An optimal transport regularized model to image reconstruction problems

Optimal transport problem has gained much attention in image processing field, such as computer vision, image interpolation and medical image registration. In this paper, we incorporate optimal transport into linear inverse problems as a regularization technique. We establish a new variational model based on Benamou-Brenier energy to regularize the evolution path from a template to latent image dynamically. The initial state of the continuity equation can be regarded as a template, which can provide priors for the reconstructed images. Also, we analyze the existence of solutions of such variational problem in Radon measure space. Moreover, the first-order primal-dual algorithm is constructed for solving this general imaging problem in a special grid strategy. Finally, numerical experiments for undersampled MRI reconstruction are presented which show that our proposed model can recover images well with high quality and structure preservation

ICON-O: The Ocean Component of the ICON Earth System Model - Global simulation characteristics and local telescoping capability

Abstract We describe the ocean general circulation model ICON-O of the Max Planck Institute for Meteorology, which forms the ocean-sea ice component of the Earth system model ICON-ESM. ICON-O relies on innovative structure-preserving finite volume numerics. We demonstrate the fundamental ability of ICON-O to simulate key features of global ocean dynamics at both uniform and non-uniform resolution. Two experiments are analyzed and compared with observations, one with a nearly uniform and eddy-rich resolution of ?10?km and another with a telescoping configuration whose resolution varies smoothly from globally ?80?km to ?10?km in a focal region in the North Atlantic. Our results show first, that ICON-O on the nearly uniform grid simulates an ocean circulation that compares well with observations and second, that ICON-O in its telescope configuration is capable of reproducing the dynamics in the focal region over decadal time scales at a fraction of the computational cost of the uniform-grid simulation. The telescopic technique offers an alternative to the established regionalization approaches. It can be used either to resolve local circulation more accurately or to represent local scales that cannot be simulated globally while remaining within a global modeling framework

A Lagrangian vertical coordinate version of the ENDGame dynamical core. Part I: Formulation, remapping strategies, and robustness

This is the author accepted manuscript. The final version is available from Wiley via the DOI in this record.Previous work provides evidence that Lagrangian conservation and related properties of a numerical model dynamical core can be improved by the use of a Lagrangian or quasi-Lagrangian vertical coordinate (LVC). Most previous model developments based on this idea have made the hydrostatic approximation. Here the LVC is implemented in a nonhydrostatic compressible Euler equation dynamical core using almost identical numerical methods to ENDGame, the operational dynamical core of the Met Office atmospheric Unified Model. This enables a clean comparison of LVCand height-coordinate versions of the dynamical core using numerical methods that are as similar as possible. Since Lagrangian surfaces distort over time, model level heights are continually reset to certain ‘target levels’ and the values of model fields are remapped onto their new locations. Different choices for these target levels are discussed, along with remapping strategies that focus on different conservation or balance properties. Sample results from a baroclinic instability test case are presented. The LVC formulation is found to be rather less robust than the height-coordinate version; some reasons for this are discussed.We are grateful to Nigel Wood for pointing out the computational mode of the LVC vertical discretization. We also thank two anonymous reviewers for their constructive comments on an earlier version of this paper. This work was funded by the Natural Environment Research Council under grant NE/H006834/1

Formulações numéricas conservativas para aproximação de modelos hiperbólicos com termos de fonte e problemas de transporte relacionados

Orientador: Eduardo Cardoso de AbreuTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O objetivo desta tese é desenvolver, pelo menos no aspecto formal, algoritmos construtivos e bem-balanceados para a aproximação de classes específicas de modelos diferenciais. Nossas principais aplicações consistem em equações de água rasa e problemas de convecção-difusão no contexto de fenômenos de transporte, relacionados a problemas de pressão capilar descontínua em meios porosos. O foco principal é desenvolver sob o framework Lagrangian-Euleriano um esquema simples e eficiente para, em nível discreto, levar em conta o delicado equilíbrio entre as aproximações numéricas não lineares do fluxo hiperbólico e o termo fonte, e entre o fluxo hiperbólico e o operador difusivo. Os esquemas numéricos são propostos para ser independentes de estruturas particulares das funções de fluxo. Apresentamos diferentes abordagens que selecionam a solução entrópica qualitativamente correta, amparados por um grande conjunto de experimentos numéricos representativosAbstract: The purpose of this thesis is to develop, at least formally by construction, conservative methods for approximating specific classes of differential models. Our major applications consist in shallow water equations and nonstandard convection-diffusion problems in the context of transport phenomena, related to discontinuous capillary pressure problems in porous media. The main focus in this work is to develop under the Lagrangian-Eulerian framework a simple and efficient scheme to, on the discrete level, account for the delicate nonlinear balance between the numerical approximations of the hyperbolic flux and source term, and between the hyperbolic flux and the diffusion operator. The proposed numerical schemes are aimed to be independent of particular structures of the flux functions. We present different approaches that select the qualitatively correct entropy solution, supported by a large set of representative numerical experimentsDoutoradoMatematica AplicadaDoutor em Matemática Aplicada165564/2014-8CNPQCAPE

A split-explizit time-stepping scheme for ICON-Ocean

The development and implementation of advantageous time-stepping schemes in existing ocean models bears the potential to improve the model performance in terms of higher numerical accuracy as well lower numerical costs in terms of increased stability (larger possible time-steps). Stability and accuracy of time-stepping schemes should be considered in coupled space-time discretization. In that respect, the derivation and analysis of a new space-time discretization especially within the novel spatial framework of ICON-O (ocean component of the ICON earth system model) is of significant interest. In this thesis, adapting and implementing a split-explicit time-stepping scheme into ICON-O, we address both accuracy and stability: (a) We reduce the propagation error of barotropic signals by up to two orders of magnitude within mainly barotropic experiments. Furthermore, choosing a more advanced baroclinic time-stepping scheme results in increased accuracy of the baroclinic signal for relevant large Courant numbers. (b) The new space-time discretization shows increased numerical stability by a factor of up to 1.3 for the analysed experiments. In addition to the new split-explicit space-time discretization based on a Leap-Frog Adams-Moulthon-3 (LF-AM3) baroclinic step, we also adapt split-explicit time-stepping for the Adams-Bashfort-2 (AB2) scheme which is originally used in ICON-O together with a semi-implicit scheme. A major effort was to bring together these time- stepping schemes with the unique spatial framework of ICON-O, which is based on a C-type staggering of variables on a triangular grid. Following this spatial framework, we preserve a mass-matrix that filters out a spurious mode and furthermore fullfill discrete conservation of volume and tracers. In experiments with increasing complexity, we compare the two new split-explicit space-time discretizations with the original AB2 semi-implicit scheme. We show higher accuracy of the barotropic mode of the split-explicit schemes within various gravity wave experiments. In a lock-exchange experiment, we find for small Courant numbers that a coupling-error of both split-explicit time-stepping schemes results in smaller accuracy in velocity compared to the AB2 semi-implicit scheme. This coupling-error can be avoided with further improvements to the split-explicit algorithm. For desired large Courant numbers, the new LF-AM3 split-explicit space-time discretization is more accurate in the velocity, even for a time step that exceeds the stability limit of both AB2 schemes. Furthermore, the new LF-AM3 space-time discretization is more accurate for tracers independent of the Courant number. LF-AM3 shows slightly larger spurious mixing which we also find for smaller time steps with both AB2 schemes. We argue that this is caused by larger noise of the velocity on grid scale due to smaller numerical velocity diffusion. This results in gain of control over the total velocity diffusion when using ICON-O. Within the coupled space-time discretizations of ICON-O, the new LF-AM3 split-explicit discretization shows a stability limit that is 1.3 times larger compared to the AB2 semi-implicit and up to 1.5 times larger stability limit compared to the new AB2 split-explicit discretization for our experiments.Die Entwicklung und Implementierung von vorteilhaften Zeitschrittverfahren für vorhandene Ozeanmodelle birgt das Potenzial, deren Ergebnis hinsichtlich höherer numerischer Genauigkeit und geringerer numerischer Kosten bezogen auf erhöhte Stabilität (größere mögliche Zeitschritte) zu verbessern. Stabilität und Genauigkeit von Zeitschrittverfahren sollten im Kontext einer gekoppelten Raum-Zeit-Diskretisierung betrachtet werden. Diesbezüglich ist die Herleitung und Analyse eines neuen Zeitschrittverfahrens innerhalb der innovativen räumlichen Diskretisierung von ICON-O (Ozeankomponente des Erdsystemmodells ICON) von bedeutendem Interesse. In dieser Dissertation passen wir ein split-explizites Zeitschrittverfahren auf ICON-O an und implementieren dieses. Damit gehen wir die beiden Punkte Genauigkeit und Stabilität an: (a) Innerhalb überwiegend barotroper Experimente verringern wir den Fehler, der durch die Ausbreitung eines barotropen Signals entsteht, um bis zu zwei Größenordnungen. Zusätzlich wählen wir ein fortschrittliches Zeitschrittverfahren für den baroklinen Zeitschritt und verbessern damit das barokline Signal für die für uns relevanten, hohen Courant-Zahlen. (b) Die neue Raum-Zeit-Diskretisierung zeigt eine 1.3-fach erhöhte numerische Stabilität für die ausgewerteten Experimente. Zusätzlich zu dem neuem split-expliziten Zeitschrittverfahren, welches auf einem Leap-Frog Adams-Moulthon-3 (LF-AM3) baroklinem Zeitschritt basiert, entwickeln wir das split-explizite Zeitschrittverfahren für das in ICON-O ursprünglich mit einem semi-impliziten Zeitschritt verwendete Adams-Bashfort-2 (AB2) Verfahren. Eine der großen Leistungen dieser Arbeit war das Entwickeln dieser Zeitschrittverfahren innerhalb der besonderen r¨aumlichen Diskretisierung von ICON-O. Dieser folgend, erhalten wir die Massen-Matrix, welche eine numerische Mode aufhebt, und erfüllen diskrete Volumen- und Tracererhaltung. In Experimenten mit ansteigender Komplexität vergleichen wir die zwei neuen split-expliziten Raum-Zeit Diskretisierungen mit dem ursprünglichen semi-impliziten AB2 Verfahren. Wir zeigen die höhere Genauigkeit der barotropen Mode beider neuer split-expliziten Verfahren anhand mehrerer Experimente von Schwerewellen. In einem Lock-exchange Experiment zeigt sich, dass für kleine Courant-Zahlen beide split- expliziten Verfahren aufgrund eines Kopplungsfehlers geringere Genauigkeit in der Geschwindigkeit haben als das semi-implizite AB2 Verfahren. Dieser Kopplungsfehler kann durch weiterführende Verbesserungen des split-expliziten Verfahrens vermieden werden. Für die üblichen großen Courant-Zahlen ist die split-explizite LF-AM3 Raum-Zeit-Diskretisierung in der Geschwindigkeit genauer, sogar außerhalb des Stabilitätslimits beider AB2 Verfahren. Zusätzlich ist LF-AM3 für Tracer, unabhängig der Courant-Zahl, genauer. LF-AM3 zeigt ein etwas höheres numerisches Vermischen von Tracern, auch beobachtbar bei der Verwendung kleinerer Zeitschritte in den AB2 Verfahren. Wir begründen dies mit erhöhtem Rauschen in der Geschwindigkeit auf Gitterskala, bedingt durch eine geringere numerische Geschwindigkeitsdiffusion. Daraus ergibt sich für den Nutzer von ICON-O wiederum eine erhöhte Kontrolle über die gesamte Diffusion der Geschwindigkeit. Innerhalb der Raum-Zeit-Diskretisierung von ICON-O zeigt die neue split-explizite LF-AM3 Diskretisierung ein 1.3-fach höheres Stabilitätslimit als das semi-implizite AB2 Verfahren und bis zu 1.5-fache erhöhte Stabilität als die neue split-explizite AB2 Diskretisierung in den durchgeführten Experimenten