Numerical and harmonic analysis of problems with anisotropic features, directional representation systems and the solution of transport dominated problems, in particular, for parameter dependent high dimensional versions

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Many important multivariate problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds. Examples are digital images as well as solutions of hyperbolic conservation laws or more generally of transport dominated equations. Over the past few years substantial progress has been made in efficiently encoding signals with anisotropic features based on new directional representation systems like shearlets. Although these developments for explicitly given signals are far from complete their understanding has by far more matured than analogous considerations for implicitly given objects like solutions to operator equations of the above type. The central objective of this project is therefore the development and understanding of new discretization concepts that are able to economically and reliably capture anisotropic phenomena in solutions governed by anisotropic phenomena. A central issue is finding suitable variational formulations that on one hand support adaptive solution concepts for transport operators and, on the other hand, are suitable for treating parameter dependent and therefore also high dimensional problems.