Multi-symplectic discretisation of wave map equations

We present a new multi-symplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. When applied to the wave map equation, this numerical scheme is explicit, preserves the constraint and can be seen as a generalisation of the Shake algorithm for constrained mechanical systems. Furthermore, numerical experiments show excellent conservation properties of the numerical solutions

Inf-Sup Stable Finite Element Methods for the Landau--Lifshitz--Gilbert and Harmonic Map Heat Flow Equations

In this paper we propose and analyze a finite element method for both the harmonic map heat and Landau–Lifshitz–Gilbert equation, the time variable remaining continuous. Our starting point is to set out a unified saddle point approach for both problems in order to impose the unit sphere constraint at the nodes since the only polynomial function satisfying the unit sphere constraint everywhere are constants. A proper inf-sup condition is proved for the Lagrange multiplier leading to the well-posedness of the unified formulation. A priori energy estimates are shown for the proposed method. When time integrations are combined with the saddle point finite element approximation some extra elaborations are required in order to ensure both a priori energy estimates for the director or magnetization vector depending on the model and an inf-sup condition for the Lagrange multiplier. This is due to the fact that the unit length at the nodes is not satisfied in general when a time integration is performed. We will carry out a linear Euler time-stepping method and a non-linear Crank–Nicolson method. The latter is solved by using the former as a non-linear solver.Ministerio de Economía y Competitividad MTM2015-69875-

Geometric partial differential equations: Theory, numerics and applications

This workshop concentrated on partial differential equations involving stationary and evolving surfaces in which geometric quantities play a major role. Mutual interest in this emerging field stimulated the interaction between analysis, numerical solution, and applications

Synergies between Numerical Methods for Kinetic Equations and Neural Networks

The overarching theme of this work is the efficient computation of large-scale systems. Here we deal with two types of mathematical challenges, which are quite different at first glance but offer similar opportunities and challenges upon closer examination. Physical descriptions of phenomena and their mathematical modeling are performed on diverse scales, ranging from nano-scale interactions of single atoms to the macroscopic dynamics of the earth\u27s atmosphere. We consider such systems of interacting particles and explore methods to simulate them efficiently and accurately, with a focus on the kinetic and macroscopic description of interacting particle systems. Macroscopic governing equations describe the time evolution of a system in time and space, whereas the more fine-grained kinetic description additionally takes the particle velocity into account. The study of discretizing kinetic equations that depend on space, time, and velocity variables is a challenge due to the need to preserve physical solution bounds, e.g. positivity, avoiding spurious artifacts and computational efficiency. In the pursuit of overcoming the challenge of computability in both kinetic and multi-scale modeling, a wide variety of approximative methods have been established in the realm of reduced order and surrogate modeling, and model compression. For kinetic models, this may manifest in hybrid numerical solvers, that switch between macroscopic and mesoscopic simulation, asymptotic preserving schemes, that bridge the gap between both physical resolution levels, or surrogate models that operate on a kinetic level but replace computationally heavy operations of the simulation by fast approximations. Thus, for the simulation of kinetic and multi-scale systems with a high spatial resolution and long temporal horizon, the quote by Paul Dirac is as relevant as it was almost a century ago. The first goal of the dissertation is therefore the development of acceleration strategies for kinetic discretization methods, that preserve the structure of their governing equations. Particularly, we investigate the use of convex neural networks, to accelerate the minimal entropy closure method. Further, we develop a neural network-based hybrid solver for multi-scale systems, where kinetic and macroscopic methods are chosen based on local flow conditions. Furthermore, we deal with the compression and efficient computation of neural networks. In the meantime, neural networks are successfully used in different forms in countless scientific works and technical systems, with well-known applications in image recognition, and computer-aided language translation, but also as surrogate models for numerical mathematics. Although the first neural networks were already presented in the 1950s, the scientific discipline has enjoyed increasing popularity mainly during the last 15 years, since only now sufficient computing capacity is available. Remarkably, the increasing availability of computing resources is accompanied by a hunger for larger models, fueled by the common conception of machine learning practitioners and researchers that more trainable parameters equal higher performance and better generalization capabilities. The increase in model size exceeds the growth of available computing resources by orders of magnitude. Since $2012$, the computational resources used in the largest neural network models doubled every $3.4$ months\footnote{\url{https://openai.com/blog/ai-and-compute/}}, opposed to Moore\u27s Law that proposes a $2$-year doubling period in available computing power. To some extent, Dirac\u27s statement also applies to the recent computational challenges in the machine-learning community. The desire to evaluate and train on resource-limited devices sparked interest in model compression, where neural networks are sparsified or factorized, typically after training. The second goal of this dissertation is thus a low-rank method, originating from numerical methods for kinetic equations, to compress neural networks already during training by low-rank factorization. This dissertation thus considers synergies between kinetic models, neural networks, and numerical methods in both disciplines to develop time-, memory- and energy-efficient computational methods for both research areas

Finite Element Methods for Geometric Problems

In the herewith presented work we numerically treat geometric partial differential equations using finite element methods. Problems of this type appear in many applications from physics, biology and engineering use. We may partition the work in two blocks. The first one, including the chapters two to five, is about the approximation of stationary points of conformally invariant, nonlinear, elliptic energy functionals. Main interest is a compactness result for accumulation points of their discrete counterparts. The corresponding Euler-Lagrange equations are nonlinear, elliptic and of second order. They contain critical nonlinearities that are quadratic in the first derivatives. Thus, accumulation points of solutions to the discrete problem are not solutions of the continuous problem in general. We deduce a weak formulation in a mixed form and chose appropriate spaces for the discretization. First we show existence of discrete solutions and then, by the use of compensated compactness and standard finite element arguments, we establish convergence. Finally we introduce an iterative algorithm for the numerical realization and run different simulations. Hereby we confirm theoretical predictions derived in the stability analysis. The second part is about the derivation of gradient flows for shape functionals and their discretization with parametric finite elements. First, we consider the Willmore energy of a twodimensional surface in the threedimensional ambient space and deduce its first variation. Afterwards we phrase the corresponding gradient flow in a weak form and discuss possible discretizations. During the further progress of the work we modell cell membranes and the effects of surface active agents on the shape of these cells. Numerical simulations with closed surface give promising results and a reason to intensify the research in this field.Finite Elemente Methoden für Geometrische Probleme In der vorliegenden Dissertationsschrift geht es um die numerische Behandlung geometrischer partieller Differentialgleichungen unter Verwendung von Finite Elemente Methoden. Probleme dieser Art treten in einer Vielzahl von physikalischen, technischen und biologischen Anwendungen auf. Thematisch lässt sich die Arbeit in zwei Blöcke aufteilen. In den Kapiteln zwei bis fünf geht es um die Approximation stationärer Punkte konform invarianter, nichtlinearer, elliptischer Energiefunktionale. Das Hauptaugenmerk liegt dabei auf einem Kompaktheitsresultat für Häufungspunkte der diskretisierten Energiefunktionale. Die Euler Lagrange Gleichungen sind elliptisch und von zweiter Ordnung. Sie beinhalten kritische Nichtlinearitäten welche quadratisch von den ersten Ableitungen abhängen. Dies f¨hrt dazu, dass Häufungspunkte von Lösungen der diskretisierten Gleichung nicht zwangsläufig Lösungen der ursprünglichen Gleichung sind. Wir leiten eine schwache Formulierung der Gleichung in gemischter Form her und wählen stabile Finite Elemente Paare für die Diskretisierung. Zunächst zeigen wir, dass Lösungen der diskreten gemischten Formulierung Sattelpunkte eines erweiterten diskreten Energiefunktionals sind und schließen daraus auf die Existenz diskreter Löosungen. Um zu beweisen, dass Häufungspunkte der diskreten Sattelpunkte tatsächlich Lösungen der schwachen Formulierung sind bedienen wir uns einigen Resultaten der kompensierten Kompaktheit sowie bekannten Techniken aus dem Bereich der Finiten Elemente. Schließlich stellen wir einen iterativen Algorithmus für die numerische Realisierung auf und föhren mehrere Simulationen durch. Theoretische Stabilitätsergebnisse für den Algorithmus werden dabei numerisch bestätigt. Im zweiten Teil stehen die Herleitung von Gradientenflüssen von Flächenfunktionalen (shape functional) sowie deren Diskretisierung unter Verwendung von Parametrischen Finite Elemente Methoden im Mittelpunkt. Wir betrachten zunächst die sogenannte Willmore Energie einer zweidimensionalen Fläche im dreidimensionalen Raum und bestimmen deren erste Variation. Anschließend formulieren wir den zugehörigen Gradientenfluss in schwacher Form und diskutieren eine Diskretisierung mittels parametrischer Finite Elemente. Im weiteren Verlauf diskutieren wir die Modellierung von Zellmembranen und die Wirkung von oberflächenaktiven Substanzen (surfactants) auf die Form von Zellen. Numerische Simulationen mit geschlossenen Flächen liefern viel versprechende Resultate und geben Anlass zu weiteren Forschungsarbeiten in diesem Bereich

Adjoint Methods as Design Tools in Thermoacoustics

In a thermoacoustic system, such as a flame in a combustor, heat release oscillations couple with acoustic pressure oscillations. If the heat release is sufficiently in phase with the pressure, these oscillations can grow, sometimes with catastrophic consequences. Thermoacoustic instabilities are still one of the most challenging problems faced by gas turbine and rocket motor manufacturers. Thermoacoustic systems are characterized by many parameters to which the stability may be extremely sensitive. However, often only few oscillation modes are unstable. Existing techniques examine how a change in one parameter affects all (calculated) oscillation modes, whether unstable or not. Adjoint techniques turn this around: They accurately and cheaply compute how each oscillation mode is affected by changes in all parameters. In a system with a million parameters, they calculate gradients a million times faster than finite difference methods. This review paper provides: (i) the methodology and theory of stability and adjoint analysis in thermoacoustics, which is characterized by degenerate and nondegenerate nonlinear eigenvalue problems; (ii) physical insight in the thermoacoustic spectrum, and its exceptional points; (iii) practical applications of adjoint sensitivity analysis to passive control of existing oscillations, and prevention of oscillations with ad hoc design modifications; (iv) accurate and efficient algorithms to perform uncertainty quantification of the stability calculations; (v) adjoint-based methods for optimization to suppress instabilities by placing acoustic dampers, and prevent instabilities by design modifications in the combustor's geometry; (vi) a methodology to gain physical insight in the stability mechanisms of thermoacoustic instability (intrinsic sensitivity); and (vii) in nonlinear periodic oscillations, the prediction of the amplitude of limit cycles with weakly nonlinear analysis, and the theoretical framework to calculate the sensitivity to design parameters of limit cycles with adjoint Floquet analysis. To show the robustness and versatility of adjoint methods, examples of applications are provided for different acoustic and flame models, both in longitudinal and annular combustors, with deterministic and probabilistic approaches. The successful application of adjoint sensitivity analysis to thermoacoustics opens up new possibilities for physical understanding, control and optimization to design safer, quieter, and cleaner aero-engines. The versatile methods proposed can be applied to other multiphysical and multiscale problems, such as fluid–structure interaction, with virtually no conceptual modification.</jats:p