Greedy Sampling in High Dimensions via the Polytope Division Method
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In the last two decades, model order reduction has been established as an important tool for the solution of high-dimensional parametrized partial differential equations. However, even with the development and success of new methods that exploit machine learning tools, the problem of offline training cost and, relatedly, offline sampling, remains. Most methods still rely on a random sampling of the parameter space, which especially in high-dimensional parameter spaces necessitates large amounts of full-order (expensive) training data. We explain the Polytope Division Method (PDM), a greedy-type method, to determine where in the parameter domain to sample, therefore reducing the offline training cost. PDM splits the parameter space into polytopes and investigates the quality of the sample in the barycenter of each polytope.