Positive Definite Kernels in Machine Learning

This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as reproducing kernel Hibert spaces, the natural extension of the set of functions $\{k(x,\cdot),x\in\mathcal{X}\}$ associated with a kernel $k$ defined on a space $\mathcal{X}$. We discuss at length the construction of kernel functions that take advantage of well-known statistical models. We provide an overview of numerous data-analysis methods which take advantage of reproducing kernel Hilbert spaces and discuss the idea of combining several kernels to improve the performance on certain tasks. We also provide a short cookbook of different kernels which are particularly useful for certain data-types such as images, graphs or speech segments.Comment: draft. corrected a typo in figure

Trefftz Difference Schemes on Irregular Stencils

The recently developed Flexible Local Approximation MEthod (FLAME) produces accurate difference schemes by replacing the usual Taylor expansion with Trefftz functions -- local solutions of the underlying differential equation. This paper advances and casts in a general form a significant modification of FLAME proposed recently by Pinheiro & Webb: a least-squares fit instead of the exact match of the approximate solution at the stencil nodes. As a consequence of that, FLAME schemes can now be generated on irregular stencils with the number of nodes substantially greater than the number of approximating functions. The accuracy of the method is preserved but its robustness is improved. For demonstration, the paper presents a number of numerical examples in 2D and 3D: electrostatic (magnetostatic) particle interactions, scattering of electromagnetic (acoustic) waves, and wave propagation in a photonic crystal. The examples explore the role of the grid and stencil size, of the number of approximating functions, and of the irregularity of the stencils.Comment: 28 pages, 12 figures; to be published in J Comp Phy

hp-Cloud Approximation Of The Dirac Eigenvalue Problem: The Way Of Stability

We apply $hp$-cloud method to the radial Dirac eigenvalue problem. The difficulty of occurrence of spurious eigenvalues among the genuine ones in the computation is resolved. The method of treatment is based on assuming $hp$-cloud Petrov-Galerkin scheme to construct the weak formulation of the problem which adds a consistent diffusivity to the variational formulation. The size of the artificially added diffusion term is controlled by a stability parameter ($\tau$). The derivation of $\tau$ assumes the limit behavior of the eigenvalues at infinity. The parameter $\tau$ is applicable for generic basis functions. This is combined with the choice of appropriate intrinsic enrichments in the construction of the cloud shape functions.Comment: 29 pages, 6 figures, 8 table

A Stabilized and Coupled Meshfree/Meshbased Method for the Incompressible Navier-Stokes Equations : Part II: Coupling

In part I of this work, meshfree Galerkin methods have been used for the approximation of the incompressible Navier-Stokes equations in Eulerian or arbitrary Lagrangian-Eulerian formulation. The problem of stabilization of meshfree methods is addressed there. Analogously, in the meshbased context, the finite element method is frequently used in similar stabilized formulations for the simulation of flow problems. In order to combine the advantages of both methods, different coupling techniques are examined in this part of the work. Standard coupling approaches are modified in order to fulfill the requirements for a reliable stabilization found in part I of this work. The resulting stabilized and coupled meshfree/meshbased flow solver employs the comparatively costly meshfree Galerkin method only where it is needed---i.e. in areas of the domain, where a mesh is difficult to maintain---, and the efficient meshbased finite element method in the rest of the domain. This enables the solution of complex flow problems, as thus involving large deformations of the physical domain and/or moving and rotating obstacles

A Stabilized and Coupled Meshfree/Meshbased Method for Fluid-Structure Interaction Problems

A method is presented which combines features of meshfree and meshbased methods in order to enable the simulation of complex flow problems involving large deformations of the domain or moving and rotating objects. Conventional meshbased methods like the finite element method have matured as standard tools for the simulation of fluid and structure problems. They offer efficient and reliable approximations, provided that a conforming mesh with sufficient quality can be maintained throughout the simulation. This, however, may not be guaranteed for complex fluid and fluid-structure interaction problems. Meshfree methods on the other hand approximate partial differential equations based on a set of nodes without the need for an additional mesh. Therefore, these methods are frequently used for problems where suitable meshes are prohibitively expensive to construct and maintain. This advantage of meshfree methods comes at the price of being considerably more time-consuming than their meshbased counterparts. A coupled meshfree/meshbased fluid solver is developed which combines the advantages of both methodologies. A meshfree method, closely related to the element-free Galerkin method, is used in small parts of the domain where a mesh is difficult to maintain, whereas the efficient meshbased finite element method is employed in the rest of the domain. Major steps in the development of the coupled flow solver are the extension of standard stabilization methods to meshfree approximations, and the realization of the coupling on the level of the shape functions, which are involved in the approximation of the governing equations. Concerning the stabilization, it is found that the same structure of the standard stabilization schemes may be used for meshfree approximations, however, the aspect of the stabilization parameter, weighing the stabilization terms, requires special care. For the coupling, these requirements for a reliable stabilization lead to modifications of the existing coupling approaches. The stabilized and coupled flow solver is verified and used for the simulation of geometrically complex fluid-structure interaction problems. Conventional meshbased methods are not suitable for the approximation of these test cases due to the prohibitively large deformations of the geometry.Es wird ein Verfahren vorgestellt, das Eigenschaften netzfreier und netzbasierter Methoden nutzt, um die Simulation komplexer Strömungsprobleme zu ermöglichen, bei denen große Verformungen des Gebietes oder sich bewegende und rotierende Objekte in der Strömung berücksichtigt werden können. Konventionelle netzbasierte Verfahren wie die Finite Element Methode haben sich zu Standardwerkzeugen bei der numerischen Analyse von Strömungen und Festkörpern entwickelt. Sie ermöglichen eine schnelle und zuverlässige Approximation, vorausgesetzt, daß ein konformes Netz mit geeigneter Qualität während der gesamten Simulation aufrecht erhalten werden kann. Dies kann jedoch bei komplexen Strömungs- und Fluid-Struktur-Interaktionsproblemen eine entscheidende Einschränkung in der Anwendbarkeit dieser Verfahren darstellen. Netzfreie Verfahren approximieren dagegen die zugrundeliegenden Modellgleichungen nur in Abhängigkeit einer gegebenen Knotenverteilung, ohne ein Netz zu erfordern, das die Konnektivität a priori festlegt. Deshalb werden diese Verfahren häufig dort eingesetzt, wo die Generierung oder Erhaltung geeigneter Netze nicht mit vertretbarem Aufwand möglich ist. Allerdings geht der Vorteil der Netzunabhängigkeit bei netzfreien Verfahren einher mit deutlich aufwendiger zu konstruierenden Ansatzfunktionen, was eine empfindliche Erhöhung des Rechenaufwandes bei der Integration der zugrundeliegenden Gleichungen erfordert. Hierin wird ein gekoppelter netzfreier/netzbasierter Strömungslöser entwickelt, der die Vorteile beider Verfahren kombiniert. Ein netzfreies Verfahren, das eng mit der "element-free" Galerkin Methode verwandt ist, wird nur in kleinen Teilen des Gebietes verwendet, wo ein Netz zu Schwierigkeiten führt, und im gesamten Rest des Gebiets wird die Finite Element Methode als effizientes netzbasiertes Standardverfahren eingesetzt. Wichtige Schritte bei der Entwicklung der gekoppelten Methode sind die Erweiterung von Standard-Stabilisierungsansätzen auf netzfreie Verfahren und die Umsetzung der Kopplung auf der Ebene der Ansatzfunktionen, die für die Approximation eingesetzt werden. Bezüglich der Stabilisierung wird gezeigt, daß die Struktur der verschiedenen Stabilisierungsmethoden direkt auf netzfreie Verfahren anwendbar ist, allerdings erfordert die Wahl geeigneter Stabilisierungsparameter, die den Stabilisierungseinfluß wichten, besondere Beachtung. Bei der Kopplung führen die Voraussetzungen, die für eine zuverlässige Stabilisierung gegeben sein müssen, zu Modifikationen der Standardansätze. Der gekoppelte Strömungslöser wird verifiziert und für die Simulation geometrisch komplexer Fluid-Struktur-Interaktionsprobleme eingesetzt. Die Fähigkeiten des gekoppelten Verfahrens werden dabei sichtbar, denn klassische netzbasierte Standardverfahren versagen bei den gezeigten Anwendungsbeispielen

Interpretable statistics for complex modelling: quantile and topological learning

As the complexity of our data increased exponentially in the last decades, so has our need for interpretable features. This thesis revolves around two paradigms to approach this quest for insights. In the first part we focus on parametric models, where the problem of interpretability can be seen as a “parametrization selection”. We introduce a quantile-centric parametrization and we show the advantages of our proposal in the context of regression, where it allows to bridge the gap between classical generalized linear (mixed) models and increasingly popular quantile methods. The second part of the thesis, concerned with topological learning, tackles the problem from a non-parametric perspective. As topology can be thought of as a way of characterizing data in terms of their connectivity structure, it allows to represent complex and possibly high dimensional through few features, such as the number of connected components, loops and voids. We illustrate how the emerging branch of statistics devoted to recovering topological structures in the data, Topological Data Analysis, can be exploited both for exploratory and inferential purposes with a special emphasis on kernels that preserve the topological information in the data. Finally, we show with an application how these two approaches can borrow strength from one another in the identification and description of brain activity through fMRI data from the ABIDE project

Remote sensing for optimal estimation of water temperature dynamics in shallow tidal environments

Given the increasing anthropogenic pressures on lagoons, estuaries, and lakes and considering the highly dynamic behavior of these systems, methods for the continuous and spatially distributed retrieval of water quality are becoming vital for their correct monitoring and management. Water temperature is certainly one of the most important drivers that influence the overall state of coastal systems. Traditionally, lake, estuarine, and lagoon temperatures are observed through point measurements carried out during field campaigns or through a network of sensors. However, sporadic measuring campaigns or probe networks rarely attain a density sufficient for process understanding, model development/validation, or integrated assessment. Here, we develop and apply an integrated approach for water temperature monitoring in a shallow lagoon which incorporates satellite and in-situ data into a mathematical model. Specifically, we use remote sensing information to constrain large-scale patterns of water temperature and high-frequency in situ observations to provide proper time constraints. A coupled hydrodynamic circulation-heat transport model is then used to propagate the state of the system forward in time between subsequent remote sensing observations. Exploiting the satellite data high spatial resolution and the in situ measurements high temporal resolution, the model may act a physical interpolator filling the gap intrinsically characterizing the two monitoring techniques