# Preprint: A probabilistic framework for approximating functions in active subspaces

The *math.NA* newsletter of arXiv announced my new preprint today.

I am writing an article on a probabilistic framework that I suggest for approximating functions in active subspaces.
It contains a proof for a result that allows further statements about approximative quantities of a conditional expectation over the complement of an active subspace, the so-called *inactive subspace*.
It is an extension to former work of Constantine et al., meaning it completes and rounds off its analysis.

I would like to acknowledge the constructive advice of my supervisor Prof. Barbara Wohlmuth. Also I have to thank the whole Chair for Numerical Mathematics for having productive disussions about that topic. Additionally, I would like to express my sincere gratitude to David Criens and Dominik Schmid (both TUM) for their support in questions of measurable functions and measurable sets.

**Abstract**. This paper develops a comprehensive probabilistic setup to compute approximating functions in active subspaces. Constantine et al. proposed the active subspace method in (Constantine et al., 2014) to reduce the dimension of computational problems. It can be seen as an attempt to approximate a high-dimensional function of interest *f* by a low-dimensional one. To do this, a common approach is to integrate *f* over the inactive, i.e. non-dominant, directions with a suitable conditional density function. In practice, this can be done with a finite Monte Carlo sum, making not only the resulting approximation random in the inactive variable for each fixed input from the active subspace, but also its expectation, i.e. the integral of the low-dimensional function weighted with a probability measure on the active variable. In this regard we develop a fully probabilistic framework extending results from (Constantine et al., 2014, 2016). The results are supported by a simple numerical example.