An algorithm for multivariate function estimation based on hierarchically refined sparse grids

Abstract:
An adaptive function estimation approach is presented to recover an unknown, multivariate functional relation from noisy data. Using a sparse grid combination approach, both discretization and Tikhonov regulariza- tion need to be selected appropriately to resolve func- tional details whilst suppressing measurement noise. An initially coarse, multivariate grid is adaptively refined using sensitivity analysis, creating a sequence of hierar- chically refined grids. The problem of choosing a multi- dimensional discretization level is thus transformed to the identification of a suitable refinement step, giving rise to a nested approach for the selection of both dis- cretization and Tikhonov regularization. Validation on multivariate test functions shows good approximation results.

Keywords:
Multivariate regression, function estimation, sparse grids, discretization, regularization, L-curve, cross validation

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