@Article{NMTMA-14-4,
author = {Yuankai, Teng and Zhu, Wang and Gruber, Anthony and Max, Gunzburger and Lili, Ju and Yuankai, Teng and Zhu, Wang},
title = {Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2021},
volume = {14},
number = {4},
pages = {839--861},
abstract = {<p style="text-align: justify;">A dimension reduction method based on the “Nonlinear Level set Learning” (NLL) approach is presented for the pointwise prediction of functions which
have been sparsely sampled. Leveraging geometric information provided by the
Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and
sensitivity analysis. Experiments and applications are presented which compare this
modified NLL with the original NLL and the Active Subspaces (AS) method. While
accommodating sparse input data, the proposed algorithm is shown to train quickly
and provide a much more accurate and informative reduction than either AS or the
original NLL on two example functions with high-dimensional domains, as well as
two state-dependent quantities depending on the solutions to parametric differential
equations.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2021-0062},
url = {https://global-sci.com/article/90322/nonlinear-level-set-learning-for-function-approximation-on-sparse-data-with-applications-to-parametric-differential-equations}
}