Contents List of Figures v Data Sets 2 Symbols 3

As the size of the data sets generated by numerical simulations and the ability to measure natural phenomena increases, new methods must be developed to represent and analyze the resulting data sets. Massive amounts of data must be made available to a user in a very concise manner, while at the same time guarantees must be ensured concerning the correctness of simplified and visualized data sets. This thesis combines both aspects by using topology to drive data simplification and analysis. Topology provides the means to analyze data in an abstract fashion well suited for large data sets describing complex phenomena. Using topology to guide the development of simplification and visualization algorithms, a theoretical framework can be used to validate results and guarantee correctness. This thesis presents a highly adaptive hierarchical representation of the topology of functions defined over 2-manifold domains. Guided by the theory of Morse-Smale complexes, topology is encoded in a dependency graph used for adaptive simplification and refinement. For given topology a corresponding approximation to the original function is constructed within a specified numerical error. Tight error bounds can be enforced and methods are developed to smooth an approximation without changing topology or violating error bounds.