# Piecewise polynomial approximation of probability density functions with application to uncertainty quantification for stochastic PDEs

## TL;DR

This paper introduces a scalable, linear polynomial-based method for approximating probability density functions from samples, applicable to uncertainty quantification in stochastic PDEs, avoiding complex computations and bandwidth tuning.

## Contribution

It presents a novel piecewise polynomial density estimator that is scalable, does not require solving linear systems, and simplifies bandwidth selection, with applications to stochastic PDEs.

## Key findings

- The estimator is consistent and enforces unit integral.
- It does not require positivity constraints.
- Validated through numerical examples with known and unknown PDFs.

## Abstract

The probability density function (PDF) associated with a given set of samples is approximated by a piecewise-linear polynomial constructed with respect to a binning of the sample space. The kernel functions are a compactly supported basis for the space of such polynomials, i.e. finite element hat functions, that are centered at the bin nodes rather than at the samples, as is the case for the standard kernel density estimation approach. This feature naturally provides an approximation that is scalable with respect to the sample size. On the other hand, unlike other strategies that use a finite element approach, the proposed approximation does not require the solution of a linear system. In addition, a simple rule that relates the bin size to the sample size eliminates the need for bandwidth selection procedures. The proposed density estimator has unitary integral, does not require a constraint to enforce positivity, and is consistent. The proposed approach is validated through numerical examples in which samples are drawn from known PDFs. The approach is also used to determine approximations of (unknown) PDFs associated with outputs of interest that depend on the solution of a stochastic partial differential equation.