Handbook of Variational Methods for Nonlinear Geometric Data

Philipp Grohs (Redaktør) ; Martin Holler (Redaktør) ; Andreas Weinmann (Redaktør)

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. Les mer
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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.

Fakta

Innholdsfortegnelse

Part I Processing geometric data




1 Geometric Finite Elements



Hanne Hardering and Oliver Sander



1.1 Introduction



1.2 Constructions of geometric finite elements



1.2.1 Projection-based finite elements



1.2.2 Geodesic finite elements



1.2.3 Geometric finite elements based on de Casteljau's algorithm



1.2.4 Interpolation in normal coordinates



1.3 Discrete test functions and vector field interpolation



1.3.1 Algebraic representation of test functions



1.3.2 Test vector fields as discretizations of maps into the tangent bundle



1.4 A priori error theory



1.4.1 Sobolev spaces of maps into manifolds



1.4.2 Discretization of elliptic energy minimization problems



1.4.3 Approximation errors . .



1.5 Numerical examples



1.5.1 Harmonic maps into the sphere



1.5.2 Magnetic Skyrmions in the plane



1.5.3 Geometrically exact Cosserat plates



2 Non-smooth variational regularization for processing manifold-valued



data



M. Holler and A. Weinmann



2.1 Introduction



2.2 Total Variation Regularization of Manifold Valued Data



vii



viii Contents



2.2.1 Models



2.2.2 Algorithmic Realization



2.3 Higher Order Total Variation Approaches, Total GeneralizedVariation



2.3.1 Models



2.3.2 Algorithmic Realization



2.4 Mumford-Shah Regularization for Manifold Valued Data



2.4.1 Models



2.4.2 Algorithmic Realization



2.5 Dealing with Indirect Measurements: Variational Regularization



of Inverse Problems for Manifold Valued Data



2.5.1 Models



2.5.2 Algorithmic Realization



2.6 Wavelet Sparse Regularization of Manifold Valued Data



2.6.1 Model



2.6.2 Algorithmic Realization



3 Lifting methods for manifold-valued variational problems



Thomas Vogt, Evgeny Strekalovskiy, Daniel Cremers, Jan Lellmann



3.1 Introduction



3.1.1 Functional liftin

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