  B-spline models are a powerful way to represent scientific data sets with a
functional approximation. However, these models can suffer from spurious
oscillations when the data to be approximated are not uniformly distributed.
Model regularization (i.e., smoothing) has traditionally been used to minimize
these oscillations; unfortunately, it is sometimes impossible to sufficiently
remove unwanted artifacts without smoothing away key features of the data set.
In this article, we present a method of model regularization that preserves
significant features of a data set while minimizing artificial oscillations.
Our method varies the strength of a smoothing parameter throughout the domain
automatically, removing artifacts in poorly-constrained regions while leaving
other regions unchanged. The proposed method selectively incorporates
regularization terms based on first and second derivatives to maintain model
accuracy while minimizing numerical artifacts. The behavior of our method is
validated on a collection of two- and three-dimensional data sets produced by
scientific simulations. In addition, a key tuning parameter is highlighted and
the effects of this parameter are presented in detail. This paper is an
extension of our previous conference paper at the 2022 International Conference
on Computational Science (ICCS) [Lenz et al. 2022].
