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Assessment of Stability in Partitional Clustering Using Resampling Techniques
2016, Mucha, Hans-Joachim
The assessment of stability in cluster analysis is strongly related to the main difficult problem of determining the number of clusters present in the data. The latter is subject of many investigations and papers considering different resampling techniques as practical tools. In this paper, we consider non-parametric resampling from the empirical distribution of a given dataset in order to investigate the stability of results of partitional clustering. In detail, we investigate here only the very popular K-means method. The estimation of the sampling distribution of the adjusted Rand index (ARI) and the averaged Jaccard index seems to be the most general way to do this. In addition, we compare bootstrapping with different subsampling schemes (i.e., with different cardinality of the drawn samples) with respect to their performance in finding the true number of clusters for both synthetic and real data.
Weak-strong uniqueness for the general Ericksen-Leslie system in three dimensions
2018, Emmrich, Etienne, Lasarzik, Robert
We study the Ericksen-Leslie system equipped with a quadratic free energy functional. The norm restriction of the director is incorporated by a standard relaxation technique using a double-well potential. We use the relative energy concept, often applied in the context of compressible Euler- or related systems of fluid dynamics, to prove weak-strong uniqueness of solutions. A main novelty, not only in the context of the Ericksen-Leslie model, is that the relative energy inequality is proved for a system with a nonconvex energy.
From Large Deviations to Semidistances of Transport and Mixing: Coherence Analysis for Finite Lagrangian Data
2018, Koltai, Péter, Renger, D.R. Michiel
One way to analyze complicated non-autonomous flows is through trying to understand their transport behavior. In a quantitative, set-oriented approach to transport and mixing, finite time coherent sets play an important role. These are time-parametrized families of sets with unlikely transport to and from their surroundings under small or vanishing random perturbations of the dynamics. Here we propose, as a measure of transport and mixing for purely advective (i.e., deterministic) flows, (semi)distances that arise under vanishing perturbations in the sense of large deviations. Analogously, for given finite Lagrangian trajectory data we derive a discrete-time-and-space semidistance that comes from the “best” approximation of the randomly perturbed process conditioned on this limited information of the deterministic flow. It can be computed as shortest path in a graph with time-dependent weights. Furthermore, we argue that coherent sets are regions of maximal farness in terms of transport and mixing, and hence they occur as extremal regions on a spanning structure of the state space under this semidistance—in fact, under any distance measure arising from the physical notion of transport. Based on this notion, we develop a tool to analyze the state space (or the finite trajectory data at hand) and identify coherent regions. We validate our approach on idealized prototypical examples and well-studied standard cases.
Bochner integrals in ordered vector spaces
2016, van Rooij, A.C.M., van Zuijlen, W.B.
We present a natural way to cover an Archimedean directed ordered vector space E by Banach spaces and extend the notion of Bochner integrability to functions with values in E. The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map.
A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions
2015, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen
A boundary control problem for the pure Cahn–Hilliard equations with possibly singular potentialsand dynamic boundary conditions is studied and rst-order necessary conditions for optimality are proved.
Large Deviations of Continuous Regular Conditional Probabilities
2016, van Zuijlen, W.
We study product regular conditional probabilities under measures of two coordinates with respect to the second coordinate that are weakly continuous on the support of the marginal of the second coordinate. Assuming that there exists a sequence of probability measures on the product space that satisfies a large deviation principle, we present necessary and sufficient conditions for the conditional probabilities under these measures to satisfy a large deviation principle. The arguments of these conditional probabilities are assumed to converge. A way to view regular conditional probabilities as a special case of product regular conditional probabilities is presented. This is used to derive conditions for large deviations of regular conditional probabilities. In addition, we derive a Sanov-type theorem for large deviations of the empirical distribution of the first coordinate conditioned on fixing the empirical distribution of the second coordinate.
Scattering matrices and Dirichlet-to-Neumann maps
2017, Behrndt, Jussi, Malamud, Mark M., Neidhardt, Hagen
A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh–Weyl m-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.
Distributed optimal control of a nonstandard nonlocal phase field system
2016, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen
We investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.
Adaptive smoothing of digital images: The R package adimpro
2007, Polzehl, J., Tabelow, K.
Digital imaging has become omnipresent in the past years with a bulk of applications ranging from medical imaging to photography. When pushing the limits of resolution and sensitivity noise has ever been a major issue. However, commonly used non-adaptive filters can do noise reduction at the cost of a reduced effective spatial resolution only. Here we present a new package adimpro for R, which implements the propagationseparation approach by (Polzehl arid Spokoiriy 2006) for smoothing digital images. This method naturally adapts to different structures of different size in the image and thus avoids oversmoothing edges and fine structures. We extend the method for imaging data with spatial correlation. Furthermore we show how the estimation of the dependence between variance and mean value can be included. We illustrate the use of the package through some examples.
Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures
2017, Liero, Matthias, Mielke, Alexander, Savaré, Giuseppe
We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger–Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger–Kakutani and Kantorovich–Wasserstein distances.
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