of computational chemistry, it has applications for a wide range of optimization problems.