This book explains how variational methods 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.

As a first feature, the book provides a comprehensive collection of works describing the state of the art in their respective discipline. The benefit is a standard reference work for researchers (such a reference is presently missing). As a second feature, there are contributions from different communities. A benefit resulting from this is that the book provides cross discipline information. A third feature is that it contains introductory contents in connection with works previously only available in research articles. As a resulting benefit it may serve as an introduction/overview for newcomers. Finally, the book addresses both theoretical aspects as well as applications

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