Linear systems are ubiquitous in the field of numerical simulation, and can be used to describe or approximate physical processes on arbitrary objects. These objects can be represented using volumetric meshes, often made out of tetrahedra. If the mesh contains many vertices, the corresponding linear system becomes too large to solve using conventional methods. Multigrid methods are a scalable alternative to iteratively approximate solutions for such large problems, but before they can be employed, a multigrid hierarchy that corresponds to the mesh must first be defined. A bad hierarchy can lead to bad convergence when solving the problem, but creating a more accurate hierarchy can be computationally expensive itself. We present an adaptation of Gravo MG, an algorithm to define such hierarchies for triangular meshes. This adaptation translates the same design principles for the hierarchy construction from the triangular to the tetrahedral case. Furthermore, we address some new challenges that arise when dealing with volumetric meshes, particularly the preservation of the boundary when coarsening the mesh.

Survival analysis revolves around studying and predicting the time it takes for a particular event to occur. In clinical trials on terminal illnesses, this is usually the time from the diagnosis of a patient until their death. Estimating the odds of survival of a new patient can be done by analyzing survival data from past patients in similar conditions. To cluster similar patients based on a set of features, survival trees may be employed, which act as decision trees that assign a survival distribution to each cluster. Many algorithms exist for creating useful survival trees, but not for creating optimal survival trees. In this paper, research on finding optimal classification trees is applied to survival analysis, by adapting the MurTree algorithm to construct survival trees. We present SurTree, an algorithm that applies many of MurTree’s techniques to create globally optimal survival trees. Furthermore, we compare the output quality and runtime performance of SurTree to a state-of-the-art method for constructing survival trees, showing its optimality and its fast computation times on smaller datasets.