Recent advances in modern technology have created a new world of enormous, high-dimensional data structures. Most types of such data are governed by anisotropic features such as edges in digital images. It is well known that wavelets - although perfectly suited for isotropic structures - do not perform equally well when dealing with anisotropic phenomena.
There has been extensive study to develop directional systems which are able to sparsely represent anisotropic features of data.
However, until now there has been little effort made in using them to solve PDEs.
In this talk, we will first present a new construction of directional systems consisting of piecewise polynomial functions and show that those systems can provide optimally sparse approximations of piecewise smooth functions.
Secondly, we will present numerical results of our Petrov-Galerkin method for first order transport equations, introduced in the talk by Gerrit Welper.
In fact, we will show that this scheme in combination with our directional systems allows effective approximations of the solutions of selected first order transport equations.