Confidence intervals for finite difference solutions

ABSTRACT

Although applications of Bayesian analysis for numerical quadrature problems have been considered before, it is 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.

KEYWORDS:

• Bayesian analysis
• Finite difference method
• Numerical analysis

MATHEMATICS SUBJECT CLASSIFICATION:

• 62P99
• 65D25

Acknowledgments

The authors thank an anonymous reviewer for helpful comments, especially for pointing them to alternate Bayesian strategies for the regression framework considered in this article.