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Geometric multigrid (GMG) methods are a fundamental tool for efficiently solving large sparse linear systems. A requirement for GMG is a hierarchy of grids; however, many practical volumetric domains are available only as single, irregular tetrahedral meshes, making the construction of a multigrid hierarchy necessary. Existing approaches often trade off speed against hierarchy quality: remeshing- or coarsening-based methods can be expensive to construct, whereas graph-based techniques are fast but often yield weaker multigrid performance. We introduce GravoTet, which bridges this gap by combining geometric structure with graph-based efficiency to construct fast and effective multigrid hierarchies. GravoTet builds a vertex hierarchy and then generates graph-Voronoi diagrams whose dual cells define coarse tetrahedra, enabling rapid construction of multigrid levels. Boundary elements are explicitly prioritized during both sampling and tet generation to preserve boundary. In our evaluation, we solve Poisson and biharmonic problems on irregular tetrahedral meshes and compare GravoTet against state-of-the-art geometric multigrid, algebraic multigrid and direct solvers, demonstrating superior performance, particularly on large meshes.
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Geometric multigrid (GMG) methods are a fundamental tool for efficiently solving large sparse linear systems. A requirement for GMG is a hierarchy of grids; however, many practical volumetric domains are available only as single, irregular tetrahedral meshes, making the construction of a multigrid hierarchy necessary. Existing approaches often trade off speed against hierarchy quality: remeshing- or coarsening-based methods can be expensive to construct, whereas graph-based techniques are fast but often yield weaker multigrid performance. We introduce GravoTet, which bridges this gap by combining geometric structure with graph-based efficiency to construct fast and effective multigrid hierarchies. GravoTet builds a vertex hierarchy and then generates graph-Voronoi diagrams whose dual cells define coarse tetrahedra, enabling rapid construction of multigrid levels. Boundary elements are explicitly prioritized during both sampling and tet generation to preserve boundary. In our evaluation, we solve Poisson and biharmonic problems on irregular tetrahedral meshes and compare GravoTet against state-of-the-art geometric multigrid, algebraic multigrid and direct solvers, demonstrating superior performance, particularly on large meshes.
Survival analysis studies and predicts the time of death, or other singular unrepeated events, based on historical data, while the true time of death for some instances is unknown. Survival trees enable the discovery of complex nonlinear relations in a compact human comprehensible model, by recursively splitting the population and predicting a distinct survival distribution in each leaf node. We use dynamic programming to provide the first survival tree method with optimality guarantees, enabling the assessment of the optimality gap of heuristics. We improve the scalability of our method through a special algorithm for computing trees up to depth two. The experiments show that our method’s run time even outperforms some heuristics for realistic cases while obtaining similar out-of-sample performance with the state-of-the-art.
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Survival analysis studies and predicts the time of death, or other singular unrepeated events, based on historical data, while the true time of death for some instances is unknown. Survival trees enable the discovery of complex nonlinear relations in a compact human comprehensible model, by recursively splitting the population and predicting a distinct survival distribution in each leaf node. We use dynamic programming to provide the first survival tree method with optimality guarantees, enabling the assessment of the optimality gap of heuristics. We improve the scalability of our method through a special algorithm for computing trees up to depth two. The experiments show that our method’s run time even outperforms some heuristics for realistic cases while obtaining similar out-of-sample performance with the state-of-the-art.
