Talks

This talk gives an overview of some classical and recent uniqueness results in Discrete Tomography. In the first part we will review the concept of J-additivity and apply it to settle a problem of Kuba on 3-dimensional lattice sets and a conjecture of Brunetti and Daurat on planar lattice convex sets. The second part of the talk deals with the computational complexity of finding a smallest set of lattice positions of a given lattice set whose disclosure yields uniqueness w.r.t. some given X-rays. It turns out that this problem is already NP-hard in the plane and for the two standard directions.

This is joint work with Peter Gritzmann and Markus Wiegelmann.

According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier-Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. Therefore, a new variational crime for the nonconforming Crouzeix-Raviart element is proposed, where divergence-free, lowest-order Raviart-Thomas velocity reconstructions reestablish L2-orthogonality. This approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.

The numerical treatment of tensors and the use of tensors for various numerical problem has rapidly increased in the last time. It is now applied to many fields in analysis (treatment of pdes, representation of multivariate functions, etc.). The key for an efficient numerical treatment is a suitable format. We discuss the various formats, their properties, and operations with tensors.

Literature: W. H.: Tensor spaces and numerical tensor calculus. Springer 2012