Construction of aggregation operators with noble reinforcement

This paper examines disjunctive aggregation operators used in various recommender systems. A specific requirement in these systems is the property of noble reinforcement: allowing a collection of high-valued arguments to reinforce each other while avoiding reinforcement of low-valued arguments. We present a new construction of Lipschitz-continuous aggregation operators with noble reinforcement property and its refinements. <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 /

Algebraic Multigrid (AMG) for Saddle Point Systems

We introduce an algebraic multigrid method for the solution of matrices with saddle point structure. Such matrices e.g. arise after discretization of a second order partial differential equation (PDE) subject to linear constraints. Algebraic multigrid (AMG) methods provide optimal linear solvers for many applications in science, engineering or economics. The strength of AMG is the automatic construction of a multigrid hierarchy adapted to the linear system to be solved. However, the scope of AMG is mainly limited to symmetric positive definite matrices. An essential feature of these matrices is that they define an inner product and a norm. In AMG, matrix-dependent norms play an important role to investigate the action of the smoother, to verify approximation properties for the interpolation operator and to show convergence for the overall multigrid cycle. Furthermore, the non-singularity of all coarse grid operators in a AMG hierarchy is ensured by the positive definiteness of the initial fine level matrix. Saddle point matrices have positive and negative eigenvalues and hence are indefinite. In consequence, if conventional AMG is applied to these matrices, the method will not always converge or may even break down if a singular coarse grid operator is computed. In this thesis, we describe how to circumvent these difficulties and to build a stable saddle point AMG hierarchy. We restrict ourselves to the class of Stokes-like problems, i.e. saddle point matrices which contain a symmetric positive definite submatrix that arises from the discretization of a second order PDE. Our approach is purely algebraic, i.e. it does not require any information not contained in the matrix itself. We identify the variables associated to the positive definite submatrix block (the so-called velocity components) and compute an inexact symmetric positive Schur complement matrix for the remaining degrees of freedom (in the following called pressure components). Then, we employ classical AMG methods for these definite operators individually and obtain an interpolation operator for the velocity components and an interpolation operator for the pressure matrix. The key idea of our method is to not just merge these interpolation matrices into a single prolongation operator for the overall system, but to introduce additional couplings between velocity and pressure. The coarse level operator is computed using this "stabilized" interpolation operator. We present three different interpolation stabilization techniques, for which we show that they resulting coarse grid operator is non-singular. For one of these methods, we can prove two-grid convergence. The numerical results obtained from finite difference and finite element discretizations of saddle point PDEs demonstrate the practical applicability of our approach

A geometric approach for the addition of nodes to an interpolatory quadrature rule with positive weights

A novel mathematical framework is derived for the addition of nodes to interpolatory quadrature rules. The framework is based on the geometrical interpretation of the Vandermonde-matrix describing the relation between the nodes and the weights and can be used to determine all nodes that can be added to an interpolatory quadrature rule with positive weights such that the positive weights are preserved. In the case of addition of a single node, the derived inequalities that describe the regions where nodes can be added or replaced are explicit. It is shown that, depending on the location of existing nodes and moments of the distribution, addition of a single node and preservation of positive weights is not always possible. On the other hand, addition of multiple nodes and preservation of positive weights is always possible, although the minimum number of nodes that need to be added can be as large as the number of nodes of the quadrature rule. Moreover, in this case the inequalities describing the regions where nodes can be added become implicit. An algorithm is presented to explore these regions and it is shown that the well-known Patter

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 /