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

Yuto Miyatake; Kaoru Irie; Takeru Matsuda

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arXiv · 2411.08338

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

Yuto Miyatake·Kaoru Irie·Takeru Matsuda

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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.

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BibTeX RIS

Yuto Miyatake, Kaoru Irie, Takeru Matsuda. 2024-11-13. Quantifying uncertainty in the numerical integration of evolution equations based on Bayesian isotonic regression. https://arxiv.org/abs/2411.08338

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