Probability Measures for Numerical Solutions of Differential Equations

TL;DR

This paper introduces a method to quantify epistemic uncertainty in numerical solutions of differential equations by randomizing solvers to induce probability measures, aiding in more accurate statistical analysis.

Contribution

It proposes a novel approach to explicitly quantify numerical uncertainty in differential equations using randomized solvers that generate probability measures over solutions.

Findings

Randomized solvers induce probability measures that contract to the true solution.

The convergence rates of these measures align with traditional deterministic methods.

The approach applies to both ordinary and elliptic partial differential equations.

Abstract

In this paper, we present a formal quantification of epistemic uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due to the finite dimensional approximation of an unknown and implicitly defined function. When statistically analysing models based on differential equations describing physical, or other naturally occurring, phenomena, it is therefore important to explicitly account for the uncertainty introduced by the numerical method. This enables objective determination of its importance relative to other uncertainties, such as those caused by data contaminated with noise or model error induced by missing physical or inadequate descriptors. To this end we show that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of such differential equations. These measures exhibit contraction to a Dirac measure around the true unknown solution, where the rates of convergence are consistent with the underlying deterministic numerical method. Ordinary differential equations and elliptic partial differential equations are used to illustrate the approach to quantifying uncertainty in both the statistical analysis of the forward and inverse problems.