The approximation of left-continuous t-norms

Abstract A discrete t-norm is a binary operation on a finite subset of the real unit interval fulfilling the same algebraic conditions as t-norms. We show that any left-continuous t-norm can, in a natural sense, be approximated by a discrete t-norm with an arbitrary precision

A General Family of Penalties for Combining Differing Types of Penalties in Generalized Structured Models

Penalized estimation has become an established tool for regularization and model selection in regression models. A variety of penalties with specific features are available and effective algorithms for specific penalties have been proposed. But not much is available to fit models that call for a combination of different penalties. When modeling rent data, which will be considered as an example, various types of predictors call for a combination of a Ridge, a grouped Lasso and a Lasso-type penalty within one model. Algorithms that can deal with such problems, are in demand. We propose to approximate penalties that are (semi-)norms of scalar linear transformations of the coefficient vector in generalized structured models. The penalty is very general such that the Lasso, the fused Lasso, the Ridge, the smoothly clipped absolute deviation penalty (SCAD), the elastic net and many more penalties are embedded. The approximation allows to combine all these penalties within one model. The computation is based on conventional penalized iteratively re-weighted least squares (PIRLS) algorithms and hence, easy to implement. Moreover, new penalties can be incorporated quickly. The approach is also extended to penalties with vector based arguments; that is, to penalties with norms of linear transformations of the coefficient vector. Some illustrative examples and the model for the Munich rent data show promising results

Error estimates for stabilized finite element methods applied to ill-posed problems

We propose an analysis for the stabilized finite element methods proposed in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput., 35(6) 2013, valid in the case of ill-posed problems for which only weak continuous dependence can be assumed. A priori and a posteriori error estimates are obtained without assuming coercivity or inf-sup stability of the continuous problem. A numerical example illustrates the theory.Comment: The theoretical part is submitted to Comptes Rendus Mathematiques and the numerical example is taken from the reference mentioned in the abstrac