Ambient Approximation of Functions and Functionals on Embedded Submanifolds Vom Fachbereich Mathematik der Technischen Universität Darmstadt zur Erlangung des akademischen Grades eines Doctor rerum naturalium (Dr. rer. nat) genehmigte Dissertation von Dipl. Math. Lars-Benjamin Maier Erster Gutacher: Prof. Dr. Ulrich Reif Zweiter Gutachter: Prof. Dr. Armin Iske Dritter Gutacher: Prof. Dr. Oleg Davydov Darmstadt 2018 Maier, Lars-Benjamin: Ambient Approximation of Functions and Functionals on Embedded Submanifolds Darmstadt, Technische Universität Darmstadt Jahr der Veröffentlichung der Dissertation auf TUprints: 2018 Tag der mündlichen Prüfung: 31.08.2018 Veröffentlicht unter CC BY-SA 4.0 International https://creativecommons.org/licenses/ 2 Contents 1 Introduction and Related Work 1 2 Embedded Submanifolds and Function Spaces 9 2.1 Embedded Submanifolds ........................... 9 2.1.1 Closed Submanifolds, Normal Foliation and Normal Extension 10 2.1.2 Open Submanifolds and General Foliations ............ 14 2.2 Function Spaces on Embedded Submanifolds ............... 18 2.2.1 Trace Theorems ............................. 21 2.2.2 Friedrichs’ Inequality .......................... 24 3 Ambient Approximation Theory 25 3.1 General Approximation Operators ...................... 26 3.2 Quasi-Projection and Polynomial Reproduction .............. 29 3.2.1 Definition and Approximation Power ................ 29 3.2.2 Interpolating Quasi-Projections ................... 32 3.3 Tensor Product B-Splines ........................... 34 3.3.1 Definition and Initial Approximation Results ............ 35 3.3.2 Locality of Approximation and the Continuity of Operators ... 37 3.3.3 Improved Approximation Results ................... 41 3.3.4 Approximation in Fractional Orders and under fixed Interpola- tion Constraints ............................. 48 3.3.5 Approximating Normal Derivatives ................. 51 3.3.6 Practical Examples ........................... 57 i 4 Tangential Calculus, Function Spaces and Functionals 63 4.1 Tangential Derivatives, Gradients and Hessians ............. 64 4.1.1 Definition and Basic Properties .................... 64 4.1.2 Intrinsic Characterisation of Sobolev Spaces ........... 70 4.2 Unisolvency in Tangential Calculus ..................... 75 4.2.1 Unisolvency and Curvature ...................... 76 4.2.2 Unisolvency and Isometry ....................... 81 4.3 Tangential Bilinear Functionals and Tangential Energies ........ 89 4.3.1 Some Specific Energy Functionals .................. 91 4.3.2 Functional Residuals and Norm Distances ............. 95 4.4 Approximately Intrinisic Functionals .................... 96 5 Ambient Functional Approximation Methods 105 5.1 Penalty Approximation in Functional Optimisation ............ 106 5.2 Penalty Approximation for Energies in Convex Sets ........... 109 5.3 Penalty Approximation for Energy Residuals ............... 115 5.4 Penalty Approximation of Energies with Linear Portion ......... 119 6 Scattered Data Problems on Embedded Submanifolds 125 6.1 Sparse Data Extrapolation .......................... 125 6.1.1 Problem Statement and Naive Approaches ............. 126 6.1.2 Extrapolation by Penalty Based Energy Minimisation ...... 128 6.2 Smoothing with Scattered Data Sites .................... 142 6.3 Irregular Samplings .............................. 149 6.3.1 A Two-Stage Approximation Approach ............... 150 6.3.2 A Bilevel Algorithm ........................... 156 7 Partial Differential Equations on Embedded Submanifolds 159 7.1 Elliptic Problems on Closed Submanifolds ................. 159 7.2 Ideas for Open Subdomains of Submanifolds ............... 170 ii 8 Conclusion and Prospects 173 9 Appendix 177 9.1 An Atlas for Embedded Submanifolds .................... 177 9.1.1 The Exponential Map .......................... 177 9.1.2 The Construction of an Atlas ..................... 178 9.2 A Theory of Function Spaces on Embedded Submanifolds ....... 181 9.2.1 Norm Equivalences for Integer Order Spaces ........... 181 9.2.2 Embeddings in Sobolev Spaces .................... 183 9.2.3 A Note on Besov Spaces ........................ 184 9.2.4 Consequences of Sobolev Space Interpolation ........... 185 9.2.5 Trace Theorems ............................. 189 9.2.6 Friedrichs’ Inequality .......................... 192 9.3 Notes on Riemannian Geometry ....................... 197 Bibliography 201 Index 209 iii iv Acknownledgments First of all, I would like to thank Prof. Ulrich Reif for supervising the voyage that ultimately culminated in this thesis. In particular, I would like to thank him for the opportunity to follow my own agenda and investigate what I found interesting. I would also like to thank Prof. Oleg Davydov and Prof. Armin Iske for agreeing to coreferee this thesis. My special thanks go to Prof. Karsten Große-Brauckmann for his support and his patience with my questions about Riemannian geometry. I would further like to thank my colleagues at TU Darmstadt for some very helpful ideas, for proofreading and their companionship, the team of the DFS section of Cenit AG for giving me a bit of distraction when my thoughts were stuck and — last but not least — my wife for her patience and understanding. v vi Abstract of the Thesis While many problems of approximation theory are already well-understood in Eu- clidean space and its subdomains, much less is known about problems on subman- ifolds of that space. And this knowledge is even more limited when the approxima- tion problem presents certain difficulties like sparsity of data samples or noise on function evaluations, both of which can be handled successfully in Euclidean space by minimisers of certain energies. On the other hand, such energies give rise to a considerable amount of techniques for handling various other approximation prob- lems, in particular certain partial differential equations. The present thesis provides a deep going analysis of approximation results on sub- manifolds and approximate representation of intrinsic functionals: It provides a method to approximate a given function on a submanifold by suitable extension of this function into the ambient space followed by approximation of this extension on the ambient space and restriction of the approximant to the manifold, and it investigates further properties of this approximant. Moreover, a differential cal- culus for submanifolds via standard calculus on the ambient space is deduced from Riemannian geometry, and various energy functionals are presented and approx- imately handled by an approximate application of this calculus. This approximate handling of functionals is then employed in several penalty-based methods to solve problems such as interpolation in sparse data sites, smoothing and denoising of function values and approximate solution of certain partial differential equations. vii viii German Summary — Deutsche Zusammenfassung Die vorliegende Dissertationsschrift befasst sich mit Problemen der Approximation auf eingebetteten Untermannigfaltigkeiten des euklidischen Raumes. Nach einer kurzen Einführung über geeignete Untermannigfaltigkeiten und über Funktionenräume auf ebensolchen erweitert sie zunächst bekannte Konvergen- zresultate für die sogenannte ambient approximation method und deren bisher wichtigsten Spezialfall, die ambient B-spline method. Insbesondere generalisiert sie diese auf Untermannigfaltigkeiten mit höherer Codimension und ergänzt die betreffenden Resultate um Ergebnisse zu Approximation unter einer endlichen, festen Anzahl an Interpolationsbedingungen und um Approximationsaussagen zur Ableitung entlang des Normalenbündels der Untermannigfaltigkeit. Im Anschluss wird ein intrinsischer, tangentialer Calculus für solche Unterman- nigfaltigkeiten eingeführt, der auf bestehenden Konzepten der riemannschen Ge- ometrie basiert. Dabei wird insbesondere die Beziehung zwischen intrinsischer, tangentialer Ableitung und der korrespondierenden euklidischen Ableitung ent- lang von Elementen des Tangentialraums beleuchtet. Außerdem werden in diesem Zusammenhang eine Reihe von intrinsischen Funktionalen eingeführt, und es wird insbesondere das Konzept der polynomiellen Unisolvenz in die Situation des tan- gentialen Calculus übertragen. Als nächstes wird eine Methodik vorgestellt, die die Approximation der Optima be- sagter Funktionale mittels im Kern extrinsischer Methoden erlaubt, die sich auf die sogenannten penalty-Verfahren beziehen. Für die resultierende ambient penalty approximation werden Konvergenzresultate präsentiert, und es werden beispiel- haft verschiedene interessante Funktionale diskutiert, die unter anderem Extrap- olation aus wenigen verstreuten Datenpunkten auf der Untermannigfaltigkeit Ϻ, Glättung von Funktionswerten über Ϻ und die näherungsweise Lösung elliptischer partieller Differentialgleichungen auf Ϻ erlauben. ix x Chapter 1 Introduction and Related Work The problem of approximation of functions on manifolds, particularly surfaces, has gained both relevance and attraction over the last years. The relevance came with the increased capability and popularity of computer systems in representation, pro- cessing and simulation of problems in engineering, manufacturing and the natural and medical sciences. Computer graphics (CG), medical imaging, computer aided design (CAD) and manufacturing (CAM) or recently industry 4.0 are just some of the keywords that one frequently encounters in this area. And the problems that need to be solved are as diverse as • simulation of biological, physical or manufacturing processes on computer