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

TL;DR

A new Bayesian framework for quantifying discretization errors in numerical solutions of ordinary differential equations by modelling the errors as random variables, referred to as discretization error variances is presented, with the use of a shrinkage prior for the variances coupled with variable transformations.

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 $\log$-$χ^2_1$ distribution, enabling the development of an efficient Gibbs sampling algorithm for the corresponding posterior.

Figures (8)

• Figure 1: The probability distribution functions of the double exponential distribution $f(\tilde{\varepsilon})$ given in \ref{['pdf:double-exponential']} and the Gaussian mixture model $g(\tilde{\varepsilon})$ give in \ref{['pdf:gmm']}.
• Figure 2: Exact solution, observations and numerical approximations to the FN model.
• Figure 3: Quantification results for $V$ in the FN model. The left figure shows the absolute value of the residual $r_i$, the quantification results in terms of $\sigma_i$ with their mean and $95\%$ credible interval, along with the quantification based on the maximum likelihood approach mm21. The right figure shows essentially the same information but with the observation removed: the error is plotted instead of the residual, and $\sqrt{\sigma_i^2 - \gamma^2}$ is plotted for the quantification results.
• Figure 4: Quantification results for $R$ in the FN model. The left figure shows the residual $r_i$, the quantification results in terms of $\sigma_i$ with their mean and $95\%$ credible interval, along with the quantification based on the maximum likelihood approach mm21. The right figure shows essentially the same information but with the observation removed: the error is plotted instead of the residual, and $\sqrt{\sigma_i^2 - \gamma^2}$ is plotted for the quantification results.
• Figure 5: These figures also present the residual and error plots for $V$. In the left figure, we sample $r_i$ from $\mathrm{N}(0,\sigma_i^2)$, where $\sigma_i^2$ is a sampled value using our Gibbs sampler, and plot the $90\%$ credible interval for $|r_i|$. Similarly, in the right figure, we sample the error $\xi_i$ from $\mathrm{N}(0,\sigma_i^2 - \gamma^2)$ and plot the $90\%$ credible interval for $|\xi_i|$. For this test, we generated 29,500 posterior samples; however, even with a much smaller sample size, the results were almost identical (posterior means showed only slight oscillations).

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6