Dirks, Carolin Maria; Rossmanith, Carolin Maria: Numerical methods for transportation networks. 2019
Inhalt
- Introduction
- Mathematical preliminaries
- Basic notation
- Functions of bounded variation
- -convergence and equi-coercivity
- The Mumford–Shah image segmentation problem
- Phase field approximation of the Mumford–Shah functional
- Functional lifting of Mumford–Shah-type problems
- Adaptive finite elements for functional lifting problems
- Review of methods for transportation network problems
- Optimal transport and transport networks - A brief overview
- Branched transport
- Urban planning
- Existence and properties of minimizers
- Numerical approaches
- Numerical optimization of transportation networks via functional lifting
- Model
- Reformulation as image inpainting problems in two dimensions
- Functional lifting of the branched transport and urban planning energy
- Analysis
- Numerical optimization with finite differences
- Numerical optimization with finite elements on adaptive triangular prism grids
- A phase field approximation approach
- Conclusion and outlook
- Appendices
- Construction of a vector field
- Bibliography