Handbook of Variational Methods for Nonlinear Geometric Data

This book covers different, current research directions in the context of variational methods for non-linear geometric data. Each chapter is authored by leading experts in the respective discipline and provides an introduction, an overview and a description of the current state of the art.

Non-linear geometric data arises in various applications in science and engineering. Examples of nonlinear data spaces are diverse and include, for instance, nonlinear spaces of matrices, spaces of curves, shapes as well as manifolds of probability measures. Applications can be found in biology, medicine, product engineering, geography and computer vision for instance.

Variational methods on the other hand have evolved to being amongst the most powerful tools for applied mathematics. They involve techniques from various branches of mathematics such as statistics, modeling, optimization, numerical mathematics and analysis. The vast majority of research on variational methods, however, is focused on data in linear spaces. Variational methods for non-linear data is currently an emerging research topic. 

As a result, and since such methods involve various branches of mathematics, there is a plethora of different, recent approaches dealing with different aspects of variational methods for nonlinear geometric data. Research results are rather scattered and appear in journals of different mathematical communities.

The main purpose of the book is to account for that by providing, for the first time, a comprehensive collection of different research directions and existing approaches in this context. It is organized in a way that leading researchers from the different fields provide an introductory overview of recent research directions in their respective discipline. As such, the book is a unique reference work for both newcomers in the field of variational methods for non-linear geometric data, as well as for established experts that aim at to exploit new research directions or collaborations.

Chapter 9 of this book is available open access under a CC BY 4.0 license at link.springer.com.

1. Geometric Finite Elements.- 2. Non-smooth variational regularization for processing manifold-valued data.- 3. Lifting methods for manifold-valued variational problems.- 4. Geometric subdivision and multiscale transforms.- 5. Variational Methods for Discrete Geometric Functionals.- 6 Variational methods for fluid-structure interactions.- 7. Convex lifting-type methods for curvature regularization.- 8. Assignment Flows.- 9. Geometric methods on low-rank matrix and tensor manifolds.- 10. Statistical Methods Generalizing Principal Component Analysis to Non-Euclidean Spaces.- 11. Advances in Geometric Statistics for manifold dimension reduction.- 12. Deep Variational Inference.­­- 13. Shape Analysis of Functional Data.- 14. Statistical Analysis of Trajectories of Multi-Modality Data.- 15. Geometric Metrics for Topological Representations.- 16. On Geometric Invariants, Learning, and Recognition of Shapes and Forms.- 17. Sub-Riemannian Methods in Shape Analysis.- 18. First order methods for optimization on Riemannian manifolds.- 19. Recent Advances in Stochastic Riemannian Optimization.- 20. Averaging symmetric positive-definite matrices.- 21. Rolling Maps and Nonlinear Data.- 22. Manifold-valued Data in Medical Imaging Applications.- 23. The Riemannian and Affine Geometry of Facial Expression and Action Recognition.- 24. Biomedical Applications of Geometric Functional Data Analysis.

Prof. Dr. Philipp Grohs was born on July 7, 1981 in Austria and has been a professor at the University of Vienna since 2016. In 2019, he also became a group leader at RICAM, the Johann Radon Institute for Computational and Applied Mathematics in the Austrian Academy of Sciences in Linz. After studying, completing his doctorate and working as a postdoc at TU Wien, Grohs transferred to King Abdullah University of Science and Technology in Thuwal, Saudi Arabia, and then to ETH Zürich, Switzerland, where he was an assistant professor from 2011 to 2016. Grohs was awarded the ETH Zurich Latsis Prize in 2014. In 2020 he was selected for an Alexander-von-Humboldt-Professorship award, the highest endowed research prize in Germany. He is a member of the board of the Austrian Mathematical Society, a member of IEEE Information Theory Society and on the editorial boards of various specialist journals.