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# DFG Project "Adaptive Anisotropic Discretization Concepts"

Duration: | October 2008 – June 2012 |

Project Heads: | W. Dahmen, G. Kutyniok and C. Schwab |

Researchers: | W.-Q Lim and G. Welper |

Support: | DFG |

# Introduction

Many important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds. Typical examples are digital images where regions of little variation are separated by possibly sharp edges. In a completely different context solutions to certain partial differential equations (PDEs) exhibit strongly anisotropic features concentrated on lower dimensional manifolds.

Here one could think of transport dominated problems like hyperbolic conservation laws where solutions exhibit shocks and singularly perturbed convection diffusion reaction equations with dominating convection resp. dominating reaction. In a way both problem settings, namely processing explicitly given data as in imaging (based on transform methods) as well as solving operator equations could be merged when employing PDE based image processing methods, e.g. based on anisotropic diffusion. In the context of PDEs, solution anisotropy is often ignored in the sense that either isotropic adaptivity is employed (such as wavelets in imaging) or it is dealt with by generating anisotropic triangulations.

While anisotropic triangulations are feasible and amenable to mathematical treatment, they pose formidable challenges when dealing with 3D problems or even 4D space-time discretizations. Significant progress can only be expected from new concepts. It is therefore tempting to employ analysis tools that are capable of extracting anisotropic structures from given data also in the context of PDEs when similar anisotropic features are expected to occur.

# Research Program

The central objective of this project is 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.

Specifically, the recent shearlet concept has in our opinion significant advantages over previous, related approaches like curvelets, ridgelets, etc. in that it offers a unified treatment of both the continuous and digital world which makes it a particularly promising candidate for the discretizations of PDEs.

The major tasks to be faced can be summarized as follows.

(I) Tool Development:

For the discretization of PDEs with anisotropic features, we propose to extend directional representation systems and, in particular, the shearlet concept, from two to higher dimensions, with particular attention to dimensions 3 and 4. In addition, spatially compactly supported versions of L^2-stable directional representation systems need to be developed for bounded domains in R^d.

(II) Approximation properties and suitable regularity classes:

The approximation properties of these shearlet systems have to be analyzed with particular attention to regularity classes of solutions to the PDE under consideration. The ultimate long term goal is to understand and characterize the inherent complexity of problems with dominant anisotropic features.

(III) Solution concepts – their analysis and implementation:

For a suitable hierarchy of model problems discretization concepts need to be developed that give rise to rigorously founded adaptive solution schemes and, in particular, can host shearlet-frame discretizations.

# Project Output

1] G. Kutyniok.

**Data Separation by Sparse Representations.**

*Compressed Sensing: Theory and Applications, *

*Cambridge University Press, 2012.*

2] G. Kutyniok, J. Lemvig, and W.-Q Lim.

**Compactly Supported Shearlets.**

*Approximation Theory XIII (San Antonio, TX, 2010), *

*Springer Proc. Math. **13**, 163-186, Springer, 2012.*

3] P. Kittipoom, G. Kutyniok, and W.-Q Lim.

**Construction of Compactly Supported Shearlet Frames.**

*Constr. Approx. 35 (2012), 21-72.*

4] P. Kittipoom, G. Kutyniok, and W.-Q Lim.

**Irregular Shearlet Frames: Geometry and Approximation Properties.**

*J. Fourier Anal. Appl. 17 (2011), 604-639.*

5] G. Kutyniok and W.-Q Lim.

**Compactly Supported Shearlets are Optimally Sparse.**

*J. Approx. Theory **163** (2011), 1564-1589. *

6] G. Kutyniok and W.-Q Lim.

**Shearlets on Bounded Domains.**

*Approximation Theory XIII (San Antonio, TX, 2010), *