(374q) Surrogate Modeling Based on Nonconventional Adaptive Sampling on Smolyak Sparse Grids

For cases where the amount of data that can be collected is small, such as computationally expensive simulations and experiments, surrogate models are a promising tool. Constructing surrogate models of these black box functions must use efficient sampling techniques and appropriate basis functions in order to maximize the accuracy of the model given a limited number of samples. Smolyak sparse grids, which use a sum of tensor products to select a subset of points assumed to contribute most to the fit, represent a promising approach to building surrogate models. Smolyak sparse grids are hierarchical in nature and are based on orthogonal basis functions that determine both the numerical values of the grid points and the terms of the surrogate function. Adaptive sampling for Smolyak grids can further improve sampling efficiency by using the current surrogate model to determine which sample points would most improve the accuracy; however existing methods for adaptive sampling on Smolyak sparse grids are limited in which points can be added at a given iteration. The ability to sample points outside the hierarchical order or entirely off the Smolyak grid would be especially useful for surrogate-based optimization, where approximation isn’t the only objective. In this work, we describe the implementation of these new adaptive strategies in the Python package smolyay. We compare the performance of surrogate models based on adaptive sampling on Smolyak sparse grids to Gaussian process models with space-filling random sampling, which is a standard approach in surrogate modeling.