Preliminary list of abstracts

Note: We are using the fantastic MathJax JavaScript library to typeset the mathematics on this web page. You can right-click on a formula to zoom in or to select the rendering engine you like (all under 'Settings'). Under Firefox the default renderer turns out to be MathML (which is quick), but you might prefer the HTML-CSS renderer for more faithful LaTeX rendering. If you encounter any problems with this setup: then please email us!

Click here to go back to the list of abstracts.

Spectral kernel methods for simulation data and the eigenvector switching problem
Tue, 14:50--15:15
  • Iza Teran, Victor Rodrigo (Numerical Software, Fraunhofer Institute for Scientific Computing, Germany)

Spectral methods are a family of methods of machine learning that have been applied for data analysis, visualization, clustering and classification. Such methods are subclasses of more general, so called manifold methods, that assume the data one observes reside at or near an intrinsic low dimensional sub-manifold in a higher-dimensional space. One does not have access to this underlying manifold but instead try to approximate it from a point cloud using point correlations. One of the most common strategy is to construct an adjacency graph with specific kernels an exploit the obtained structure, which is found through the eigenvalues and eigenvectors of the corresponding matrix. The graph is considered as a proxy for the manifold and the analysis based on this structure corresponds to the desired analysis based on the geometric structure of the manifold. The theoretical justification of this intuition is a research work developed in the last few years. Nevertheless the application of it has shown very interesting results in many areas. For the development of new products in industry, especially in the car industry, computer simulated models of physical process are used, an example of it is the model of a car impacting against a wall at a certain velocity. The high deformation produced can be model on a computer and each model is very fine (several million data points) and detailed (several hundred car components are in the model) that means, we have a high extrinsic dimensionality. A car company generates several hundred simulations, with a small geometrical or constructive parameter change each day, in order to achieve a desired design configuration with a lower cost but at the same time maintaining the security standards. There is a strong need for the engineers to organize the information about all changes globally, including its effect on some output quantity of interest like for example the Head Injury Index. The application of spectral kernel methods for the analysis of this type of data show very interesting results and it is demonstrated that a global organization of the datasets is possible, based on the fact that the intrinsic dimensionality of such datasets is actually very low. Nevertheless, the usefulness of it depends on the type of kernel and the corresponding spectral gap. If eigenvalues get close, the eigenvector switching problem appears and the use of eigenvectors is not justified anymore. In this talk we present the results of our work on the use of spectral kernel methods for data from engineering simulations, we describe several applications and give some theoretical inside that justify the results. After this we describe the eigenvector switching problem, including a practical example of it. Considering the underlying theory it is possible to analyze the wobbly behavior of eigenvectors and we contribute to a possible control of this behavior. It is shown that at least for perturbations caused by geometrical changes, the suggested control effectively eliminate it.