Identification of general and double aggregation operators using monotone smoothing

Aggregation operators model various operations on fuzzy sets, such as conjunction, disjunction and averaging. Recently double aggregation operators have been introduced; they model multistep aggregation process. The choice of aggregation operators depends on the particular problem, and can be done by fitting the operator to empirical data. We examine fitting general aggregation operators by using a new method of monotone Lipschitz smoothing. We study various boundary conditions and constraints which determine specific types of aggregation.<br /

Pointwise construction of Lippschitz aggregation operators with specific properties

This paper describes an approach to pointwise construction of general aggregation operators, based on monotone Lipschitz approximation. The aggregation operators are constructed from a set of desired values at certain points, or from empirically collected data. It establishes tight upper and lower bounds on Lipschitz aggregation operators with a number of different properties, as well as the optimal aggregation operator, consistent with the given values. We consider conjunctive, disjunctive and idempotent n-ary aggregation operators; p-stable aggregation operators; various choices of the neutral element and annihilator; diagonal, opposite diagonal and marginal sections; bipolar and double aggregation operators. In all cases we provide either explicit formulas or deterministic numerical procedures to determine the bounds. The findings of this paper are useful for construction of aggregation operators with specified properties, especially using interpolation schemata.<br /

Construction of aggregation operators for automated decision making via optimal interpolation and global optimization

This paper examines methods of point wise construction of aggregation operators via optimal interpolation. It is shown that several types of application-specific requirements lead to interpolatory type constraints on the aggregation function. These constraints are translated into global optimization problems, which are the focus of this paper. We present several methods of reduction of the number of variables, and formulate suitable numerical algorithms based on Lipschitz optimization.<br /

Absorbent tuples of aggregation operators

We generalize the notion of an absorbent element of aggregation operators. Our construction involves tuples of values that decide the result of aggregation. Absorbent tuples are useful to model situations in which certain decision makers may decide the outcome irrespective of the opinion of the others. We examine the most important classes of aggregation operators in respect to their absorbent tuples, and also construct new aggregation operators with predefined sets of absorbent tuples.<br /

Iterative regularization in nonparametric instrumental regression

We consider the nonparametric regression model with an additive error that is correlated with the explanatory variables. We suppose the existence of instrumental variables that are considered in this model for the identification and the estimation of the regression function. The nonparametric estimation by instrumental variables is an ill-posed linear inverse problem with an unknown but estimable operator. We provide a new estimator of the regression function using an iterative regularization method (the Landweber-Fridman method). The optimal number of iterations and the convergence of the mean square error of the resulting estimator are derived under both mild and severe degrees of ill-posedness. A Monte-Carlo exercise shows the impact of some parameters on the estimator and concludes on the reasonable finite sample performance of the new estimator.nonparametric estimation, instrumental variable, ill-posed inverse problem, iterative method, estimation by projection

On Multilevel Methods Based on Non-Nested Meshes

This thesis is concerned with multilevel methods for the efficient solution of partial differential equations in the field of scientific computing. Further, emphasis is put on an extensive study of the information transfer between finite element spaces associated with non-nested meshes. For the discretization of complicated geometries with a finite element method, unstructured meshes are often beneficial as they can easily be adjusted to the shape of the computational domain. Such meshes, and thus the corresponding discrete function spaces, do not allow for straightforward multilevel hierarchies that could be exploited to construct fast solvers. In the present thesis, we present a class of "semi-geometric" multilevel iterations, which are based on hierarchies of independent, non-nested meshes. This is realized by a variational approach such that the images of suitable prolongation operators in the next (finer) space recursively determine the coarse level spaces. The semi-geometric concept is of very general nature compared with other methods relying on geometric considerations. This is reflected in the relatively loose relations of the employed meshes to each other. The specific benefit of the approach based on non-nested meshes is the flexibility in the choice of the coarse meshes, which can, for instance, be generated independently by standard methods. The resolution of the boundaries of the actual computational domain in the constructed coarse level spaces is a characteristic feature of the devised class of methods. The flexible applicability and the efficiency of the presented solution methods is demonstrated in a series of numerical experiments. We also explain the practical implementation of the semi-geometric ideas and concrete transfer concepts between non-nested meshes. Moreover, an extension to a semi-geometric monotone multigrid method for the solution of variational inequalities is discussed. We carry out the analysis of the convergence and preconditioning properties, respectively, in the framework of the theory of subspace correction methods. Our technical considerations yield a quasi-optimal result, which we prove for general, shape regular meshes by local arguments. The relevant properties of the operators for the prolongation between non-nested finite element spaces are the H1-stability and an L2-approximation property as well as the locality of the transfer. This thesis is a contribution to the development of fast solvers for equations on complicated geometries with focus on geometric techniques (as opposed to algebraic ones). Connections to other approaches are carefully elaborated. In addition, we examine the actual information transfer between non-nested finite element spaces. In a novel study, we combine theoretical, practical and experimental considerations. A thourough investigation of the qualitative properties and a quantitative analysis of the differences of individual transfer concepts to each other lead to new results on the information transfer as such. Finally, by the introduction of a generalized projection operator, the pseudo-L2-projection, we obtain a significantly better approximation of the actual L2-orthogonal projection than other approaches from the literature.Nicht-geschachtelte Gitter in Multilevel-Verfahren Diese Arbeit beschäftigt sich mit Multilevel-Verfahren zur effizienten Lösung von Partiellen Differentialgleichungen im Bereich des Wissenschaftlichen Rechnens. Dabei liegt ein weiterer Schwerpunkt auf der eingehenden Untersuchung des Informationsaustauschs zwischen Finite-Elemente-Räumen zu nicht-geschachtelten Gittern. Zur Diskretisierung von komplizierten Geometrien mit einer Finite-Elemente-Methode sind unstrukturierte Gitter oft von Vorteil, weil sie der Form des Rechengebiets einfacher angepasst werden können. Solche Gitter, und somit die zugehörigen diskreten Funktionenräume, besitzen im Allgemeinen keine leicht zugängliche Multilevel-Struktur, die sich zur Konstruktion schneller Löser ausnutzen ließe. In der vorliegenden Arbeit stellen wir eine Klasse "semi-geometrischer" Multilevel-Iterationen vor, die auf Hierarchien voneinander unabhängiger, nicht-geschachtelter Gitter beruhen. Dabei bestimmen in einem variationellen Ansatz rekursiv die Bilder geeigneter Prolongationsoperatoren im jeweils folgenden (feineren) Raum die Grobgitterräume. Das semi-geometrische Konzept ist sehr allgemeiner Natur verglichen mit anderen Verfahren, die auf geometrischen Überlegungen beruhen. Dies zeigt sich in der verhältnismäßig losen Beziehung der verwendeten Gitter zueinander. Der konkrete Nutzen des Ansatzes mit nicht-geschachtelten Gittern ist die Flexibilität der Wahl der Grobgitter. Diese können beispielsweise unabhängig mit Standardverfahren generiert werden. Die Auflösung des Randes des tatsächlichen Rechengebiets in den konstruierten Grobgitterräumen ist eine Eigenschaft der entwickelten Verfahrensklasse. Die flexible Einsetzbarkeit und die Effizienz der vorgestellten Lösungsverfahren zeigt sich in einer Reihe von numerischen Experimenten. Dazu geben wir Hinweise zur praktischen Umsetzung der semi-geometrischen Ideen und konkreter Transfer-Konzepte zwischen nicht-geschachtelten Gittern. Darüber hinaus wird eine Erweiterung zu einem semi-geometrischen monotonen Mehrgitterverfahren zur Lösung von Variationsungleichungen untersucht. Wir führen die Analysis der Konvergenz- bzw. Vorkonditionierungseigenschaften im Rahmen der Theorie der Teilraumkorrekturmethoden durch. Unsere technische Ausarbeitung liefert ein quasi-optimales Resultat, das wir mithilfe lokaler Argumente für allgemeine, shape-reguläre Gitterfamilien beweisen. Als relevante Eigenschaften der Operatoren zur Prolongation zwischen nicht-geschachtelten Finite-Elemente-Räumen erweisen sich die H1-Stabilität und eine L2-Approximationseigenschaft sowie die Lokalität des Transfers. Diese Arbeit ist ein Beitrag zur Entwicklung schneller Löser für Gleichungen auf komplizierten Gebieten mit Schwerpunkt auf geometrischen Techniken (im Unterschied zu algebraischen). Verbindungen zu anderen Ansätzen werden sorgfältig aufgezeigt. Daneben untersuchen wir den Informationsaustausch zwischen nicht-geschachtelten Finite-Elemente-Räumen als solchen. In einer neuartigen Studie verbinden wir theoretische, praktische und experimentelle Überlegungen. Eine sorgfältige Prüfung der qualitativen Eigenschaften sowie eine quantitative Analyse der Unterschiede verschiedener Transfer-Konzepte zueinander führen zu neuen Ergebnissen bezüglich des Informationsaustauschs selbst. Schließlich erreichen wir durch die Einführung eines verallgemeinerten Projektionsoperators, der Pseudo-L2-Projektion, eine deutlich bessere Approximation der eigentlichen L2-orthogonalen Projektion als andere Ansätze aus der Literatur

On a Cahn–Hilliard–Keller–Segel model with generalized logistic source describing tumor growth

We propose a new type of diffuse interface model describing the evolution of a tumor mass under the effects of a chemical substance (e.g., a nutrient or a drug). The process is described by utilizing the variables $\varphi$, an order parameter representing the local proportion of tumor cells, and $\sigma$, representing the concentration of the chemical. The order parameter $\varphi$ is assumed to satisfy a suitable form of the Cahn-Hilliard equation with mass source and logarithmic potential of Flory-Huggins type (or generalizations of it). The chemical concentration $\sigma$ satisfies a reaction-diffusion equation where the cross-diffusion term has the same expression as in the celebrated Keller-Segel model. In this respect, the model we propose represents a new coupling between the Cahn-Hilliard equation and a subsystem of the Keller-Segel model. We believe that, compared to other models, this choice is more effective in capturing the chemotactic effects that may occur in tumor growth dynamics (chemically induced tumor evolution and consumption of nutrient/drug by tumor cells). Note that, in order to prevent finite time blowup of $\sigma$, we assume a chemical source term of logistic type. Our main mathematical result is devoted to proving existence of weak solutions in a rather general setting that covers both the two- and three- dimensional cases. Under more restrictive assumptions on coefficients and data, and in some cases on the spatial dimension, we prove various regularity results. Finally, in a proper class of smooth solutions we show uniqueness and continuous dependence on the initial data in a number of significant cases.Comment: 38 page