Quantifying uncertainty in the numerical integration of evolution equations based on Bayesian isotonic regression

TL;DR

This paper introduces a Bayesian approach with isotonic regression and mixture models to quantify discretization errors in numerical solutions of differential equations, providing a probabilistic error estimate.

Contribution

It extends Bayesian isotonic regression techniques with a novel shrinkage prior and Gaussian mixture models to estimate discretization error variances in ODE solutions.

Findings

Develops a Gibbs sampling algorithm for error variance estimation.

Provides a probabilistic framework for discretization error quantification.

Enhances accuracy of numerical ODE solutions with Bayesian error modeling.

Abstract

This paper presents a new Bayesian framework for quantifying discretization errors in numerical solutions of ordinary differential equations. By modelling the errors as random variables, we impose a monotonicity constraint on the variances, referred to as discretization error variances. The key to our approach is the use of a shrinkage prior for the variances coupled with variable transformations. This methodology extends existing Bayesian isotonic regression techniques to tackle the challenge of estimating the variances of a normal distribution. An additional key feature is the use of a Gaussian mixture model for the $\log$-$\chi^2_1$ distribution, enabling the development of an efficient Gibbs sampling algorithm for the corresponding posterior.