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Computational Complexity Theory for Advanced Function Spaces in Analysis

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Computational Complexity Theory for Advanced Function Spaces in Analysis Computational Complexity Theory for Advanced Function Spaces in Analysis vom Fachbereich Mathematik der Technischen Universit¨atDarmstadt zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation Tag der Einreichung: 07.09.2016 Tag der m¨undlichen Pr¨ufung: 02.11.2016 Referent: Prof. Dr. Martin Ziegler 1. Korreferent: Prof. Dr. Ulrich Kohlenbach 2. Korreferent: Prof. Dr. Akitoshi Kawamura von Diplom-Mathematiker Florian Steinberg aus Frankfurt a. M. Darmstadt, D 17 2017 Acknowledgments I would like to thank my supervisors Martin Ziegler, Akitoshi Kawamura and Ulrich Kohlenbach for making this dissertation possible. I would like to thank Martin Ziegler for insightful discussion of the contents of my research, for many suggestions of further research and for keeping me from straying to far. I would like to thank Akitoshi Kawa- mura for being an excellent host during my stays in Japan and for providing many useful tips and valuable ideas and inspirations regarding my research. I would like to thank Ulrich Kohlenbach for making me see the big picture and pointing out many connections to other fields of research as well as in depth discussion of my research. I would like to thank the IRTG 1529 and all the people involved with it for giving me the opportunity to pursue the research mostly without additional duties and while being able to gather new experiences abroad. In particular I would like to thank the IRTG 1529 for paying for my research stays in Japan and Korea and the travel expenses for many conferences. I would like to thank the community of my field of research for providing a healthy academic environment. I would like to thank Arno Pauly, Holger Thies and Matthias Schr¨oderfor the many hours of discussion. Some others whose input I deeply appreci- ate are: Hugo F´er´ee,Eike Neumann, Norbert M¨uller,Franz Brause, Patrick Tolksdorf, Martin Rapp/Bolkart, and those I forgot. I would like to thank Christoph Dittman and Friederike Steglich for reading through parts of my thesis while not being from the field. I would like to thank the working group of logics at TU Darmstadt for the sup- port. I appreciate that I got the opportunity to organize a conference toghether with the other PhD students: Angeliki Koutsoukou-Argyraki, Julian Bitterlich, Felix Canavoi and Daniel K¨ornlein. These people also patiently listened to my ramblings during the sessions of the "Doktorandentreff” which I deeply enjoyed. Thanks for that, and also for interesting discussion of other research topics. Special thanks go to Angeliki for starting the "Doktorandentreff”. I would like to thank my friends both in Germany and Japan, and my family for giving me the support I needed to finish this project. Darmstadt, February 1, 2017 I Deutsche Zusammenfassung Motivation fur¨ diese Doktorarbeit ist das Vorantreiben der Suche nach einem auf ei- nem realistisches Maschinenmodell basierenden, mathematisch rigorosen Rahmen fur¨ Effizienzuberlegungen¨ uber¨ die Numerik partieller Differenzialgleichungen. W¨ahrend die Berechenbarkeitstheorie fur¨ kontinuierliche Strukturen auf die meisten Situationen an- wendbar ist und best¨andig weiter entwickelt wird, ist es selbst in simplen F¨allen oft noch unklar, welche Algorithmen als effizient zu gelten haben. In dieser Arbeit wird das Problem im Rahmen der Darstellungen zweiter Ordnung angegangen. Diese beziehen sowohl ihren Namen, als auch ihren Begriff von Polyno- mialzeitberechenbarkeit aus der Komplexit¨atstheorie der Funktionale auf dem Baire Raum, auch Komplexit¨atstheorie zweiter Ordnung genannt. Als Maschinenmodell die- nen Orakel-Turingmaschinen. Die Darstellungen zweiter Ordnung bieten viel Freiraum in der Gestaltung von Rechenmodellen und auch einen Grundstock an Konstruktuionen und behandelten Problemen. Insbesondere existiert eine etablierte Darstellung fur¨ das Rechnen mit stetigen Funktionen auf dem Einheitsintervall. Diese ist als die schw¨achste charakterisiert, die eine Auswertung in Polynomialzeit m¨oglich macht. Als ein erster Schritt wird die schw¨achste Darstellung zweiter Ordnung fur¨ die in- tegrierbaren Funktionen angegeben, die es erm¨oglicht in Polynomialzeit Integrale aus- zuwerten. Unglucklicherweise¨ erweist sich diese als unstetig und damit ungeeignet. Auf der Suche nach einer geeigneteren Darstellung werden die generellen Beschr¨ankungen des Rechnens mit Darstellungen zweiter Ordnung auf metrischen R¨aumen untersucht. Dabei spielt die von Kolmogorov eingefuhrte¨ metrische Entropie eines kompakten metrischen Raumes eine tragende Rolle. Es stellt sich heraus, dass es einen direkten Zusammenhang zwischen dieser und der Existenz `kurzer' Darstellungen gibt, bezuglich¨ derer sich die Me- trik `schnell' berechnen l¨asst. Um diese Resultate auf R¨aume integrierbarer Funktionen anwendbar zu machen, werden quantitative Versionen der Klassifikationsresultate kom- pakter Teilmengen, bekannt als die S¨atze von Arzel`a-Ascoli und Fr´echet-Kolmogorov, untersucht. Auf die gewonnen Erkenntnisse gestutzt¨ wird eine Familie von Darstellungen fur¨ Lp- R¨aume konstruiert. Es wird beweisen, dass diese berechenbar ¨aquivalent zu den Darstel- lungen selbiger R¨aume als metrische R¨aume sind und gezeigt, dass sie die Lp-Norm in Exponentialzeit berechnenbar machen. Dies ist auch die Komplexit¨at der Supremums Norm auf den stetigen Funktionen. Eine ¨ahnliche Konstruktion fuhrt¨ zu Darstellungen von Sobolev-R¨aumen. Die Berechenbarkeit unterschiedlichster Operationen in Polyno- mialzeit wird gezeigt, aus technischen Grunden¨ nur fur¨ den eindimensionalen Fall. Abschließend wird ein Resultat pr¨asentiert, das die Schwierigkeit des Berechnens des L¨osungsoperators zum Dirichlet-Problem fur¨ Poisson's Gleichung auf der Einheitskugel gleichsetzt mit der Schwierigkeit eine stetigen Funktion zu integrieren. Zum Verglei- chen von Schwierigkeiten von Aufgaben werden Polynomialzeit-Weihrauch-Reduktionen verwendet. II Translation of the German Abstract This PhD thesis presents progress in the search for a mathematical rigorous framework for efficient numerics of partial differential equations based on a realistic machine model. While the computability theory of continuous structures is well developed and still an active field of research, in most settings it remains unclear what computations should be considered feasible. This problem is tackled within the framework of second-order representations. The name as well as the notion of polynomial-time computability of the framework is inherited from the complexity theory of functionals on the Baire space, also called second-order complexity theory. As model of computation we use oracle Turing machines. Second- order representations offer a great deal of freedom in developing models of computation and a supply of well investigated structures. In particular, there exists an established second-order representation of the set of continuous functions on the unit interval. This representation has been classified as the weakest representation such that evaluation is possible in polynomial time. As a first step we specify the weakest representation of the integrable functions such that integration is polynomial-time computable. This representation turns out to be discontinuous and therefore not suitable. We go on to explore the general restrictions of bounded-time computations on metric spaces within the framework of second-order representations. The notion of metric entropy, originally introduced by Kolmogorov, is used to classify those compact metric spaces that allow for a short representation such that the metric is computable efficiently. To be able to apply the results to spaces of integrable functions we investigate quantitative versions of classification results of their compact subsets called the Arzel`a-AscoliTheorem and the Fr´echet-Kolmogorov Theorem. We use the above to propose a family of representations for Lp-spaces. These are shown to be computably equivalent to the standard representations of the same spaces as metric spaces. Furthermore, we prove that the norm can be computed in exponential time. This is also the case for the supremum norm on the continuous functions on the unit interval. A similar family of representations is presented for Sobolev spaces. These representations are investigated in some detail. Several operators on Sobolev spaces are proven to be polynomial-time computable with respect to these representations. For technical reasons only the one-dimensional case is discussed. Finally, we present a result that classifies the computational complexity of the solution operator of the Dirichlet Problem for Poisson's Equation on the unit disk as that of inte- grating a continuous function. As a tool for the comparison, polynomial-time Weihrauch reductions are used. III Contents 1 Introduction1 1.1 Background . .1 1.1.1 Historical digest and references . .4 1.1.2 Organization of the thesis . .5 1.1.3 Basic notational conventions . 11 1.2 The model of computation . 13 1.2.1 Representations . 13 1.2.2 Second order complexity theory . 14 1.2.3 Second-order representations . 16 1.2.4 Representations of compact spaces and length . 18 2 Minimal representations 21 2.1 Representing continuous functions . 21 2.1.1 Moduli of continuity . 21 2.1.2 Representing the continuous functions . 22 2.1.3 Some complexity theoretical results . 24 2.2 Representing integrable functions . 26 2.2.1 Singularity moduli . 26 2.2.2 The singular representation in one dimension . 29 2.2.3 Higher dimensions . 29 2.2.4 Discontinuity . 31 2.3 Spreads . 33 2.3.1 The spread of a representation . 34 2.3.2 Joins . 34 2.3.3 Applications . 36 3 Metric spaces 39 3.1 Metric entropies and spanning bounds . 39 3.1.1 Metric entropy in normed spaces . 41 3.1.2 Moduli of continuity . 44 3.2 Metric entropy and complexity . 45 3.2.1 From complexity to metric entropy . 46 3.2.2 From metric entropy to complexity . 51 3.3 Variations of the results . 53 3.3.1 Cantor space representations . 54 3.3.2 Space-bounded computation . 57 V Contents 4 Arzel`a-Ascoliand Fr´echet-Kolmogorov 63 4.1 Arzel`a-Ascoli . 63 4.2 Lp-spaces . 66 4.2.1 Lp-moduli . 67 4.2.2 Examples . 70 4.2.3 Sobolev spaces . 73 4.3 The Fr´echet-Kolmogorov Theorem . 75 4.3.1 The upper bound .
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