This thesis addresses the challenge of segmenting ultra-high-resolution images. Limitations of current approaches to segment these are that either detailed spatial contextual information is lost or many redundant computations are necessary. To overcome these issues, we propose a novel approach combining the U-Net architecture with domain decomposition strategies to balance incorporating spatial context and maintaining computational efficiency. Our proposed method partitions input images into non-overlapping patches, each processed independently on a separate device. A communication network facilitates the exchange of information between patches, enhancing the model's understanding of the spatial context.

Through theoretical analysis and practical experimentation on device memory usage during training, we demonstrate that our approach incurs minimal additional memory overhead for inter-device communication. Evaluation on synthetic and realistic datasets, including the Inria Aerial Image and DeepGlobe Satellite Segmentation datasets, demonstrates the effectiveness of our approach. Our model achieves competitive performance compared to the baseline U-Net model, with consistent class predictions around boundaries. Visualization of feature maps highlights the role of the communication network in transferring contextual information. Furthermore, it is shown that our approach remains scalable even when trained on limited subdomains.

In conclusion, our proposed model offers an intuitive solution for segmenting ultra-high-resolution images by effectively incorporating spatial context. Future research could explore variations of our model, such as overlapping subdomains or communication on different levels of the U-Net, to further enhance boundary consistency and information transfer.

This work aims at finding a second order accurate level-set method which solves the Stefan problem with non-homogeneous Dirichlet boundary conditions in one dimension. The numerical accuracy of the FTCS-scheme, BTCS-scheme and Crank-Nicolson scheme for the discretization of the heat equation was considered, as well as the accuracy of the first order Upwind method, Leapfrog method and the Lax-Wendroff method for the discretization of the advection equation. A level-set method was developed using a finite volume Crank-Nicolson scheme for the discretization of the moving boundary. A second order accurate scheme for solving the advection equation was developed using Lagrange extrapolation polynomials. The moving boundary velocity was estimated using second order Lagrange polynomials. The developed method was found to be second order accurate for a specific range of ratios between time step size and spatial step size.