hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity

A considerable amount of attention has been given to discontinuous Galerkin methods for hyperbolic problems in numerical relativity, showing potential advantages of the methods in dealing with hydrodynamical shocks and other discontinuities. This paper investigates discontinuous Galerkin methods for the solution of elliptic problems in numerical relativity. We present a novel hp-adaptive numerical scheme for curvilinear and non-conforming meshes. It uses a multigrid preconditioner with a Chebyshev or Schwarz smoother to create a very scalable discontinuous Galerkin code on generic domains. The code employs compactification to move the outer boundary near spatial infinity. We explore the properties of the code on some test problems, including one mimicking Neutron stars with phase transitions. We also apply it to construct initial data for two or three black holes

PDE-based Group Equivariant Convolutional Neural Networks

We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these networks to be designed with built-in symmetries such as rotation in addition to the standard translation equivariance of CNNs. Having all the desired symmetries included in the design obviates the need to include them by means of costly techniques such as data augmentation. We will discuss our PDE-based G-CNNs (PDE-G-CNNs) in a general homogeneous space setting while also going into the specifics of our primary case of interest: roto-translation equivariance. We solve the PDE of interest by a combination of linear group convolutions and non-linear morphological group convolutions with analytic kernel approximations that we underpin with formal theorems. Our kernel approximations allow for fast GPU-implementation of the PDE-solvers, we release our implementation with this article in the form of the LieTorch extension to PyTorch, available at https://gitlab.com/bsmetsjr/lietorch . Just like for linear convolution a morphological convolution is specified by a kernel that we train in our PDE-G-CNNs. In PDE-G-CNNs we do not use non-linearities such as max/min-pooling and ReLUs as they are already subsumed by morphological convolutions. We present a set of experiments to demonstrate the strength of the proposed PDE-G-CNNs in increasing the performance of deep learning based imaging applications with far fewer parameters than traditional CNNs.Comment: 27 pages, 18 figures. v2 changes: - mentioned KerCNNs - added section Generalization of G-CNNs - clarification that the experiments utilized automatic differentiation and SGD. v3 changes: - streamlined theoretical framework - formulation and proof Thm.1 & 2 - expanded experiments. v4 changes: typos in Prop.5 and (20) v5/6 changes: minor revisio

Nonlocal smoothing and adaptive morphology for scalar- and matrix-valued images

In this work we deal with two classic degradation processes in image analysis, namely noise contamination and incomplete data. Standard greyscale and colour photographs as well as matrix-valued images, e.g. diffusion-tensor magnetic resonance imaging, may be corrupted by Gaussian or impulse noise, and may suffer from missing data. In this thesis we develop novel reconstruction approaches to image smoothing and image completion that are applicable to both scalar- and matrix-valued images. For the image smoothing problem, we propose discrete variational methods consisting of nonlocal data and smoothness constraints that penalise general dissimilarity measures. We obtain edge-preserving filters by the joint use of such measures rich in texture content together with robust non-convex penalisers. For the image completion problem, we introduce adaptive, anisotropic morphological partial differential equations modelling the dilation and erosion processes. They adjust themselves to the local geometry to adaptively fill in missing data, complete broken directional structures and even enhance flow-like patterns in an anisotropic manner. The excellent reconstruction capabilities of the proposed techniques are tested on various synthetic and real-world data sets.In dieser Arbeit beschäftigen wir uns mit zwei klassischen Störungsquellen in der Bildanalyse, nämlich mit Rauschen und unvollständigen Daten. Klassische Grauwert- und Farb-Fotografien wie auch matrixwertige Bilder, zum Beispiel Diffusionstensor-Magnetresonanz-Aufnahmen, können durch Gauß- oder Impulsrauschen gestört werden, oder können durch fehlende Daten gestört sein. In dieser Arbeit entwickeln wir neue Rekonstruktionsverfahren zum zur Bildglättung und zur Bildvervollständigung, die sowohl auf skalar- als auch auf matrixwertige Bilddaten anwendbar sind. Zur Lösung des Bildglättungsproblems schlagen wir diskrete Variationsverfahren vor, die aus nichtlokalen Daten- und Glattheitstermen bestehen und allgemeine auf Bildausschnitten definierte Unähnlichkeitsmaße bestrafen. Kantenerhaltende Filter werden durch die gemeinsame Verwendung solcher Maße in stark texturierten Regionen zusammen mit robusten nichtkonvexen Straffunktionen möglich. Für das Problem der Datenvervollständigung führen wir adaptive anisotrope morphologische partielle Differentialgleichungen ein, die Dilatations- und Erosionsprozesse modellieren. Diese passen sich der lokalen Geometrie an, um adaptiv fehlende Daten aufzufüllen, unterbrochene gerichtet Strukturen zu schließen und sogar flussartige Strukturen anisotrop zu verstärken. Die ausgezeichneten Rekonstruktionseigenschaften der vorgestellten Techniken werden anhand verschiedener synthetischer und realer Datensätze demonstriert

Truncated Nonsmooth Newton Multigrid for phase-field brittle-fracture problems

We propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems in roughly the time of solving one linear system of equations. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without penalization or the introduction of a local history field. Memory consumption is significantly lower compared to approaches based on direct solvers. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler-Lagrange equations of the spatial problem for any load and initial iterate. Numerical comparisons to an operator-splitting algorithm show a speed increase of more than one order of magnitude, without loss of robustness

Refresher course in maths and a project on numerical modeling done in twos

These lecture notes accompany a refresher course in applied mathematics with a focus on numerical concepts (Part I), numerical linear algebra (Part II), numerical analysis, Fourier series and Fourier transforms (Part III), and differential equations (Part IV). Several numerical projects for group work are provided in Part V. In these projects, the tasks are threefold: mathematical modeling, algorithmic design, and implementation. Therein, it is important to draw interpretations of the obtained results and provide measures (Parts I-IV) how to build confidence into numerical findings such intuition, error analysis, convergence analysis, and comparison to manufactured solutions. Both authors have been jointly teaching over several years this class and bring in a unique mixture of their respective teaching and research fields

Compatible finite element methods for geophysical fluid dynamics

This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces which are linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge-Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretising the transport terms that arise in dynamical core equation systems, and provide some example discretisations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretisations, Poisson bracket discretisations, and consistent vorticity transport.Comment: correction of some typo

Refresher course in maths and a project on numerical modeling done in twos

These lecture notes accompany a refresher course in applied mathematics with a focus on numerical concepts (Part I), numerical linear algebra (Part II), numerical analysis, Fourier series and Fourier transforms (Part III), and differential equations (Part IV). Several numerical projects for group work are provided in Part V. In these projects, the tasks are threefold: mathematical modeling, algorithmic design, and implementation. Therein, it is important to draw interpretations of the obtained results and provide measures (Parts I-IV) how to build confidence into numerical findings such intuition, error analysis, convergence analysis, and comparison to manufactured solutions. Both authors have been jointly teaching over several years this class and bring in a unique mixture of their respective teaching and research fields

Nonlocal Graph-PDEs and Riemannian Gradient Flows for Image Labeling

In this thesis, we focus on the image labeling problem which is the task of performing unique pixel-wise label decisions to simplify the image while reducing its redundant information. We build upon a recently introduced geometric approach for data labeling by assignment flows [ APSS17 ] that comprises a smooth dynamical system for data processing on weighted graphs. Hereby we pursue two lines of research that give new application and theoretically-oriented insights on the underlying segmentation task. We demonstrate using the example of Optical Coherence Tomography (OCT), which is the mostly used non-invasive acquisition method of large volumetric scans of human retinal tis- sues, how incorporation of constraints on the geometry of statistical manifold results in a novel purely data driven geometric approach for order-constrained segmentation of volumetric data in any metric space. In particular, making diagnostic analysis for human eye diseases requires decisive information in form of exact measurement of retinal layer thicknesses that has be done for each patient separately resulting in an demanding and time consuming task. To ease the clinical diagnosis we will introduce a fully automated segmentation algorithm that comes up with a high segmentation accuracy and a high level of built-in-parallelism. As opposed to many established retinal layer segmentation methods, we use only local information as input without incorporation of additional global shape priors. Instead, we achieve physiological order of reti- nal cell layers and membranes including a new formulation of ordered pair of distributions in an smoothed energy term. This systematically avoids bias pertaining to global shape and is hence suited for the detection of anatomical changes of retinal tissue structure. To access the perfor- mance of our approach we compare two different choices of features on a data set of manually annotated 3 D OCT volumes of healthy human retina and evaluate our method against state of the art in automatic retinal layer segmentation as well as to manually annotated ground truth data using different metrics. We generalize the recent work [ SS21 ] on a variational perspective on assignment flows and introduce a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as nonlocal reparametrization of the assignment flow approach that was introduced in J. Math. Imaging & Vision 58(2), 2017. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with re- spect to a nonconvex potential. We devise an entropy-regularized difference-of-convex-functions (DC) decomposition of this potential and show that the basic geometric Euler scheme for inte- grating the assignment flow is equivalent to solving the G-PDE by an established DC program- ming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments