# Confidence Intervals for Finite Difference Solutions

## TL;DR

This paper introduces a Bayesian regression approach to derive confidence intervals for finite difference solutions of differential equations, linking statistical methods with numerical analysis to quantify solution uncertainty.

## Contribution

It presents a novel Bayesian framework for finite difference methods, providing confidence intervals that relate to truncation errors in differential equation solutions.

## Key findings

- Confidence intervals accurately reflect truncation errors.
- The Bayesian approach enhances understanding of solution uncertainty.
- Framework applicable to various differential equations.

## Abstract

Although applications of Bayesian analysis for numerical quadrature problems have been considered before, it's only very recently that statisticians have focused on the connections between statistics and numerical analysis of differential equations. In line with this very recent trend, we show how certain commonly used finite difference schemes for numerical solutions of ordinary and partial differential equations can be considered in a regression setting. Focusing on this regression framework, we apply a simple Bayesian strategy to obtain confidence intervals for the finite difference solutions. We apply this framework on several examples to show how the confidence intervals are related to truncation error and illustrate the utility of the confidence intervals for the examples considered.