Optimal Sampling of Random Fields for Topological Analysis




Cochran, Gregory Scott

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Algebraic topology is becoming an increasing important tool in applied mathematics. In particular, homology theory allows one to distinguish different topologies while being tractable to compute. An important application is the study of nodal domains for solutions to stochastic partial differential equations. These are the sets where the function value is greater than zero and less than zero. In order to compute the homology of the nodal sets computationally, we must discretize the domains. However, in this discretization process, we can make mistakes in the topology. Can we develop a method that will allow us to determine a proper discretization size a priori? One approach is to use an algorithm that is guaranteed to return the correct homology. The original algorithm has a few shortcomings. We will present these shortcomings and develop methods to overcome these issues. The other approach is to establish explicit probability bounds for the making the correct homology. This is an a priori approach that will returns the probability for a fixed discretization size and also determines the optimal location of the sampled points.



Topology, Homology, Applied Mathematics, Sampling