A Higher-Order VOF Interface Reconstruction Scheme for Non-Orthogonal Structured Grids - with Application to Surface Tension Modelling

The volume-of-fluid (VOF) method [24] is widely used to track the interface for the purpose of simulating liquid-gas interfacial flows numerically. The key strength of VOF is its mass conserving property. However, interface reconstruction is required when geometric properties such as curvature need to be accurately computed. For surface tension modelling in particular, computing the interface curvature accurately is crucial to avoiding so-called spurious or parasitic currents. Of the existing VOF-based schemes, the height-function (HF) method [10, 16, 18, 42, 46, 53] allows accurate interface representation on Cartesian grids. No work has hitherto been done to extend the HF philosophy to non-orthogonal structured grids. To this end, this work proposes a higher-order accurate VOF interface reconstruction method for non-orthogonal structured grids. Higher-order in the context of this work denotes up to 4 th-order. The scheme generalises the interface reconstruction component of the HF method. Columns of control volumes that straddle the interface are identified, and piecewise-linear interface constructions (PLIC) are computed in a volume-conservative manner in each column. To ensure efficiency, this procedure is executed by a novel sweep-plane algorithm based on the convex decomposition of the control volumes in each column. The PLIC representation of the interface is then smoothed by iteratively refining the PLIC facet normals. Rapid convergence of the latter is achieved via a novel spring-based acceleration procedure. The interface is then reconstructed by fitting higher-order polynomial curves/surfaces to local stencils of PLIC facets in a least squares manner [29]. Volume conservation is optimised for at the central column. The accuracy of the interface reconstruction procedure is evaluated via grid convergence studies in terms of volume conservation and curvature errors. The scheme is shown to achieve arbitrary-order accuracy on Cartesian grids and up to fourth-order accuracy on non-orthogonal structured grids. The curvature computation scheme is finally applied in a balanced-force continuum-surface-force (CSF) [4] surface tension scheme for variable-density flows on nonorthogonal structured grids in 2D. Up to fourth-order accuracy is demonstrated for the Laplace pressure jump in the simulation of a 2D stationary bubble with a high liquid-gas density ratio. A significant reduction in parasitic currents is demonstrated. Lastly, second-order accuracy is achieved when computing the frequency of a 2D inviscid oscillating droplet in zero gravity. The above tools were implemented and evaluated using the Elemental®multi-physics code and using a vertex-centred finite volume framework. For the purpose of VOF advection the algebraic CICSAM scheme (available in Elemental®) was employed

Topics in learning sparse and low-rank models of non-negative data

Advances in information and measurement technology have led to a surge in prevalence of high-dimensional data. Sparse and low-rank modeling can both be seen as techniques of dimensionality reduction, which is essential for obtaining compact and interpretable representations of such data. In this thesis, we investigate aspects of sparse and low-rank modeling in conjunction with non-negative data or non-negativity constraints. The first part is devoted to the problem of learning sparse non-negative representations, with a focus on how non-negativity can be taken advantage of. We work out a detailed analysis of non-negative least squares regression, showing that under certain conditions sparsity-promoting regularization, the approach advocated paradigmatically over the past years, is not required. Our results have implications for problems in signal processing such as compressed sensing and spike train deconvolution. In the second part, we consider the problem of factorizing a given matrix into two factors of low rank, out of which one is binary. We devise a provably correct algorithm computing such factorization whose running time is exponential only in the rank of the factorization, but linear in the dimensions of the input matrix. Our approach is extended to noisy settings and applied to an unmixing problem in DNA methylation array analysis. On the theoretical side, we relate the uniqueness of the factorization to Littlewood-Offord theory in combinatorics.Fortschritte in Informations- und Messtechnologie führen zu erhöhtem Vorkommen hochdimensionaler Daten. Modellierungsansätze basierend auf Sparsity oder niedrigem Rang können als Dimensionsreduktion betrachtet werden, die notwendig ist, um kompakte und interpretierbare Darstellungen solcher Daten zu erhalten. In dieser Arbeit untersuchen wir Aspekte dieser Ansätze in Verbindung mit nichtnegativen Daten oder Nichtnegativitätsbeschränkungen. Der erste Teil handelt vom Lernen nichtnegativer sparsamer Darstellungen, mit einem Schwerpunkt darauf, wie Nichtnegativität ausgenutzt werden kann. Wir analysieren nichtnegative kleinste Quadrate im Detail und zeigen, dass unter gewissen Bedingungen Sparsity-fördernde Regularisierung - der in den letzten Jahren paradigmatisch enpfohlene Ansatz - nicht notwendig ist. Unsere Resultate haben Auswirkungen auf Probleme in der Signalverarbeitung wie Compressed Sensing und die Entfaltung von Pulsfolgen. Im zweiten Teil betrachten wir das Problem, eine Matrix in zwei Faktoren mit niedrigem Rang, von denen einer binär ist, zu zerlegen. Wir entwickeln dafür einen Algorithmus, dessen Laufzeit nur exponentiell in dem Rang der Faktorisierung, aber linear in den Dimensionen der gegebenen Matrix ist. Wir erweitern unseren Ansatz für verrauschte Szenarien und wenden ihn zur Analyse von DNA-Methylierungsdaten an. Auf theoretischer Ebene setzen wir die Eindeutigkeit der Faktorisierung in Beziehung zur Littlewood-Offord-Theorie aus der Kombinatorik