Freie Universitat¨ Berlin Random partial differential equations on evolving hypersurfaces Vorgelegt von Ana Djurdjevac Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) am Fachbereich Mathematik und Informatik der Freien Universitat¨ Berlin Berlin, 2018 Betreuer: Prof. Dr. Ralf Kornhuber Freie Universitat¨ Berlin Fachbereich Mathematik und Informatik Gutachter: Prof. Dr. Ralf Kornhuber Prof. Dr. Charles Elliott (University of Warwick) Tag der Disputation: 27. November 2018 Selbststandigkeitserkl¨ arung¨ Hiermit erklare¨ ich, dass ich diese Arbeit selbststandig¨ verfasst und keine anderen als die angegebe- nen Hilfsmittel und Quellen verwendet habe. Ich erklare¨ weiterhin, dass ich die vorliegende Arbeit oder deren Inhalt nicht in einem fruheren¨ Promotionsverfahren eingereicht habe. Ana Djurdjevac Berlin, den 24.07.2018 iii Acknowledgments Firstly, I would like to express my sincere gratitude to my Ph.D. supervisor Ralf Kornhuber. He gave me the freedom and support that allowed me to broaden my knowledge, while at the same time he guided me during my graduate studies and helped me to stay on track. We had many helpful discussions and he provided a stimulating research environment. I would also like to express my gratitude to Charles M. Elliott for many helpful suggestions and insightful conversations, but also for the hard questions which motivated me to widen my research from various perspectives. My sincere thanks goes to Thomas Ranner for his numerous discussions. I was lucky to have Elias Pipping as my office mate. We had various mathematical blackboard discussions during which I learned to present my problems and solve most of them. He was my great teacher and friend. I would like to thank Gojko Kalajdziˇ c,´ who helped me during all these years with problems concerning algebra, particularly with understanding the complicated concept of tensor spaces. Moreover, I am thankful to Milivoje Lukic´ for all our discussions concerning functional analysis and Carsten Graser¨ for his help in improving my results concerning random moving domains. Many other outstanding mathematicians supported my research and improved my understand- ing and results. In particular: Amal Alphonse, Markus Bachmayr, Max Gunzburger, Helmut Harbrecht, Tim Sullivan, Claudia Schillings and Aretha Teckentrup. I also wish to thank my working group for their contribution in my everyday study and making the university a pleasant working environment. I would especially like to mention Evgenia, Jonathan, Hanne, Max and Tobias, as well as Marco, Milica and Peter,´ who encouraged and helped me through my PhD years. I am thankful to Mirjana Djoric´ and Boskoˇ Francuski for taking on the challenging task of reading my thesis. Last but not least, I would like to thank my family for their love and support throughout my work on this thesis and my life in general. I acknowledge the support of the Berlin Mathematical School (BMS) and the Hilda Geiringer scholarship. This research has also benefited from fruitful discussions within the project A07 of CRC 1114 and my stay at the Isaac Newton Institute in Cambridge during the Uncertainty quantification for complex systems: theory and methodologies programme. v Contents 1. Introduction1 2. Preliminaries7 2.1. Probability spaces . .7 2.2. Bochner spaces . 10 2.3. Hypersurfaces . 13 2.4. Moving surfaces . 17 2.5. Tensor products . 19 2.6. Karhunen-Loeve` expansion . 25 2.7. Gaussian and log-normal fields . 36 3. Function spaces 45 3.1. Gelfand triple . 45 3.2. Compatibility of spaces . 48 3.3. Bochner-type spaces . 51 3.4. Material derivative . 52 3.5. Solution space . 56 4. Uniformly bounded random diffusion coefficient 61 4.1. Formulation of the problem . 64 4.2. Existence and uniqueness . 66 4.3. Regularity . 72 5. Log-normal random diffusion coefficient 75 5.1. Transformation to parametrized discrete formulation and auxiliary measures . 76 5.2. Path-wise formulation of the problem . 81 5.3. Integrability of the solution . 84 6. Evolving surface finite element methods 87 6.1. Evolving simplicial surfaces . 88 6.2. Finite elements on simplicial surfaces . 91 6.3. Lifted finite elements . 96 7. Error estimates 99 7.1. Interpolation and geometric error estimates . 99 7.2. Ritz projection . 100 7.3. Discretization error estimates for the evolving surface finite element . 103 vii 8. Numerical Experiments 107 8.1. Computational aspects . 107 8.2. Moving curve . 108 8.3. Moving surface . 112 9. Random moving domains 115 9.1. Random tubes . 117 9.2. Heat equation on a random domain . 121 9.3. Well-posedness of the transformed equation . 124 A. Appendix 129 A.1. Dual operator . 129 A.2. Duality pairing and the inner product . 130 A.3. Doob–Dynkin lemma . 130 A.4. Kolmogorov test . 131 Summary 133 Zusammenfassung 135 Bibliography 137 viii List of Figures 2.1. Fermi coordinates x = a(x) + d(x)ν(a(x)).................... 16 2.2. Example of a space-time domain . 18 4.1. Probability density functions of the uniform distributions . 62 4.2. Realizations of the random checkerboard model . 63 6.1. Example of an approximation of Γ by Γh .................... 89 8.1. Polygonal approximation Γh;0 of Γ(0) ...................... 109 8.2. Realizations of diffusion coefficient . 112 8.3. Triangular approximation Γh;0 of Γ(0) ..................... 113 9.1. Cylindrical domain, realizations of a random cylindrical domain and of a random non-cylindrical domain . 117 ix List of Tables 8.1. Error table for a moving curve for a spatially smooth coefficient . 109 8.2. Error table for a moving curve for a spatially less smooth coefficient . 110 8.3. Error table for a moving curve for non-linear randomness . 110 8.4. Error table for a moving curve, when the assumptions are violated . 111 8.5. Error table for a moving curve for the test case with more RVs . 112 8.6. Error table for a moving surface . 113 xi 1. Introduction Partial differential equations (PDEs) appear in the mathematical modelling of a great variety of processes. Most of these equations contain various parameters that describe some physical properties, for example permeability or thermal conductivity. Usually it is presumed that these parameters are precisely given and a PDE is considered in a deterministic manner. However, often this is not the case, but there is a degree of uncertainty regarding the given data. Clearly, one would like to quantify the effect of uncertain parameters.. First, let us comment on various causes of uncertainty in model inputs. Generally, we can separate those causes into two main categories. The first category consists of uncertainty due to incomplete knowledge. This means that, in principle, it could be removed by performing additional measurements or having complete information. However, those measurements are typically very costly or impractical. This type of uncertainty is known as epismetic uncertainty. The second type is the so-called aleatoric uncertainty and it refers to the uncertainty of a phe- nomenon that comes from its own nature. It appears due to some unexpected or uncontrolled circumstances and cannot be reduced or removed by additional measurements. Thus, it relates to those quantities that are different every time we run the experiment due to information that cannot be controlled or measured, such as wind vibration. For a more detailed discussion about the types and causes of uncertainty we refer the reader to [36, 69]. We will mainly concentrate on the epismetic type of uncertainty. Thus, we will think about uncertainty in the way it is interpreted in [36]: ”uncertainty may be thought of as a measure of the incompleteness of one’s knowledge or information about an unknown quantity whose true value could be established if a perfect measuring device were available.” On this background, uncertainty quantification (UQ) has developed to a flourishing and very active mathematical field. We would refer to Sullivan [118] for the underlying mathematical concepts, typical UQ objectives and numerous examples. Concerning other basic references on UQ, we would point out [13, 69, 94, 96]. The overall goal is to identify and quantify uncertainty. In particular, given some information about the uncertainty of input data, we want to study the uncertainty of the system output, which is a solution function in the PDE setting. There are several approaches to the quantification of uncertainty, what is meant exactly by UQ and the corresponding mathematical framework. Some common approaches are: the worst case scenario, the probabilistic approach, Bayesian interface, the measure-theoretic approach, etc. For more details on these approaches see [69] and the references therein. This thesis will concentrate on the probabilistic approach which characterizes uncertainty by statistical information, such as probability density function, mean value, variance etc. Thus, we interpret the input data of a PDE as random fields. This results in a PDE with random coefficients, also known as a random partial differential equation (RPDE). Hence, the solution is also a random field and the aim is to determine its statistics or the statistics of some functional 1 applied to it. Furthermore, we would like to analyse the impact of a given uncertainty of random input data on the solution. There is a growing interest in RPDEs as these equations occur in many applications, such as hydrogeology, material science, fluid dynamics, biological fluids etc. This in turn leads especially to the growing development of numerical analysis and numerical methods for solving RPDEs, [10, 11, 14, 15, 27, 29, 32, 33, 69, 77]. Note that most of those papers deal with elliptic RPDEs. Parabolic PDEs with random coefficients, specifically, have so far been studied in the following papers , [10, 28, 77, 88]. All these papers have considered equations on a bounded flat fixed domain in Rd. However, it is known and well studied that, in a variety of applications, these models can be better for- mulated on both stationary and evolving curved domains, cf., e.g.